On the enumerative geometry of branchedcovers of curves
Carl Lian
Submitted in partial fulfillment of therequirements for the degree of
Doctor of Philosophyin the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2020
c© 2020
Carl Lian
All rights reserved
Abstract
On the enumerative geometry of branched covers of curves
Carl Lian
In this thesis, we undertake two computations in enumerative geometry involving
branched covers of algebraic curves.
Firstly, we consider the general problem of enumerating branched covers of the
projective line from a fixed general curve subject to ramification conditions at possibly
moving points. Our main computations are in genus 1; the theory of limit linear series
allows one to reduce to this case. We first obtain a simple formula for a weighted count
of pencils on a fixed elliptic curve E, where base-points are allowed. We then deduce,
using an inclusion-exclusion procedure, formulas for the numbers of maps E → P1 with
moving ramification conditions. A striking consequence is the invariance of these counts
under a certain involution. Our results generalize work of Harris, Logan, Osserman,
and Farkas-Moschetti-Naranjo-Pirola. The content of this chapter is essentially that
of [L19b].
Secondly, we consider the loci of curves of genus 2 and 3 admitting a d-to-1 map
to a genus 1 curve. After compactifying these loci via admissible covers, we obtain
formulas for their Chow classes, recovering results of Faber-Pagani and van Zelm when
d = 2. The answers exhibit quasimodularity properties similar to those in the Gromov-
Witten theory of a fixed genus 1 curve; we conjecture that the quasimodularity persists
in higher genus, and indicate a number of possible variants. The content of this chapter
is essentially that of [L19a].1
1At the time of its writing, this thesis contains improved results and proofs from the preprint[L19a]. The preprint will be updated soon after the publication of this thesis. In particular, the titleof the preprint will be changed to that of Chapter 3.
Contents
Acknowledgments v
1 Vorspiel 1
1.1 Enumerative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Moduli spaces of curves . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Moduli spaces of branched covers of curves . . . . . . . . . . . . 6
1.3 Branched cover loci on moduli spaces of curves . . . . . . . . . . . . . . 8
1.4 The questions of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Enumerating pencils with moving ramification on curves 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Numerology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.3 Schubert Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 The weighted count in genus 1: Proof of Theorem 2.1.2 . . . . . . . . . 21
2.3.1 The weighted count Nd1,d2,d3,d4 . . . . . . . . . . . . . . . . . . . 21
2.3.2 Outline of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Pencils with fixed underlying line bundle . . . . . . . . . . . . . 24
i
2.3.4 The ramification loci on G× E . . . . . . . . . . . . . . . . . . 24
2.3.5 Imposing ramification at additional points . . . . . . . . . . . . 28
2.3.6 Proof of Theorem 2.1.2 . . . . . . . . . . . . . . . . . . . . . . . 34
2.3.7 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.4 Base-point-free pencils in genus 1: Proof of Theorems 2.1.3 and 2.1.4 . 41
2.4.1 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4.2 Schubert cycle formula . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Laurent polynomial formula . . . . . . . . . . . . . . . . . . . . 47
2.4.4 Explicit formula, and proof of Theorem 2.1.4 . . . . . . . . . . . 53
2.5 The general case via limit linear series . . . . . . . . . . . . . . . . . . 58
2.5.1 The degeneration formula . . . . . . . . . . . . . . . . . . . . . 58
2.5.2 Weighted counts via degeneration . . . . . . . . . . . . . . . . . 61
2.A Brill-Noether curves in M1,4 via admissible covers and Hurwitz numbers 63
2.B Positivity of enumerative counts . . . . . . . . . . . . . . . . . . . . . . 67
3 d-elliptic loci in genus 2 and 3 71
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.2 Intersection numbers on moduli spaces of curves . . . . . . . . . 77
3.2.2.1 M1,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.2.2 M1,3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2.2.3 M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.2.4 M2,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.2.5 M3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2.3 Admissible covers . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2.4 Quasimodular forms . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Auxiliary computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
ii
3.3.1 Counting isogenies . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.2 The 2-pointed d-elliptic locus on M1,2 . . . . . . . . . . . . . . 86
3.3.3 Doubly totally ramified covers of P1 . . . . . . . . . . . . . . . . 88
3.4 The d-elliptic locus on M2 . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.1 Classification of Admissible Covers . . . . . . . . . . . . . . . . 92
3.4.1.1 [Y ] ∈ ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.1.2 [Y ] ∈ ∆0 . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.2 Intersection numbers: the case [C] ∈ ∆1 . . . . . . . . . . . . . 95
3.4.3 Intersection numbers: the case [C] ∈ ∆0 . . . . . . . . . . . . . 97
3.4.3.1 Contribution from type (∆0,∆1) . . . . . . . . . . . . 98
3.4.3.2 Contribution from type (∆0,∆0) . . . . . . . . . . . . 99
3.4.3.3 Contribution from type (∆00,∆0) . . . . . . . . . . . . 101
3.4.4 The class of the admissible locus . . . . . . . . . . . . . . . . . 103
3.5 Variants in genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.5.1 Covers of a fixed elliptic curve . . . . . . . . . . . . . . . . . . . 105
3.5.2 Interlude: quasimodularity for correspondences . . . . . . . . . 106
3.5.3 The d-elliptic locus on M2,1 . . . . . . . . . . . . . . . . . . . . 107
3.5.3.1 The case [S] ∈ ∆1 . . . . . . . . . . . . . . . . . . . . 108
3.5.3.2 The case [S] ∈ ∆0. . . . . . . . . . . . . . . . . . . . . 109
3.5.3.3 Final computation . . . . . . . . . . . . . . . . . . . . 111
3.6 The d-elliptic locus on M3 . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.6.1 Classification of Admissible Covers . . . . . . . . . . . . . . . . 114
3.6.2 The case [S] ∈ ∆1 . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.6.2.1 Type (∆1,∆1,2) . . . . . . . . . . . . . . . . . . . . . . 116
3.6.2.2 Type (∆1,∆1,3) . . . . . . . . . . . . . . . . . . . . . . 118
3.6.2.3 Type (∆11,∆1,4) . . . . . . . . . . . . . . . . . . . . . 118
3.6.3 The case [S] ∈ ∆0 . . . . . . . . . . . . . . . . . . . . . . . . . . 119
iii
3.6.4 The class of the admissible locus . . . . . . . . . . . . . . . . . 121
3.A Quasi-modularity on M2,2 . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.B An enumerative application . . . . . . . . . . . . . . . . . . . . . . . . 125
3.B.1 The Class of µP . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3.B.2 Genus 2 curves with split Jacobian . . . . . . . . . . . . . . . . 127
3.B.3 Intersection of µP and π2/1,d . . . . . . . . . . . . . . . . . . . . 128
Bibliography 133
iv
Acknowledgments
First and foremost, I thank my advisor, Johan de Jong for his invaluable guidance
and encouragement throughout my time at Columbia. I cannot say that I expected
to write a thesis about curves and enumerative geometry when I first started working
with the author of a 6000 page book on stacks, but Johan encouraged me to cultivate
my own interests, and was always infectiously enthusiastic about what I was thinking
about, despite its distance from his own mathematics. Our many conversations were
crucial in the formation of the ideas in this thesis, even (especially) when I thought
I had nothing interesting to say. I cannot have asked for a better advisor, and I
constantly look to his example when serving as a mentor myself.
I thank Angela Gibney, Melissa Liu, Andrei Okounkov, and Michael Thaddeus for
agreeing to serve on my thesis committee, and for helpful comments and discussions. I
particularly thank Michael for being my first point of contact with the math department
at Columbia in July 2014.
I thank all of the senior mathematicians who invited me to speak about my results
and listened to my ideas in the later stages of this work: Aaron Bertram, Dawei
Chen, Izzet Coskun, David Eisenbud, Angela Gibney, Joe Harris, Danny Krashen,
Eric Larson, Ravi Vakil, Isabel Vogt, and Shing-Tung Yau.
I have learned a lot from the many algebraic geometers at Columbia, junior and
senior, especially through the learning seminars with my academic siblings Raymond
Cheng, Remy van Dobben de Bruyn, Shizhang Li, Qixiao Ma, Monica Marinescu, Noah
v
Olander, and Dmitrii Pirozhkov.
The mathematical content of this thesis benefited from comments from and dis-
cussions with a number of people, including Amol Aggarwal, Jim Bryan, Bong Lian,
Henry Liu, Georg Oberdieck, and Nicola Pagani, and many others already mentioned
above. I especially thank Nicola Tarasca for pointing me to the paper [Har84], which
led to the main results of Chapter 2. I consider this the most significant turning point
in the progress of this thesis, and by extension, in my overall level of satisfaction with
said progress as a graduate student.
I am grateful to have been part of such a supportive group of graduate students
over the last five years. I haven’t always been the most active participant in the
department’s social life, but the community has always been there when I needed it. I
feel blessed to be among such welcoming peers, especially my dear friends Clara Dolfen
and Renata Picciotto.
I thank all of my friends and colleagues at the Columbia New Opera Workshop, for
giving me a home away from math, and for helping me discover more about myself then
I could ever have imagined. I especially thank Julian Vleeschhouwer for believing in
me and for putting up with my neuroticism and propensity for post-rehearsal drinking.
I thank my friend Lucy Zhang for all of the adventures and bizarre conversations
that have kept me sane over the years.
Finally, I think my family for their unending love and support.
vi
Chapter 1
Vorspiel
“Nun! Ich warte noch! Es sei – bis
man zahlet: eins, zwei, drei”
Wolfgang Amadeus
Mozart/Emanuel Schikaneder, Die
Zauberflote (1791)
1.1 Enumerative geometry
Broadly speaking, enumerative geometry studies questions of the form: how many
geometric objects of some type satisfy a certain list of properties? Such questions
have fascinated mathematicians for centuries. In modern mathematics, enumerative
geometry may be approached from a number of different perspectives, for example,
coming from topology, symplectic or differential geometry, or combinatorics; in this
thesis, we will approach the subject from the lens of algebraic geometry.
Notable advances in enumerative geometry have included:
• The enumeration of the 27 lines on a smooth cubic surface in P3 due to Cayley
and Salmon [Cay49]
1
• The introduction of Schubert Calculus [Sch74], giving a framework for solving
enumerative problems concerning linear spaces
• Castelnuovo’s enumeration of minimal degree covers of P1 [Cas89], which may be
considered the starting point for Chapter 2 of this thesis
• The recursive formulas counting plane curves with incidence conditions at general
points due to Kontsevich [Kon95] and Caporaso-Harris [CH98], and the ensuing
development of Gromov-Witten theory
A standard approach to problems in enumerative geometry is as follows. Suppose
that one is interested in the determining the number of elements of a set S of geometric
objects satisfying constraints Ci. One first constructs a moduli space M, itself a
geometric object (in this thesis, a scheme or Deligne-Mumford stack) whose points,
in a precise sense, naturally correspond to the elements of S. It is often the case
that one needs to enlarge S in order for M to be compact and thus be amenable
to the tools of intersection theory. Next, the constraints Ci must be understood as
subspaces (closed subschemes or substacks) Γi ⊂M, and their associated cycle classes
[Γi] ∈ H(M) in a suitable cohomology theory must be computed. (In this thesis, we
deal primarily with H = A∗, the Chow functor.) Finally, computing the intersection
product of the [Γi] and integrating the resulting class yields a number, which provides
at least a candidate answer to the original question.
Subtle issues may arise at any of these steps. A moduli spaceM or a compactifica-
tion thereof may not have the desired geometric properties, or may not exist altogether.
The classes [Γi] or the cohomology object H(M) itself may be difficult to compute.
Finally, the number obtained by integration may not correspond to the enumerative
count originally sought, due to considerations of transversality.
In many instances, confronting these obstacles individually already leads to ques-
tions interesting in their own right, and has spurned important theoretical advances
2
orthogonal to strictly enumerative problems. In the somewhat rare case that all of
these obstacles are surmountable, the resulting enumerative counts may have numer-
ical properties that suggest surprising connections to other parts of mathematics, or
may otherwise merit further investigation. The computations of this thesis, which we
outline in more detail in §1.4, uncover such numerological phenomena, see Theorems
2.1.4 and 3.1.2, suggesting directions for future work.
1.2 Moduli spaces
We now give an overview of the moduli spaces that play a substantial role in this
thesis, and the salient aspects of their intersection theory. While much of the ensuing
discussion is valid over any base scheme, we work over C for concreteness.
1.2.1 Grassmannians
A reference for this section is [EH16, Chapter 4] or [Ful84, Chapter 14]. Let V be
an n-dimensional vector space, and let k ≤ n be a positive integer. Consider the
contravariant functor
Gr(k, n) : {Schemes/ Spec(C)} → {Sets}
sending a base scheme B to the set of rank k subbundles W ⊂ V ⊗C OB (by which
we mean we require the cokernel to be a locally free sheaf on B). Then, Gr(k, n) is
representable by a smooth, projective variety of dimension k(n − k), which we also
denote Gr(k, n).
We have a tautological short exact sequence on Gr(k, n)
0→W → V ⊗C OGr(k,n) → Q→ 0,
3
where W ,Q are locally free of ranks k, n− k, respectively.
Let ci = ci(W) ∈ Ai(Gr(k, n)) for i = 1, 2, . . . , k denote the Chern classes of W .
Then, the Whitney Sum Formula implies that
c(Q) =1
1 + c1 + · · ·+ ck.
Moreover, we have ci(Q) = 0 for i > n− k, giving an ideal I of relations in the ci. We
then have:
Theorem 1.2.1. [EH16, Theorem 5.26]
A∗(Gr(k, n)) ∼= Z[c1, . . . , ck]/I.
It is often more useful to work with an explicit, additive basis of cycles inA∗(Gr(k, n)).
Fix a flag 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn−1 ⊂ Vn = V of subspaces of V , where dim(Vi) = i.
Let a1, . . . , ak be a sequence of integers satisfying
n− k ≥ a1 ≥ · · · ≥ ak ≥ 0.
Then, we define the Schubert cycle σa1,...,ak ∈ A∗(Gr(k, n)) to be the class of the
subscheme of Gr(k, n) parametrizing W ⊂ V such that
dim(Vn−k+i−ai ∩W ) ≥ i
for i = 1, 2, . . . , k. More precisely, Schubert cycles may be expressed in terms of
degeneracy loci for the natural maps W → V/Vj ⊗k OGr(k,n). We then have:
Theorem 1.2.2. [EH16, §4.1] The Schubert cycles σa1,...,ak form an additive basis of
A∗(Gr(k, n)).
Multiplication of Schubert cycles is determined combinatorially by the Pieri Rule
4
([EH16, Theorem 4.14] or [Ful84, Lemma 14.5.2]) and Giambelli Formula ([EH16,
Proposition 4.16] or [Ful84, Proposition 14.6.4]), or by the Littlewood-Richardson Rule
[Ful84, Lemma 14.5.3].
1.2.2 Moduli spaces of curves
Fix integers g, n ≥ 0 satisfying 2g − 2 + n > 0. Consider the category fibered in
groupoids Mg.n sending a base scheme B to the groupoid of families of n-pointed,
smooth, projective, connected curves over B, that is, smooth proper morphisms ϕ :
X → B whose geometric fibers are connected curves of genus g, along with pairwise
disjoint sections σi : B → X for i = 1, 2, . . . , n.
We also consider the enlargement Mg,n sending B to the groupoid of stable n-
pointed projective, connected, nodal curves ϕ : X → B; we require the marked points
to lie in the smooth locus of π, and the geometric fibers of ϕ to have finitely many
automorphisms (as pointed curves). We then have:
Theorem 1.2.3. [DM69] Mg,n is a smooth, proper, and irreducible Deligne-Mumford
stack of dimension 3g − 3 + n containing Mg,n as a dense open substack.
We have the following natural morphisms between moduli spaces of curves:
• u : Mg,n+1 → Mg,n, forgetting the last marked point. When taken with the
rational sections σi : Mg,n → Mg,n+1 attaching a 2-pointed rational tail at the
i-th marked point, u may be regarded as the universal family over Mg,n.
• ξ0 :Mg−1,n+2 →Mg,n gluing the last two marked points.
• ξh,S : Mh,|S|+1 ×Mg−h,n−|S|+1 → Mg,n, gluing the last marked point on each
component, where |S| ⊂ {1, 2, . . . , n}.
The boundaryMg,n−Mg,n is covered by the images of the maps ξ0 and ξh,S. More
generally, the construction of these maps may be iterated to obtain a stratification
5
of Mg,n by topological type. The resulting combinatorial structure is well-suited to
inductive arguments, as we will see in Chapter 3 of this thesis.
The usual tools of intersection theory do not immediately carry over to the stack-
theoretic setting, but Mumford [Mum83] showed that one can in fact perform intersection-
theoretic calculations on Mg,n in a robust way, at least with rational coefficients. Al-
ternatively, one can employ the equivariant methods of Edidin-Graham [EG98].
The full Chow ring A∗(Mg,n) is, in general, very large and difficult to understand.
However, it contains a subring R∗(Mg,n) of tautological classes, which arise naturally
in many geometric calculations. By definition, R∗(Mg,n) is the smallest system of
subrings of A∗(Mg,n) containing the ψ-classes ψi = N∗σi(Mg,n)/Mg,n+1
∈ A1(Mg,n) for
i = 1, 2, . . . , n, and that is closed under pushforwards by all morphisms of the form
u, ξ0, ξh,S.
The tautological ring contains many classes that arise naturally in geometric cal-
culations: in addition to boundary classes defined by strata, the tautological ring also
contains λ-classes and κ-classes, see, §3.2.2.5 for definitions. It is fairly (but not com-
pletely) well-understood, see, for example, [Loo95, Fab99a, Fab99b, PPZ15, Pan15]. In
particular, there are algorithms, which have been implemented in [DSvZ20], to intersect
any collection of tautological classes, making many concrete intersection-theoretic cal-
culations tractable. While the tautological ring agrees with the Chow ring in a small
range [Mum83, Fab90a, Fab90b, Iza95, PV15], in general the Chow ring is strictly
larger, see, for example, [GP03, vZ18b].
1.2.3 Moduli spaces of branched covers of curves
Fix integers g ≥ h ≥ 0, as long as an integer d ≥ 1. We will consider the moduli of
morphisms f : X → Y , where X, Y are smooth, projective, and connected curves of
genus g, h, respectively, and f has degree d. In this section, we assume for simplicity
that f is simply branched, that is, f is branched over b = (2g − 2) − d(2h − 2)
6
distinct points y1, . . . , yb ∈ Y , but the discussion carries over to more general covers
(which we will consider later in this thesis). We will also label the ramification points
x1, . . . , xb ∈ X such that f(xi) = yi.
We then have a Deligne-Mumford stack Hg/h,d parametrizing such f : X → Y ,
along with the data of the marked ramification and branch points. We have a diagram
Hg/h,d
πg/h,d //
ψg/h,d
��
Mg,b
Mh,b
where the maps πg/h,d, ψg/h,d remember X, Y , respectively, along with the (ordered)
marked points.
Over C, the Riemann Existence Theorem implies that ψg/h,d is finite etale, and
that the geometric fibers of ψg/h,d are in bijection with collections of transpositions
σ1, . . . , σb ∈ Sd, such that σ1 · · ·σb = 1 and the σi generate a transitive subgroup of Sd,
considered up to the action of simultaneous conjugation on σ1, . . . , σb ∈ Sd. The same
eis true in the case of non-simple ramification, where the σi are constrained to have
particular cycle types. The resulting degrees of ψg/h,d are Hurwitz numbers, see [Cav10]
for a survey. While Hurwitz numbers can in principle be accessed combinatorially,
in recent years substantial progress toward their computation has been made using
Gromov-Witten theory, see for example [ELSV01, GV03, OP06]
The components of Hg/h,d, on the other hand, are indexed by orbits under a natural
braid group action on such collections σ1, . . . , σb. Determining the number of compo-
nents of Hurwitz spaces is in general a subtle combinatorial problem, but a classical
theorem of Severi shows that Hg/0,d is connected for all g, d, which in particular implies
the connectedness of moduli spaces of curves, see [Ful69]. (Individual components of
Hurwitz spaces will not play a role in this thesis.)
For enumerative applications, it is necessary to compactify the space Hg/h,d. We
7
will employ the Harris-Mumford stack Admg/h,d of admissible covers, which contains
Hg/h,d as a dense open substack, see [HM82] or §3.2.3. The maps πg/h,d, ψg/h,d may be
extended over Admg/h,d to land in Mg,b,Mh,b, respectively.
1.3 Branched cover loci on moduli spaces of curves
While the geometry of the map ψg/h,d is essentially governed by the combinatorics of
permutations, the geometry of πg/h,d is more subtle, and is the main object of study of
this thesis. Here, we make a few remarks on the role of branched cover loci (πg/h,d)∗(1)
on Mg,n and Mg,n in understanding the geometry of these moduli spaces.
In the series of papers [HM82, Har84, EH87], Eisenbud, Harris, and Mumford prove
that Mg is of general type for g ≥ 24; these results were later generalized to Mg,n
by Logan in [Log03]. The method is to produce effective divisors E on Mg for which
KMg= A + E, for some ample divisor A. The locus E is typically taken to be one
arising from branched covers. For example, when g = 2k + 1 is odd, one can take E
to be the locus of curves admitting a degree k + 1 cover of P1, that is, E is the image
of the map π(2k+1)/0,k+1. One must then compute the class of E in A1(Mg,n), which
is achieved either by degenerations to nodal curves, either through admissible covers
([HM82, Har84]) or through the theory of limit linear series ([EH87]).
A more refined question asks for the slope of Mg, see [CFM13] for a detailed
discussion. The result of Eisenbud, Harris, and Mumford implies that the slope of
Mg is at most 13/2 for g ≥ 24, an upper bound which continues to resist substantial
improvements, at least as g → ∞. On the other hand, Chen [Che10] has exhibited
lower bounds for the slope of Mg via moving curves, in the form of one-parameter
families of genus g curves admitting a cover of an elliptic curve with certain ramification
properties.
Another point of interest in the birational geometry of moduli spaces of curves is
8
the construction of extremal classes, many of which arise from branched cover loci: see,
for instance, [CC14, CC15, CP16, CT16].
Finally, some branched cover loci are known to lie outside the tautological ring, see
[GP03, vZ18b], and conjecturally this phenomenon persists in many more examples.
Thus, sufficiently strong results on the intersection-theoretic properties of branched
cover loci would enlarge the subring of A∗(Mg,n) which is understood in principle.
Recent work of Schmitt-van Zelm [SvZ18] gives an algorithm for intersecting admissible
cover loci with tautological classes in the setting of Galois curves of curves, whose
moduli spaces are smooth. Extending these results is the subject of ongoing work.
1.4 The questions of this thesis
In this thesis, we study the numerical invariants of the map πg/h,d in two settings. The
branched covers that arise are, in general, not Galois, and thus lie outside the realm of
the results of [SvZ18].
Let g ≥ 0, d ≥ 1 be integers, and let−→di = (d1, . . . , dN) be a tuple of integers
satisfying 2 ≤ di ≤ d and∑
i(di − 1) = 2g + 2d − 2. In Chapter 2, we consider
the space H−→dig/0,d parametrizing degree d covers f : C → P1, where C is a smooth,
connected curve of genus g and f is ramified to order di at distinct points xi ∈ C
with distinct images under f . We will be primarily interested in the degree of the
natural morphism π−→dig/0,d : H
−→dig/0,d →Mg,n, where n = N − 3g. That is, we enumerate
rank 1 linear series (pencils) on general pointed curves with ramification conditions
imposed at possibly variable (moving) points. We give explicit formulas in the case
(g, n,N) = (1, 1, 4) (Theorems 2.1.2 and 2.1.3), and deduce a “duality” that appears
to be a new phenomenon (Theorem 2.1.4). We also explain how to reduce the general
computation to the genus 1 case (Theorem 2.1.5).
In Chapter 3, we turn to the case h = 1, studying the so-called d-elliptic loci
9
(πg/1,d)∗(1) ∈ Ag−1(Mg) (in the setting of simply branched covers). We give formulas
in genus 2 (Theorem 3.1.3) and genus 3 (Theorem 3.1.4) and conjecture that for any
fixed g ≥ 2, the generating function
∑d≥1
(πg/1,d)∗(1)qd
is a cycle-valued quasimodular form (Conjecture 1). An important corollary of this
conjecture would be that “most” of the classes (πg/1,d)∗(1) are non-tautological.
10
Chapter 2
Enumerating pencils with moving
ramification on curves
“Cinque, dieci, venti, trenta,
trentasei, quarantatre”
Wolfgang Amadeus Mozart/Lorenzo
da Ponte, Le Nozze di Figaro (1786)
2.1 Introduction
Question 1. Let (C, p1, . . . , pn) be a general pointed curve of genus g, where 2g−2+n >
0. Let d, d1, d2, . . . , dn+m be integers such that 2 ≤ di ≤ d for all i. How many (m+ 1)-
tuples (pn+1, . . . , pn+m, f) are there, where pi ∈ C are pairwise distinct points, and
f : C → P1 is a morphism of degree d with ramification index at least di at each pi?
In other words, we count f : C → P1 (up to automorphisms of the target) subject
to ramification conditions at n fixed points and m moving points. According to a naıve
11
dimension count, we should expect the answer to be a positive integer when
g + 2(d− g − 1) =n∑i=1
(di − 1) +n+m∑i=n+1
(di − 2). (2.1)
Indeed, under the genericity assumption, the associated moduli problem has dimension
zero if and only if (2.1) holds. Comparison with the Riemann-Hurwitz formula shows
that m ≤ 3g; in fact, by introducing additional moving simple ramification points, one
may assume m = 3g, at the cost of multiplying the answer to Question 1 by (3g−m)!.
Various special cases of Question 1 have been addressed in the literature, and arise
naturally in the study of cycles on moduli spaces of curves. Formulas were given in
the case m = 0 by Osserman [Oss03], and the case (n,m) = (1, 1) by Logan [Log03,
Theorem 3.2]. The case g = 1, n = 1, m = 3, (d1, d2, d3, d4) = (d, d−1, 3, 2) was estab-
lished by Harris [Har84, Theorem 2.1(f)]. Most recently, Farkas-Moschetti-Naranjo-
Pirola [FMNP19] introduced alternating Catalan numbers, counting minimal degree
covers f : C → P1 with alternating monodromy group: this is the case where n = 0,
d = 2g + 1, and di = 3 for i = 1, 2, . . . ,m = 3g.
In this chapter, we give an essentially complete answer to Question 1. First, we
record the well-known answer when g = 0, in which case we must have m = 0:
Theorem 2.1.1 (cf. [Oss03]). Let p1, . . . , pn be general points on P1. Let d, d1, . . . , dn
be integers satisfying d1+· · ·+dn = 2d−2+n. Then, the number of degree d morphisms
f : P1 → P1 with ramification index at least di at pi is equal to the intersection number
∫Gr(2,d+1)
σd1−1 · · ·σdn−1.
The condition of ramification of order di at a general point pi is parametrized by a
Schubert cycle of class σdi−1 ∈ A∗(Gr(2, H0(P1,O(d)))). Thus, the content of Theorem
2.1.1 is that the Schubert cycles associated to general points pi intersect transversely,
which follows from [MTV09].
12
Our main new results are on an elliptic curve (E, p1), where we adapt the method
of [Har84]. The principal difficulty is the possibility that the moving points may be-
come equal, producing high-dimensional excess loci. We circumvent this problem by
imposing the ramification conditions one at a time, and in two steps: first, impose the
divisorial condition of simple ramification at pi. Then, subtract the “diagonal” excess
divisors where pj = pi, where j < i, and express the condition of higher ramification
in terms of a contact condition of the residual divisor in the universal family of pencils
on E.
This process introduces contributions from pencils with base-points, with multiplic-
ities equal to certain intersection numbers on a Grassmannian. We are led to a natural
weighting on the set of pencils on E, and obtain:
Theorem 2.1.2. Let (E, p1) be a general elliptic curve. Let d, d1, d2, d3, d4 be integers
such that d ≥ 2, 1 ≤ di ≤ 2d + 1 and d1 + d2 + d3 + d4 = 2d + 4. Then, the weighted
number of 4-tuples (V, p2, p3, p4), where the pi ∈ E are pairwise distinct points, and V
is a pencil on E of degree d with total vanishing at least di at pi, is
Nd1,d2,d3,d4 =12Cd−2
d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1),
where
Cn =1
n+ 1
(2n
n
)denotes the n-th Catalan Number.
In order to extract the answer to Question 1 when (g, n,m) = (1, 1, 3), we carry
out a delicate inclusion-exclusion procedure, and obtain:
Theorem 2.1.3. Let (E, p1) be a general elliptic curve. Let d, d1, d2, d3, d4 be integers
such that d ≥ 2, 1 ≤ di ≤ d and d1 +d2 +d3 +d4 = 2d+4. Then, the number Nd1,d2,d3,d4
of 4-tuples (p2, p3, p4, f), where pi ∈ E are pairwise distinct points, and f : E → P1 is
a morphism of degree d with ramification index at least di at each pi, is equal to:
13
(a) The intersection number
∫Gr(2,d+1)
(4∏i=1
∑ai+bi=di−2
σaiσbi
)(8σ11 − 2σ2
1).
(b) The constant term of the Laurent polynomial
Pd1−1Pd2−1Pd3−1Pd4−1,
where
Pr = rqr + (r − 2)qr−2 + · · ·+ (2− r)q2−r + (−r)q−r.
(c) An explicit piecewise polynomial function of degree 7 in d1, d2, d3, d4, see (2.9)
and (2.10) of §2.4.4.
When (d1, d2, d3, d4) = (d, d, 2, 2), we recover the familiar fact that the number of
covers f : E → P1 of degree d, totally ramified at the origin and one other point, are in
bijection with the d2−1 elements of E[d]−{p1}. When (d1, d2, d3, d4) = (d, d−1, 3, 2),
we recover [Har84, Theorem 2.1(f)]. When (d1, d2, d3, d4) = (3, 3, 3, 3), (5, 3, 3, 3), we
recover [FMNP19, Theorem 4.1] and [FMNP19, Theorem 4.8], respectively.
We may then deduce the following “duality.”
Theorem 2.1.4. Let (E, p1), d, d1, d2, d3, d4 be as in Theorem 2.1.3. Then, we have
Nd1,d2,d3,d4 = Nd+2−d1,d+2−d2,d+2−d3,d+2−d4 .
A similar duality was observed by Liu-Osserman in genus 0, see [LO06, Question
5.1]. We are not aware of a geometric explanation for this phenomenon in genus 0 or
genus 1, nor whether it generalizes in any way to higher genus.
Finally, we consider the general case. As in [Log03, Oss03, FMNP19], we degenerate
to a comb curve in which the p1, . . . , pn specialize to general points on the rational spine,
14
and obtain:
Theorem 2.1.5. For any g, d, d1, . . . , dn+m, the answer to Question 1 is determined
explicitly by the formulas given in Theorems 2.1.1 and 2.1.3, see Proposition 2.5.4.
While the idea is simple, the resulting degeneration formula is complicated, be-
cause in general, there are many ways to assign ramification sequences at the nodes of
the comb. As a result, this approach has not yet yielded simple formulas answering
Question 1, as in the case of genus 1.
We also remark that the methods in the proof of Theorem 2.1.2 work in the general
case: one can define a weighted count as in genus 1, and proceed in a similar way.
However, combinatorial difficulties again arise from the fact that the number of number
of moving points is linear in g. Thus, when g ≥ 2, obtaining answers to Question 1 in
the spirit of Theorems 2.1.2 and 2.1.3 remains open.
The structure of this chapter is as follows. We collect a series of preliminary facts
in §2.2. We develop the main geometric input in §2.3, proving Theorem 2.1.2. §2.4 is
purely combinatorial: here we deduce Theorem 2.1.3 and Theorem 2.1.4 from Theorem
2.1.2. Finally, we explain the degeneration method in §2.5, giving a precise version of
Theorem 2.1.5.
2.2 Preliminaries
2.2.1 Conventions
We work over an algebraically closed field k of characteristic zero.
If V is a vector space, PV denotes the variety Proj(Sym∗ V ∨), parametrizing lines in
V . More generally, if V is a vector bundle over a scheme, we follow the same convention.
Similarly, Gr(r, V ) is the Grassmannian of r-planes in V .
Let V be a linear series on a smooth curve C; in this paper V will always have rank
15
1, that is, dimk V = 2. The vanishing sequence of V at a point p ∈ C is the pair
(a0, a1) such that, in terms of some analytic local coordinate x around p, the sections
of V are xa0 , xa1 , where a1 > a0 ≥ 0 are integers. The total vanishing of V at p is
the integer a0 + a1, and V has a base-point at p if and only if a0 ≥ 1. If a0 = 0, the
ramification index of V at x is a1; we also say that V is ramified to order a1 at p.
These same definitions make sense when V is a limit linear series on a compact type
curve C, and p ∈ C is a smooth point.
The Brill-Noether number of V respect to marked points pi ∈ C at which V has
vanishing sequence (ai, bi) for i = 1, 2, . . . , n is
ρ(V, {pi}) = g + 2(d− g − 2)−n∑i=1
(ai + bi − 1).
If V is a limit linear series on a compact type curve C on which the pi are smooth points,
the same definition makes sense. Then, we will denote the Brill-Noether number of the
C0-aspect of V with respect to the marked points and nodes on C0 by ρ(V, {pi})C0 . A
straightforward computation shows that when V is a crude limit linear series (in the
sense of [EH86]), we have
ρ(V, {pi}) ≥∑C0⊂C
ρ(V, {pi})C0 , (2.2)
with equality if and only if V is a refined limit linear series.
We consider counts of morphisms f : C → P1 up to automorphisms of the target.
Thus, it is equivalent to count isomorphism classes of base-point-free pencils (linear
series of rank 1) on the fixed curve C.
If F (q) is a power series in q, we denote the coefficient of qd by F (q)[qd]. If α ∈
A∗(X) is a Chow class on some variety X, then {α}d denotes its projection to Ad(X).
16
2.2.2 Numerology
Here, we collect the numerical conditions in order for Question 1 to have interesting
answers.
The celebrated Brill-Noether theorem states that the moduli space of linear series
of degree d and rank r and on a general curve of rank C has dimension ρ(d, g, r) =
g + (r + 1)(d − g − r), and moreover that loci determined by ramification conditions
at fixed general points of C have the expected codimension, see [EH86, Theorem 4.5].
However, ramification conditions at moving points may fail to impose the expected
number of conditions, that is, Brill-Noether loci inMg,n may have lower-than-expected
codimension, see [EH89, §2].
On the other hand, owing to the existence of well-behaved Hurwitz spaces, moving
ramification conditions impose the correct number of conditions in the case r = 1. We
summarize this in the following well-known proposition:
Proposition 2.2.1. Let (C, p1, . . . , pn) be a general marked curve of genus g, where
2g− 2 + n > 0. Let d ≥ 2 be an integer, and let (ai0, ai1), i = 1, 2, . . . , n+m be ordered
pairs of integers satisfying 0 ≤ ai0 < ai1 ≤ d for i = 1, 2, . . . , n + m and ai1 > 1 for
i = n+ 1, . . . , n+m. Let G be the moduli space of tuples (V, pn+1, . . . , pn+m), where the
pi ∈ C are pairwise distinct points, and V is a pencil with vanishing sequence at least
(ai0, ai1) at pi for i = 1, 2, . . . , n+m. Then, G is pure of the expected dimension
ρ′ = g + 2(d− g − 1)−n∑i=1
(ai0 + ai1 − 1)−n+m∑i=n+1
(ai0 + ai1 − 2).
In particular, if ρ′ < 0, then G is empty.
Proof. We may assume by twisting V and decreasing d that the ai0 = 0 for all i. Then,
the proposition is an immediate consequence of the classical fact that Hurwitz spaces
of covers C → P1 with prescribed ramification profiles are etale over the spaces M0,r
17
parametrizing branch divisors on P1, and in particular have the expected dimension.
We omit the details.
Thus, in Question 1, we impose the condition (2.1).
Corollary 2.2.2. Suppose that (2.1) holds. Then, all of of the morphisms counted
in Question 1 are pairwise distinct, have ramification index exactly di at pi, and have
ramification index at most 2 away from the pi.
Proposition 2.2.3. Suppose that (2.1) holds. Then, m ≤ 3g.
Proof. By Riemann-Hurwitz, we have
2d+ 2g − 2 ≥n+m∑i=1
(di − 1)
= g + 2(d− g − 1) +m,
where we have applied (2.1) in the second line. Rearranging yields m ≤ 3g.
By the last part of Corollary 2.2.2, we may add additional moving points pi with
di = 2, where m+n+ 1 ≤ i ≤ m+ 3g, without changing condition 2.1. From the proof
of Proposition 2.2.3, f : C → P1 is unramified away from the pi. With these additional
moving points, the answer to Question 1 is multiplied by a factor of (3g − m)!, the
number of ways to label the additional simple ramification points. We will therefore
assume throughout the rest of the paper that m = 3g, and that all ramification of f
occurs at the pi.
We will also need a version of Proposition 2.2.1 for pencils with restricted underlying
line bundle. For simplicity, we stick to the following special case.
Proposition 2.2.4. Let E be a general curve of genus 1. Let d ≥ 2 be an integer, and
let (ai0, ai1), i = 1, 2, . . . ,m be ordered pairs of integers satisfying 0 ≤ ai0 < ai1 ≤ d and
ai1 > 1 for i = 1, 2, . . . , n+m. Let GL be the moduli space of tuples (V, p1, . . . , pm), where
18
the pi ∈ C are pairwise distinct points, and V is a pencil with vanishing sequence at
least (ai0, ai1) at pi for i = 1, 2, . . . ,m, and the underlying line bundle of V is isomorphic
to L. Then, GL is pure of the expected dimension
ρ′ = g + 2(d− g − 1)−m∑i=1
(ai0 + ai1 − 2)− 1.
In particular, if ρ′ < 0, then G is empty.
Proof. Fix a general point p′1 ∈ E. Let G be the moduli space of tuples (V, p2, . . . , pm)
with the same vanishing conditions as before at p2, . . . , pm, and the vanishing condi-
tions at p1 imposed at p′1, with no condition on the underlying line bundle of V . By
Proposition 2.2.1, G is pure of the expected dimension ρ′.
We have a map ϕ : GL → G sending (V, p1, . . . , pm) to t∗p1(V, p2, . . . , pm), where tp1
denotes the translation by p1 according to the group law on the elliptic curve (E, p′1).
We have that ϕ is a E[d]-torsor: indeed, if L′ is the underlying line bundle of V , then
ϕ−1(V, p2, . . . , pm) = {p1 ∈ E|t∗p1L′ ∼= L}.
In particular, dim(GL) = dim(G) = ρ′.
2.2.3 Schubert Calculus
Let V be a vector space of dimension n, and fix a complete flag 0 = V0 ⊂ V1 ⊂ · · · ⊂
Vn = V , where dimVk = k. On the Grassmannian Gr(2, n), let σa,b ∈ Aa+b(Gr(2, n))
denote the class of the subscheme parametrizing two-dimensional subspaces W ⊂ V
satisfying W ∩ Vn−1−a 6= {0} and W ⊂ Vn−b. As is conventional, we denote σa = σa,0.
The classes σa,b, where 0 ≤ b ≤ a ≤ n−2, form a Z-basis for the Chow ring A∗(Gr(2, n)).
The following is a consequence of the Pieri Rule and Hook Length Formula:
19
Lemma 2.2.5. We have
σk1 =∑a+b=k
ca,bσa,b,
where
ca,b =
(a+ b
a
)· a− b+ 1
a+ 1
is the number of Standard Young Tableaux (SYT) of shape (a, b).
We also have the following generating function formula for the ca,b:
Lemma 2.2.6. For t ≥ 1, we have
ft(z) =∞∑
mi=0
ct+mi−1,mizmi =
(1−√
1− 4z
2z
)t.
Proof. We proceed by induction on t. When t = 1, we have that cmi,miis the Catalan
number Cmi, and
∞∑mi=0
cmi,mizmi =
1−√
1− 4z
2z,
see, for example, [Sta99, Example 6.2.6]. When t = 2, we have that cmi+1,miis the
Catalan number Cmi+1, so
∞∑mi=0
cmi+1,mizmi =
1
z
(1−√
1− 4z
2z− 1
)
=
(1−√
1− 4z
2z
)2
.
When t ≥ 3, we have
ct+mi−1,mi= ct+mi−1,mi+1 − ct+mi−2,mi+1,
as a SYT of shape (t + mi − 1,mi + 1) has its largest entry in the right-most box of
20
either the top or bottom row. Therefore,
ft(z) =ft−1(z)− 1
z− ft−2(z)− 1
z,
as cn,0 = 1 for all n. The lemma now follows from the fact that α = 1−√
1−4z2
satisfies
the quadratic equation zα2 − α + 1 = 0.
2.3 The weighted count in genus 1: Proof of Theo-
rem 2.1.2
In this section, we consider Question 1.1 in the case g = n = 1, so that m = 3: we refer
to the fixed curve as (E, p1) to emphasize that its genus is 1. Fix integers d, d1, d2, d3, d4
such that 2 ≤ di ≤ 2d−2 and d1 +d2 +d3 +d4 = 2d+4 (we comment on the additional
boundary cases allowed in Theorem 2.1.2 at the end of this section).
2.3.1 The weighted count Nd1,d2,d3,d4
Let us now define the weighted count appearing in Theorem 2.1.2.
Definition 2.3.1. We define Nd1,d2,d3,d4 to be the number of 4-tuples (p2, p3, p4, V ),
where pi ∈ E are pairwise distinct points, and V is a pencil with total vanishing at
least (and thus, by Corollary 2.2.2, exactly) di at pi for i = 1, 2, 3, 4, such if V is a
pencil with vanishing sequence (ki, di − ki) at pi, then V is counted with multiplicity
Cd1,d2,d3,d4k1,k2,k3,k4
=4∏i=1
cdi−ki−1,ki .
Here, the ca,b are as in Lemma 2.2.5.
Remark 2.3.2. We digress here to illustrate the role of the weights in Definition 2.3.1
in genus 0. Consider the weighted number of pencils with total vanishing di at general
21
points p1, . . . , pn on P1, with weights defined analogously as in Definition 2.3.1. By
Theorem 2.1.1, this is
∑0≤ki<di/2
∫Gr(2,d+1)
n∏i=1
cdi−ki−1,kiσdi−ki−1,ki
=
∫Gr(2,d+1)
∑0≤ki<di/2
n∏i=1
cdi−ki−1,kiσdi−ki−1,ki
=
∫Gr(2,d+1)
n∏i=1
∑0≤ki<di/2
cdi−ki−1,kiσdi−ki−1,ki
=
∫Gr(2,d+1)
n∏i=1
σdi−11
=
∫Gr(2,d+1)
σ2d−21
=Cd−1,
Thus, the weighted count of pencils produces a considerably simpler answer than the
unweighted count of base-point free pencils; we will find a similar phenomenon in genus
1. More generally, in the weighted setting, vanishing conditions at multiple fixed points
may be combined in to a vanishing condition at a single point, see Proposition 2.5.5.
2.3.2 Outline of Proof
We briefly summarize the method to compute Nd1,d2,d3,d4 . We first change the problem
slightly: fix a line bundle L on E. Up to a factor of d2, it suffices to enumerate pencils
on E with underlying line bundle L and the same ramification conditions, but where
p1 is allowed to move (Proposition 2.3.3).
We then work on the parameter space T = Gr(2, H0(L)) × E1 × E2 × E3 × E4,
where the Ei are all isomorphic to E. We would like to consider the locus of 5-tuples
(V, p1, p2, p3, p4) where V is ramified to order di. The main difficulty is to remove the
excess loci where the pi become equal to each other: we do this as follows. First, let
22
T1 be the (closure of the) codimension 1 locus where V is simply ramified at p1; its
class expressed using Porteous’s formula. We show in Lemma 2.3.7 that the locus Td1−1
where T1 has contact order at least d1− 1 with E1 is, set-theoretically, the locus where
V has total vanishing at least d1 at p1. Moreover, we show in Lemma 2.3.8 that the
components of Td1−1 parametrizing pencils with vanishing sequence at least (k1, d1−k1)
appear with multiplicity is cd1−k1−1,k1 , as defined in Definition 2.3.1.
Next, on Td1−1, we impose the condition of simple ramification at p2, which defines
a Cartier divisor Td1−1,1 ⊂ Td1−1. We find in Lemma 2.3.10 that Td1,1 contains the
diagonal locus ∆12, where p1 = p2, with multiplicity d1 − 1. The residual divisor
Td1−1,1 = Td1−1,1 − (d1 − 1)∆12 is the closure of the locus of (V, p1, p2, p3, p4) where
p1 6= p2, and V has total vanishing at least d1 at p1 and at least 2 at p2.
As in the construction of Td1−1, we now let Td1−1,d2−1 be the locus on Td1−1 where
Td1−1,1 intersects E2 with multiplicity at least d2−1. Set-theoretically, Td1−1,d2−1 is the
locus where V has total vanishing at least di at pi for i = 1, 2. We then repeat this
procedure at p3, p4.
In the end, we obtain the zero-dimensional subscheme
Td1−1,d2−1,d3−1,d4−1 ⊂ Gr(2, H0(L))× E1 × E2 × E3 × E4
where V has total vanishing at least di at pi, for i = 1, 2, 3, 4. The theory of limit linear
series guarantees that Td1−1,d2−1,d3−1,d4−1 is disjoint from all diagonals, as we see in Lem-
mas 2.3.11 and 2.3.14. The multiplicity of a component of Td1−1,d2−1,d3−1,d4−1 is exactly
its weight, as defined in Definition 2.3.1, and integrating the class of Td1−1,d2−1,d3−1,d4−1
over T yields Theorem 2.1.2.
23
2.3.3 Pencils with fixed underlying line bundle
For the rest of this section, we fix a line bundle L of degree d on E. Let G =
Gr(2, H0(E,L)) ∼= Gr(2, d).
Proposition 2.3.3. Nd1,d2,d3,d4 is equal to the product of 1/d2 and the weighted number
of 5-tuples (V, p′1, p2, p3, p4), where p′1, p2, p3, p4 ∈ E are pairwise distinct, and V is a
pencil on E with total vanishing d1 at p′1 and di at pi for i = 2, 3, 4. Here, the weighting
in the latter count is the same as in the definition of Nd1,d2,d3,d4.
Proof. This is immediate from the proof of Proposition 2.2.4, as the fibers of ϕ : GL → G
have size d2 when the (expected) dimension of the source and target are both equal to
zero.
In light of Proposition 2.3.3, we will drop the fixed point p1 from E, and by abuse
of notation, count 5-tuples (V, p1, p2, p3, p4) where p1 is allowed to move, but the un-
derlying line bundle of V is constrained to be isomorphic to L, that is, V ⊂ H0(L).
2.3.4 The ramification loci on G× E
Definition 2.3.4. For non-negative integers 0 ≤ b ≤ a ≤ d − 1, let Σa,b ⊂ G × E
be the closed subscheme parametrizing pairs (V, p) where V ⊂ H0(L) is a pencil and
p ∈ E is a point at which the vanishing sequence of V is at least (b, a+ 1). When a, b
fail to satisfy 0 ≤ b ≤ a ≤ d − 1, we declare Σa,b to be empty, and when b = 0, we
denote Σa = Σa,b.
We construct Σa,b as follows. Let
Fk = p2∗(p∗1L ⊗OE×E/Ik∆),
where pi : E×E → E are the projection maps and I∆ is the ideal sheaf of the diagonal
24
∆ ⊂ E × E. Note that Fk is locally free of rank k. We have natural maps
ϕk : pr∗GP → pr∗E Fk
evaluating sections of L to order k. Then, Σa,b is the scheme-theoretic intersection
M0(ϕb) ∩M1(ϕa+1), where Mi(ϕk) is the degeneracy locus where ϕk has rank at most
i.
Lemma 2.3.5.
(a) Suppose that 0 ≤ b ≤ a ≤ d−2. Then, Σa,b is integral of the expected codimension
a+ b.
(b) Suppose that 0 ≤ b ≤ a = d − 1. Then, Σa,b, as a set, is the disjoint union of
Schubert cycles σd−2,b on the fibers G× {q}, for all q such that L ∼= OE(dq). In
particular, Σa,b again has the expected dimension a+ b.
Proof. In both cases, Proposition 2.2.4 implies that Σa,b has the expected codimension.
When a < d − 1, the restriction to the fiber of prE : G × E → E over any q ∈ E
is the usual Schubert cycle σa,b with respect to the flag consisting of the subspaces
H0(E,L(−rq)) ⊂ V , r = 0, 1, . . . , d− 1, which is integral of the expected codimension.
Therefore, Σa,b has the same properties.
When a = d−1, a section s ∈ H0(L) can only vanish at q to order d if L ∼= OE(dq),
in which case the condition of vanishing to order d−1 is equivalent to that of vanishing
to order d. Part (b) follows.
Lemma 2.3.6. Fix (V, q) ∈ G×E, and suppose that V has vanishing sequence (a0, a1)
at q. Then, the multiplicity of the intersection of Σ1 with EV = pr−1G (V ) is a0 + a1− 1.
Proof. Let x be an analytic local coordinate on EV near q = V (x), so that P|EPis
freely generated by the sections xa0 , xa1 . After restriction to EP , we have that Σ1 is
25
the vanishing locus of
det
xa0 xa1
a0xa0−1 a1x
a0−1
= (a1 − a0)xa0+a1−1,
which vanishes to order exactly a0 + a1 − 1 at q because a0 6= a1.
Definition 2.3.7. For integers r ≥ 1, Let Tr be the (scheme-theoretic) locus of points
(V, q) ∈ G× E where Σ1 intersects EV with multiplicity at least r.
We construct Tr as follows. LetWr be the vector bundle of rank r on G×E whose
fiber over (P, q) is
H0(EP ,O(Σ1)|EV)/mr
(V,q)H0(EP ,O(Σ1)|EV
).
Globally,
Wr = p2∗(p∗1O(Σ1)⊗OG×E×E/Ir∆),
where pi : G×E ×E → G×E are the two projection maps and I∆ is the ideal sheaf
of the pullback of the diagonal under G×E ×E → E ×E. Then, the effective divisor
Σ1 defines a tautological section of Wr, and we define Tr ⊂ G×E to be the vanishing
locus of this section. In particular, T1 = Σ1.
As a set, Lemma 2.3.6 implies that Tr is the locus where V has total vanishing at
least r + 1. Thus, it is the union of the subschemes Σa,b with a+ b = r, and in partic-
ular has the expected codimension r. Scheme-theoretically, the following proposition
identifies the scheme-theoretic multiplicities with which the Σa,b appear in Tr.
Lemma 2.3.8. We have
[Σ1]r = [Tr] =∑a+b=r
ca,b[Σa,b] (2.3)
26
in Ar(G× E), where the ca,b are as in Lemma 2.2.5.
Proof. Because I∆/I2∆∼= N∨∆/G×E×E is trivial, we may filter OG×E×E/Ir∆ by r trivial
line bundle quotients on G× E × E. As Tr has expected codimension, we get
[Tr] = cr(Wr) = {(1 + [Σ1])r}r = [Σ1]r.
establishing the first equality.
By the set-theoretic description of Tr, we have
[Tr] =∑a+b=r
c′a,b[Σa,b] (2.4)
for some integers c′a,b > 0. We wish to show that c′a,b = ca,b for all a, b. First, consider
the case in which r < d − 1. We restrict (2.4) to the fibers Gq over points q ∈ E.
As we have already seen, Σ1 and the Σa,b restrict to the usual Schubert cycles σ1 and
σa,b with respect to the flag of sections of L vanishing to varying orders at q, so [Tr]
restricts to σr1 ∈ A∗(Gq). On the other hand, in Ar(Gq), we have the formula
σr1 =∑a+b=r
ca,bσa,b,
by definition. Because the σa,b are linearly independent in Ar(Gq), we conclude ca,b =
c′a,b for all a, b.
In the case r ≥ d− 1, the above argument fails because Σd−1,r−d+1 vanishes under
pullback to Gq. We instead argue as follows. Fix a non-trivial translation τ on E. Let
G be the Gr(2, d + 1)-bundle over E whose fiber over q is H0(E,L((r − d + 2)τ(q))).
On this bundle, we may define the cycles Σa,b in terms of vanishing conditions at q in
exactly the same way as before. We then have a closed embedding ι : G×E → G over
E, sending a pencil (P, q) to the pencil (P (τ(q)), q) – that is, ι adds a base point of
order r − d + 2 at τ(q) 6= q to P , increasing the degree of the underlying line bundle
27
by the same amount.
We then obtain the formula (2.3) on G in the same way we did above, as we now
have r < deg(L(r− d+ 2)τ(q))− 1. The cycles Σa,b are stable under pullback by ι, so
we then obtain the same formula (2.3) on G× E, as desired.
2.3.5 Imposing ramification at additional points
We now impose the vanishing conditions at the points p2, p3, p4 one at a time. We work
on the subscheme Td1−1 × E2 ⊂ G × E1 × E2, where the superscripts denote different
copies of E parametrizing the pi.
Definition 2.3.9. Let Td1−1,1 ⊂ Td1−1 × E2 denote the subscheme parametrizing
(V, p1, p2) where (V, p1) ⊂ Td1−1, and additionally V has total vanishing at least 2
at p2.
More precisely, we construct Td1−1,1 by repeating the construction of T1 ⊂ G× E1
on G × T 2, and pulling back to Td1−1 × E2. As a set, Td1−1,1 includes the diagonal
∆12, that is, the locus where p1 = p2, which has codimension 1 on every component
of Td1−1 × E2. Off of the diagonal, Td1−1,1 parametrizes (V, p1, p2) where V has total
vanishing at least d1 at p1 and at least 2 at p2. It thus follows from Proposition 2.2.4
that Td1−1,1 is a Cartier divisor on Td1−1 × E2.
Lemma 2.3.10. As Cartier Divisors on Td1−1 × E2, we have
Td1−1,1 = (d1 − 1)∆12 + Td1−1,1,
where ∆12 is the pullback to Td1−1 × E2 of the diagonal in E1 × E2, and Td1−1,1 is the
scheme-theoretic closure of the locus on Td1−1×E2 where V has total vanishing at least
2 at p2.
Proof. It suffices to show that the multiplicity of ∆12 in Td1−1,1 is d1−1. In an analytic
local neighborhood of a point of G×E1×E2, let f(g, e1) be the equation cut out by T1
28
on G×E1×E2, where g is a vector of local coordinates on G and e1 is a local coordinate
on E1. Then the equations cutting out Td1−1 are ∂i
∂(e1)if(g, e1), for i = 0, 1, . . . , d1 − 2.
Now, the additional equation f(g, e2) cuts out Td1−1,1 on Td1−1×E2 ⊂ G×E1×E2,
where e2 is a coordinate on E2. Taylor expanding in an analytic local neighborhood of
a point in ∆12, we have
f(g, e2) = f(g, e1 − (e1 − e2))
=∞∑i=0
(∂i
∂(e1)if(g, e1)
)(e1 − e2)i
= (e1 − e2)d1−1
∞∑i=d1−1
(∂i
∂(e1)if(g, e1)
)(e1 − e2)i−(d1−1),
because on T 1d1−1, we have ∂i
∂(e1)if(g, e1) = 0 for i = 0, 1, . . . , d1 − 2. Because e1 − e2
is exactly the equation cutting out ∆12, it is left to check that ∂d1−1
∂(e1)d1−1f(g, e1) is not
identically zero on T 1d1−1. This follows from Proposition 2.2.4, as the locus of triples
(V, p1, p2) with total vanishing d1 + 1 at p1 is pure of dimension strictly less than that
of Td1−1.
Lemma 2.3.11.
(a) As a set, Td1−1,1 ∩ ∆12 ⊂ G × E1 × E2 is equal to the locus of triples (V, p1, p2)
where V has total vanishing at least d1 + 1 at p = p1 = p2.
(b) Suppose that V ∈ G has vanishing sequence (a0, a1) at p. Then, the multiplicity of
the intersection of Td1−1,1 with {V }×{p}×E2 at (V, p, p) is equal to a0 +a1−d1.
Proof. We first prove (a). As a set,
Td1−1,1 =⋃j
(pr∗1 Σd1−1−j,j ∩ pr∗2 Σ1)−∆12,
where pri : G × E1 × E2 → G × Ei are the projection maps, and the closure is taken
in G× E1 × E2 (equivalently, in Td1−1 × E2). Let Sj = (pr∗1 Σd1−1−j,j ∩ pr∗2 Σ1)−∆12.
29
Identifying ∆12 ⊂ G×E1×E2 with G×E, we claim that the set-theoretic restriction
of Si to ∆12 is Σd1−j,j ∪ Σd1−1−j,j+1.
It suffices to check the claim pointwise, after further restriction to G × {q} × {q},
for a fixed q ∈ E. Consider the one-parameter family p1 : X = Blq×q E × E → E,
with sections σ1, σ2 equal to the proper transforms of {q} × E and ∆, respectively. If
p2 : X → E is the second projection, the line bundle p∗2L restricts to L on the general
fiber of p1, and, over q, to L on the elliptic component and to OP1 on the rational
component.
Let GL,j be the moduli space of limit linear series on the fibers of p1 with underlying
line bundle p∗2L and with vanishing sequence at least (j, d1 − j) along σ1 and at least
(0, 2) along σ2. Following [EH86], GL,j may be constructed as a closed subscheme of a
product of Grassmannian bundles over E, and carries a projection map π : GL,j → G×E
remembering the aspects of limit linear series on the elliptic components. In particular,
π is proper, so the image of π, when restricted to G× {q}, contains Si.
The fiber of GL,j over q is the space of limit linear series V on E ∪ P1, where the
E-aspect of V has underlying line bundle L, and V has vanishing at least (j, d1 − j)
and (0, 2) at p1, p2 ∈ P1, respectively. A straightforward calculation shows that the
E-aspect of V has vanishing at least (i, d1 − i + 1) or (j + 1, d1 − j) at q. Thus, we
conclude that Sj ⊂ Σd1−j,j ∪ Σd1−1−j,j+1.
In fact, this inclusion must be an equality, because the cycle class of Sj when
restricted to general fiber of G× E → E is
σ1σd1−1−j,j = σd1−j,j + σd1−1−j,j+1,
and thus the same is true over q. Taking the union over all i yields (a).
The statement in part (b) follows from Lemmas 2.3.6 and 2.3.10. Namely, the
same proof from Lemma 2.3.6 shows that Td1−1 × E2 intersects {P} × {q1} × E2 at
30
(P, q1, q2) with multiplicity a0 +a1−1. By Lemma 2.3.10, the contribution from Td1−1,1
is a0 + a1 − 1− (d1 − 1) = a0 + a1 − d1.
We now proceed as in Definition 2.3.7 and Lemma 2.3.8. Let Td1−1,d2−1 be the
locus on Td1−1 × E2 where the divisor Td1−1,1 intersects the fibers of the projection
Td1−1 × E2 → Td1−1 with multiplicity at least d2 − 1.
On ∆12, by Lemma 2.3.11, the underlying set of Td1−1,d2−1 is the locus of pencils
with total vanishing at least d1+d2−1. Away from ∆12, the underlying set of Td1−1,d2−1
is the locus of pencils with total vanishing at least di at pi for i = 1, 2. It follows from
Proposition 2.2.4, that Td1−1,d2−1 has the expected dimension. We therefore obtain the
following analogue of Lemma 2.3.8:
Lemma 2.3.12. We have
[Td1−1,2]d2−1 = [Td1−1,d2−1] =∑
a+b=d2−1
ca,b pr∗2[Σa,b]
in Ar(Td1−1 × E2).
By the push-pull formula, we conclude:
Corollary 2.3.13. We have
[Td1−1,d2−1] = pr∗1[Σ1]d1−1 · (pr∗2[Σ1]− (d1 − 1)[∆12])d2−1
in Ar(G× E1 × E2).
We may repeat this procedure with the additional conditions at p3, p4 to obtain a
subscheme
Td1−1,d2−1,d3−1,d4−1 ⊂ G× E1 × E2 × E3 × E4
31
of class
pr∗1[Σ1]d1−1 · (pr∗2[Σ1]− (d1 − 1)[∆12])d2−1
· (pr∗3[Σ1]− (d1 − 1)[∆13]− (d2 − 1)[∆23])d3−1
· (pr∗4[Σ1]− (d1 − 1)[∆14]− (d2 − 1)[∆24]− (d3 − 1)[∆34])d4−1 (2.5)
in A∗(G×E1 ×E2 ×E3 ×E4), where pri : G×E1 ×E2 ×E3 ×E4 → G×Ei denotes
the projection as before. By construction, on the locus where the pi ∈ E are pairwise
disjoint, Td1−1,d2−1,d3−1,d4−1 is the subscheme parametrizing (V, p1, p2, p3, p4) such that
V has total vanishing at least di at pi. The following two lemmas show that in fact
Td1−1,d2−1,d3−1,d4−1 has the desired structure to obtain the weighted counts Nd1,d2,d3,d4 .
Lemma 2.3.14. The subscheme Td1−1,d2−1,d3−1,d4−1 has dimension 0, and is disjoint
from all diagonals of G× E1 × E2 × E3 × E4.
Proof. The expected dimension of Td1−1,d2−1,d3−1,d4−1 is 0, and the natural extensions
of Lemma 2.3.11, along with Proposition 2.2.4 imply the first statement. Along the
diagonals, an limit linear series similar to that in the proof of Lemma 2.3.11 shows that
the underlying set of Td1−1,d2−1,d3−1,d4−1 consists of pencils on E whose ramification is
concentrated at three or fewer points. Proposition 2.2.4 implies that for a general E,
there are no such pencils, so the lemma follows.
Lemma 2.3.15. We have
Nd1,d2,d3,d4 =1
d2
∫G×E1×E2×E3×E4
[Td1−1,d2−1,d3−1,d4−1]
Proof. By Proposition 2.3.3, it suffices to show that each point of Td1−1,d2−1,d3−1,d4−1
appears with multiplicity equal to that given in Definition 2.3.1. Because the pi
are pairwise distinct and V must vanish to order exactly di at the pi, we may re-
gard (V, p1, p2, p3, p4) as arising locally from the intersections of pr∗i [Σdi−ki−1,ki ], with
32
multiplicities as dictated by Lemma 2.3.8. Therefore, it suffices to prove that the
pr∗i Σdi−ki−1,ki intersect transversely in G× E1 × E2 × E3 × E4.
Suppose that this were not the case. We would then have a non-trivial deformation
(V , p1, p2, p3, p4) of (V, p1, p2, p3, p4). Letting B = Spec k[ε]/ε2, this means explicitly
that V ⊂ H0(E,L) ⊗k B is a linear system on E × B with underlying line bundle
LB = L⊗k B, and has vanishing (ki, di− ki) along sections p1, p2, p3, p4 of E ×B → B
restricting to p1, p2, p3, p4.
We now remove the base-points of V by twisting, and apply a translation so that
pi becomes the identity section. Let τ : E × B → E × B be the translation by p1.
Explicitly, have a new quintuple (V ′, p, p′2, p′3, p′4) with
V ′ = τ ∗
(V
(−∑i
kipi
))
p′i = τ ∗(pi);
in particular, p′1 is just the identity section p. Note that the underlying line bundle of
V ′ is
L′ = τ ∗
(LB
(−∑i
kipi
)).
Let H be the Hurwitz space parametrizing degree d −∑4
i=1 ki covers f : X → P1
ramified to order di−2ki−1 at pairwise distinct marked points pi ∈ X for i = 1, 2, 3, 4,
where X is a smooth curve of genus 1, and let ψ : H →M1,1 be the map remembering
the elliptic curve (X, p1). We claim that V ′ gives rise to a non-trivial tangent vector
v of H in the kernel of dψ. It suffices to prove that, with E fixed, we can recover the
deformation of (V, p1, p2, p3, p4) from the data of (V ′, p, p′2, p′3, p′4). Indeed, we have
τ ∗LB = L′(∑
i
kip′i
).
We may recover the translation τ , by the etaleness of the group scheme K(LB) over B,
33
and thus the section p1. Now, by inverting the formulas for P ′ and p′i, we may recover
P and pi as well.
Finally, H and M1,1 are smooth, hence the map H →M1,1 is generically smooth.
Thus, v can only map to special (E, p) ∈M1,1. Because E is general, we have reached
a contradiction, completing the proof of the lemma.
2.3.6 Proof of Theorem 2.1.2
By (2.5) and Lemma 2.3.15, to prove Theorem 2.1.2 in the case di ≥ 2, we need to
compute the integral of the class
pr∗1[Σ1]d1−1 · (pr∗2[Σ1]− (d1 − 1)[∆12])d2−1
· (pr∗3[Σ1]− (d1 − 1)[∆13]− (d2 − 1)[∆23])d3−1
· (pr∗4[Σ1]− (d1 − 1)[∆14]− (d2 − 1)[∆24]− (d3 − 1)[∆34])d4−1
on G× E1 × E2 × E3 × E4. It suffices to work in numerical equivalence; we will do so
throughout this section.
Lemma 2.3.16. In Num(G× Ei), we have
[Σ1] = σ1 + 2dxi,
where σ1 ∈ Num(G) is the usual Schubert cycle and xi ∈ Num(Ei) is the class of a
point.
Proof. We first compute the classes of c(Fk) and c(V), as defined in §2.3.4. Because
I∆/I2∆∼= N∨Ei/Ei×Ei
is trivial, we may filter OEi×Ei/Ik∆ by k trivial line bundle quotients
on E, and thus
c(Fk) = (1 + c1(L))k = 1 + dkxi.
34
Next, let V → H0(E,L) ⊗k OG be the tautological inclusion. Let W ⊂ H0(E,L)
be a subspace of codimension k + 1. By defintion, the first degeneracy locus of the
composition P → (H0(E,L)/W )⊗k OG is σk, and by Porteous, its class is also equal
to {c(V)−1}k. We thus conclude that
1
c(V)=
d−2∑i=0
σi.
Now, Σk,0 is the first degeneracy locus of ϕk+1, so by Porteous, we have
[Σk,0] =
{c(Fk+1) · 1
c(V)
}k
= σk + d(k + 1)σk−1z.
We thus have
Nd1,d2,d3,d4 =1
d2
∫G×E1×E2×E3×E4
R1R2R3R4,
where
R1 = (σ1 + 2dx1)d1−1,
R2 = (σ1 + 2dx2 − (d1 − 1)∆12)d2−1,
R3 = (σ1 + 2dx3 − (d1 − 1)∆13 − (d2 − 1)∆23)d3−1,
R4 = (σ1 + 2dx4 − (d1 − 1)∆14 − (d2 − 1)∆24 − (d3 − 1)∆34)d4−1.
35
Here, all classes are regarded as pulled back to the ambient space. We have
R2 = σd2−11
+ σd2−21 (d2 − 1)(2dx2 − (d1 − 1))∆12
+ σd2−31 (d2 − 1)(d2 − 2) · (−2d(d1 − 1)x1x2)
and
R3 = σd3−11
+ σd3−21 (d3 − 1)(2dx3 − (d1 − 1)∆13 − (d2 − 1)(d3 − 1)∆23)
+ σd3−31 (d3 − 1)(d3 − 2)(−2d(d1 − 1)x1x3 − 2d(d2 − 1)x2x3 + (d1 − 1)(d2 − 1)∆123)
+ σd3−41 (d3 − 1)(d3 − 2)(d3 − 3) · 2d(d1 − 1)(d2 − 1).
Multiplying,
R2R3 = σd2+d3−21
+ σd2+d3−31 (2d(d2 − 1)x2 + 2d(d3 − 1)x3
− (d1 − 1)(d2 − 1)∆12 − (d1 − 1)(d3 − 1)∆13 − (d2 − 1)(d3 − 1)∆23)
+ σd2+d3−41 (2d(d1 − 1)(d2 − 1)(d2 − 2)x1x2 − 2d(d1 − 1)(d3 − 1)(d3 − 2)x1x3
+ 2d(d2 − 1)(d3 − 1)(2d− d2 − d3 + 3)x2x3
− 2d(d1 − 1)(d2 − 1)(d3 − 1)∆12x3 − 2d(d1 − 1)(d2 − 1)(d3 − 1)∆13x2
+ (d1 − 1)(d2 − 1)(d3 − 1)(d1 + d2 + d3 − 4)∆123)
+ σd2+d3−51 (−2d(d1 − 1)(d2 − 1)(d3 − 1)d4(d2 + d3 − 4)x1x2x3).
36
We next multiply with
R4 = σd4−11
+ σd4−21 (d4 − 1)(2dx4 − (d1 − 1)∆14 − (d2 − 1)∆24 − (d3 − 1)∆34)
+ σd4−31 (d4 − 1)(d4 − 2)(−2d(d1 − 1)x1x4 − 2d(d2 − 1)x2x4 − 2d(d3 − 1)x3x4
+(d1 − 1)(d2 − 1)∆124 + (d1 − 1)(d3 − 1)∆134 + (d2 − 1)(d3 − 1)∆234)
+ σd4−41 (d4 − 1)(d4 − 2)(d4 − 3)(2d(d1 − 1)(d2 − 1)x1x2x4 + 2d(d1 − 1)(d3 − 1)x1x3x4
+2d(d2 − 1)(d3 − 1)x2x3x4 − (d1 − 1)(d2 − 1)(d3 − 1)∆1234)
+ σd4−51 (d4 − 1)(d4 − 2)(d4 − 3)(d4 − 4)(−2d(d1 − 1)(d2 − 1)(d3 − 1)x1x2x3x4).
In the product R2R3R4, we only wish to extract the terms that will be non-zero after
multiplying by
R1 = σd1−11 + σd1−2
1 (d1 − 1)(2dx1)
and integrating. These are the terms of R2R3R4 that have factors of exactly σd2+d3+d4−61
and σd2+d3+d4−71 .
First, we extract the terms having a factor of σd2+d3+d4−61 , and multiply by x1.
There are three non-zero contributions: the product of the σd2+d3−i1 term of R2R3 and
the σd4−(6−i)1 for i = 2, 3, 4 (when i = 5, multiplying by x1 kills the term coming from
R2R3). These are listed below; we suppress the factor of σd2+d3+d4−61 x1x2x3x4 appearing
in all three.
• i = 2: (d2 − 1)(d3 − 1)(d4 − 1)(d4 − 2)(d4 − 3)(d2 + d3 + d4 − 3)
• i = 3: −2(d2 − 1)(d3 − 1)(d4 − 1)2(d4 − 2)(d2 + d3 + d4 − 3)
• i = 4: (d2 − 1)(d3 − 1)(d4 − 1)2d4(d2 + d3 + d4 − 3)
37
The sum of these contributions is
(d2 − 1)(d3 − 1)(d4 − 1)(d2 + d3 + d4 − 3)((d4 − 2)(d4 − 3)− 2(d4 − 1)(d4 − 2) + d4(d4 − 1))
= 2(d2 − 1)(d3 − 1)(d4 − 1)(d2 + d3 + d4 − 3).
Therefore, we get a total contribution to Nd1,d2,d3,d4 of
4d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)(d2 + d3 + d4 − 3) ·∫G
σ2d−41 .
Now, we consider the contribution from terms with a factor of σd2+d3+d4−71 . Here,
there are four non-zero contributions: the product of the σd2+d3−i1 term of R2R3 and
the σd4−(7−i)1 for i = 2, 3, 4, 5. Suppressing the factors of σd2+d3+d4−7
1 x1x2x3x4, they are:
• i = 2: −2d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)(d4 − 2)(d4 − 3)(d4 − 4)
• i = 3: 2d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)(d4 − 2)(d4 − 3)(2d4 − d2 − d3)
• i = 4: 2d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)2(d4 − 2)(2d2 + 2d3 − d4 − 6)
• i = 5: −2d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)2d4(d2 + d3 − 4)
The sum of these contributions is
2d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)(−(d4 − 2)(d4 − 3)(d4 − 4) + (d4 − 2)(d4 − 3)(2d4 − d2 − d3)
+(d4 − 1)(d4 − 2)(2d2 + 2d3 + d4 − 6) + (d4 − 1)(d2 + d3 − 4))
= −4d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)(d2 + d3 + d4 − 6).
The corresponding contribution to Nd1,d2,d3,d4 is then
−4d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)(d2 + d3 + d4 − 6) ·∫G
σ2d−41 .
Summing the two contributions, we conclude:
38
d2Nd1,d2,d3,d4 = 12d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)Cd−2, (2.6)
Proof of Theorem 2.1.2. When d1, d2, d3, d4 ≥ 2, we have (2.6). When di = 1, the right
hand side of (2.6) becomes zero, and indeed, Proposition 2.2.1 implies that Nd1,d2,d3,d4 =
0.
2.3.7 Variants
Here, we make some auxiliary remarks on variants of the method of computation
above. First, note that in the first step of the proof, instead of imposing the condition
Td1−1 ⊂ G× E, we could have directly imposed the condition that V has ramification
index at least d1 at p1, that is, computed the locus Σd1−1 ⊂ G×E by Porteous’s formula.
Then, as we need to subtract excess loci in the subsequent steps, the remainder of the
computation will remain the same. Carrying out the computation in this way yields
the following:
Proposition 2.3.17. Let (E, p1), d, d1, d2, d3, d4 be as above. Then, the weighted num-
ber N◦d1,d2,d3,d4 of tuples (V, p2, p3, p4) of pencils with vanishing (0, d1) at p1 and total
vanishing di at pi for i = 2, 3, 4 is
N◦d1,d2,d3,d4 = 2d1(d1 + 1)(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)
(2d− d1 − 2
d− d1
)· 1
d(d− 1).
Here, the multiplicity of (V, p2, p3, p4) in the weighted count is
Cd2,d3,d4k2,k3,k4
=4∏i=2
cdi−ki−i,ki ,
where (ki, di − ki) is the vanishing sequence of V at pi.
Instead of working on G × E1 × E2 × E3 × E4, one can also prove Theorem 2.1.2
via an analogous computation on the smooth moduli variety G ×ME,5, where ME,5
39
denotes the fiber of the forgetful mapM1,5 →M1,1 over (E, p1). The ramification loci
may then be expressed in terms of tautological classes on M1,5. One pleasant feature
is that the analogues of “diagonal” loci ∆ij appear with multiplicity 1, and in all of
the classes pr∗i Σ1 (not just those with j < i), so the class of Td1,d2,d3,d4 in this setting
is clearly symmetric under permutation of the pi.
In fact, in this setting, it is natural to perform the computation in smooth families,
for instance, over the universal family C1,1 →M1,1 of elliptic curves. While it would be
desirable for the method to extend further to the singular fiber in the family M1,2 →
M1,1, for instance, to compute certain pure-cycle Hurwitz numbers, it breaks down at
singular points.
In either setting, one can extend the technique to higher genus curves C, and allow
the line bundle L to vary. For example, let J = Picd(C), and assume for simplicity
that d > 2g − 2. Let prJ : C × J → J be the projection map, and let E = (prJ)∗P ,
where P is the Poincare bundle. Then, one can define the ramification loci as before
on Gr(2, E)× C3g. There are no obstructions to generalizing Theorem 2.1.2 to higher
genus except for the combinatorial difficulty of having 3g copies of C. Thus, to answer
Question 1 in the case g > 1, we instead use the degeneration approach in §2.5.
Finally, the method we have developed also works in enumerating higher rank linear
systems, with two caveats. First, as Proposition 2.2.1 fails in higher rank, it is necessary
to restrict to cases in which the expected dimension statements are guaranteed to hold,
for instance, if m is small (see [EH89, Edi93, Far13]). Second, one can again obtain
counts of linear systems with imposed conditions of total vanishing, but unlike in rank
1, it is not possible to recover the counts of linear systems with prescribed vanishing
sequences simply by twisting away base-points. Thus, results such as that of Farkas-
Tarasca [FT16] remain out of reach of our techniques.
40
2.4 Base-point-free pencils in genus 1: Proof of
Theorems 2.1.3 and 2.1.4
As in the previous section, let (E, p1) be a general elliptic curve, let d, d1, d2, d3, d4
be integers such that 1 ≤ di ≤ d and d1 + d2 + d3 + d4 = 2d + 4. Let Nd1,d2,d3,d4
be the number of 4-tuples (p2, p3, p4, f), where pi ∈ E are pairwise distinct points,
and f : E → P1 is a morphism of degree d with ramification index at least (and,
by Corollary 2.2.2, exactly) di at each pi. In this section, we use the fact that the
Nd1,d2,d3,d4 are determined by the Nd1,d2,d3,d4 to prove Theorems 2.1.3 and 2.1.4.
Proposition 2.4.1. We have:
Nd1,d2,d3,d4 =∑
k1,k2,k3,k4≥0
Cd1,d2,d3,d4k1,k2,k3,k4
Nd1−2k1,d2−2k2,d3−2k3,d4−2k4 .
Proof. After adding base-points of order ki, each term on the left hand side counts
the number of pencils of degree d on E with vanishing sequence (ki, di − ki) at pi for
i = 1, 2, 3, 4, with the appropriate multiplicity as in the definition of Nd1,d2,d3,d4 .
2.4.1 Generating functions
It is natural to package the numbers Nd1,d2,d3,d4 , Nd1,d2,d3,d4 into generating functions;
after doing so, we obtain a formula for Nd1,d2,d3,d4 as a particular coefficient in a power
series in one variable, see Proposition 2.4.8. It will be convenient to treat d as an
independent variable from the di, so we first extend the definitions of Nd1,d2,d3,d4 and
Nd1,d2,d3,d4 .
Definition 2.4.2. For any integers d, d1, d2, d3, d4 with di ≥ 1 and d ≥ 2, define
Ndd1,d2,d3,d4
=12Cd−2
d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1).
41
Then, define Ndd1,d2,d3,d4
inductively as the unique integers satisfying
Ndd1,d2,d3,d4
=∑
k1,k2,k3,k4≥0
Cd1,d2,d3,d4k1,k2,k3,k4
Ndd1−2k1,d2−2k2,d3−2k3,d4−2k4
for all di ≥ 1, d ≥ 2.
Clearly, when d1 + d2 + d3 + d4 = 2d + 4, we have Nd1,d2,d3,d4 = Ndd1,d2,d3,d4
and
Nd1,d2,d3,d4 = Ndd1,d2,d3,d4
.
Definition 2.4.3. Define the generating functions
N(x1, x2, x3, x4, q) =∑
d≥2,di≥1
Ndd1,d2,d3,d4
xd11 xd22 x
d33 x
d44 q
d
N(x1, x2, x3, x4, q) =∑
d≥2,di≥1
Ndd1,d2,d3,d4
xd11 xd22 x
d33 x
d44 q
d
Lemma 2.4.4. We have
N(x1, x2, x3, x4, q) = [(6q − 1) + (1− 4q)3/2] ·4∏i=1
(xi
1− xi
)2
Proof. Indeed,
N(x1, x2, x3, x4, q)
=∑di,d
12
d(d1 − 1)(d2 − 1)(d3 − 1)(d4 − 1)Cd−2x
d11 x
d22 x
d33 x
d44 q
d
=∞∑d=2
12Cd−2
dqd ·
4∏i=1
(∞∑di=1
(di − 1)xdii
)
= [(6q − 1) + (1− 4q)3/2] ·4∏i=1
(xi
1− xi
)2
,
where in the last step we obtain the generating function for the sequence bd = 12Cd−2/d
by integrating that of the Catalan numbers, see [Sta99, Example 6.2.6].
42
Proposition 2.4.5. The generating functions N(x1, x2, x3, x4, q) and N(x1, x2, x3, x4, q)
are related by the following formulas:
N(x1, x2, x3, x4, q) = N
(1−
√1− 4x2
1q
2x1q,1−
√1− 4x2
2q
2x2q,1−
√1− 4x2
3q
2x3q,1−
√1− 4x2
4q
2x4q, q
)
N(x1, x2, x3, x4, q) = N
(x1
1 + x21q,
x2
1 + x22q,
x3
1 + x23q,
x4
1 + x24q, q
)
Proof. The second formula will follow directly from the first. We have
N(x1, x2, x3, x4, q)
=∑d,di
Ndd1,d2,d3,d4
xd11 xd22 x
d33 x
d44 q
d
=∑d,di,ki
cd1−k1−1,d1 · · · cd4−k4−1,d4 ·Nd−k1−k2−k3−k4d1−2k1,d2−2k2,d3−2k3,d4−2k4
xd11 xd22 x
d33 x
d44 q
d
=∑d,di
[xd11 x
d22 x
d33 x
d44 q
d
4∏i=1
(∞∑
mi=0
cdi+mi−1,mi(x2
i q)mi
)]Ndd1,d2,d3,d4
=∑d,di
xd11 xd22 x
d33 x
d44 q
d
4∏i=1
(1−
√1− 4x2
i q
2x2i q
)diNd
d1,d2,d3,d4
=∑d,di
qd 4∏i=1
(1−
√1− 4x2
i q
2xiq
)diNd
d1,d2,d3,d4
= N
(1−
√1− 4x2
1q
2x1q,1−
√1− 4x2
2q
2x2q,1−
√1− 4x2
3q
2x3q, 1−
√1− 4x2
4q
2x4q, q
),
where in the fifth line we have applied Lemma 2.2.6.
Combining Lemma 2.4.4 and the second part of Proposition 2.4.5, we obtain:
Corollary 2.4.6. We have
N(x1, x2, x3, x4, q) = [(6q − 1) + (1− 4q)√
1− 4q] ·4∏i=1
(xi
1− xi + x2i q
)2
(2.7)
43
Lemma 2.4.7. We have
(x
1− x+ x2q
)2
=∞∑n=0
1
1− 4q
( ∑k+`=n
αk − βk
α− β· α
` − β`
α− β
)xn
where
α =1 +√
1− 4q
2,
β =1−√
1− 4q
2.
Furthermore, the coefficient of xn above is a polynomial in q of degree⌊n
2
⌋− 1.
Proof. One verifies by a straightforward computation that
(x
1− x+ x2q
)2
=1
1− 4q
[(1
1− αx
)2
+
(1
1− βx
)2
−(
1
1− αx+
1
1− βx
)− 1√
1− 4q
(1
1− αx− 1
1− βx
)]
=∞∑n=0
1
1− 4q
[n(αn + βn)− 1√
1− 4q(αn − βn)
]xn.
Then, note that
1
1− 4q
[n(αn + βn)− 1√
1− 4q(αn − βn)
]=
1
(α− β)2
[n(αn + βn)− (α + β) · α
n − βn
α− β
]=∑k+`=n
αk − βk
α− β· α
` − β`
α− β,
which is a symmetric polynomial of degree n− 2 in α and β. Because α + β = 1 and
αβ = 1− 4q, the coefficient of xn is thus a polynomial of degree
⌊n− 2
2
⌋=⌊n
2
⌋− 1
44
in q, as claimed.
Corollary 2.4.8. We have
Nd1,d2,d3,d4 = (1− 4q)3/2 ·4∏i=1
(d1−2∑j=0
sj(α, β)sdi−2−j(α, β)
)[qd],
where
sj(x, y) =xj+1 − yj+1
x− y
is a Schur polynomial in two variables.
Proof. This is a consequence of Corollary 2.4.6 and Lemma 2.4.7. We may ignore the
contribution of the (6q − 1) term appearing on the right hand side of (2.7), because,
by the last statement in Lemma 2.4.7, the degree of the coefficient of xd11 xd22 x
d33 x
d44 in
4∏i=1
(xi
1− xi + x2i q
)2
as a polynomial in q is4∑i=1
(⌊di2
⌋− 1
)≤ d− 2,
and thus contributes nothing to the qd coefficient after multiplication by (6q − 1).
2.4.2 Schubert cycle formula
We now relate the formula in Corollary 2.4.8 to intersection numbers on the Grass-
mannian to prove Theorem 2.1.3(a).
Lemma 2.4.9. Let d be a positive integer, and let f(x, y) be a homogeneous symmetric
polynomial with deg(f) ≤ 2d− 2. Then, we have
(−1
2(1− 4q)1/2 · f(α, β)
)[qd] =
∫Gr(2,d+1)
f(x, y) · (x+ y)2d−2−deg(f),
45
where as before we put
α =1 +√
1− 4q
2,
β =1−√
1− 4q
2,
and the integrand on the right hand side is viewed as a top cohomology class on Gr(2, d+
1) via the identification of Schur polynomials sj and Schubert cycles σj.
Proof. The vector space of symmetric polynomials f(x, y) is spanned by polynomials
of the form
f(x, y) = (xy)m(x+ y)n = s11(x, y)m · s1(x, y)n,
where 2m+n = 2d−2; it suffices to prove the claim for such f . Note that f(α, β) = qm.
Now,
∫Gr(2,d+1)
f(x, y) · (x+ y)2d−2−deg(f) =
∫Gr(2,d+1)
(xy)m · (x+ y)2d−2−2m
=
∫Gr(2,d+1)
σm11 · σ2d−2−2m1
= Cd−m−1
= −1
2(1− 4q)1/2[qd−m]
= −1
2f(α, β)(1− 4q)1/2[qd−m],
where we have applied the Pieri Rule and Lemma 2.2.6.
Proposition 2.4.10. We have
Nd1,d2,d3,d4 =
∫Gr(2,d+1)
(4∏i=1
∑ai+bi=di−2
σaiσbi
)(8σ11 − 2σ2
1).
46
Proof. Applying Corollary 2.4.8,
Nd1,d2,d3,d4 = (1− 4q)3/2
4∏i=1
(d1−2∑j=0
sj(α, β)sdi−2−j(α, β)
)[qd]
= (−2 + 8q) ·
[−1
2(1− 4q)1/2
4∏i=1
(d1−2∑j=0
sj(α, β)sdi−2−j(α, β)
)][qd]
= 8 ·
[−1
2(1− 4q)1/2
4∏i=1
(d1−2∑j=0
sj(α, β)sdj−2−j(α, β)
)][qd−1]
− 2 ·
[−1
2(1− 4q)1/2
4∏i=1
(d1−2∑j=0
sj(α, β)sdi−2−j(α, β)
)][qd]
= 8
∫Gr(2,d)
(4∏i=1
∑ai+bi=di−2
σaiσbi
)− 2
∫Gr(2,d+1)
(4∏i=1
∑ai+bi=di−2
σaiσbi
)σ2
1
=
∫Gr(2,d+1)
(4∏i=1
∑ai+bi=di−2
σaiσbi
)(8σ11 − 2σ2
1),
where in the second to last step we have applied Lemma 2.4.9 to the polynomial
f(x, y) =4∏i=1
di−2∑j=0
sj(x, y)sdi−2−j(x, y)
of degree (d1 − 2) + · · ·+ (d4 − 2) = 2(d− 1)− 2.
2.4.3 Laurent polynomial formula
Here, we expand the formula in Proposition 2.4.10 to prove Theorem 2.1.3(b).
Lemma 2.4.11. Let n1, n2, n3, n4 be integers satisfying 0 ≤ ni ≤ d− 1 and n1 + n2 +
n3 + n4 = 2d− 4. Then, we have:
∫Gr(2,d)
σn1σn2σn3σn4 = min(d− n1 − 1, n4 + 1)
Proof. Without loss of generality, suppose that n1 ≥ n2 ≥ n3 ≥ n4, so that n1 + n2 ≥
d−2. If n1 = d1−1, then σn1 = 0 = min(d−n1−1, n4 +1), so assume that n1 ≤ d1−2.
47
By the Pieri Rule, we have
σn1σn2 = σn1,n2 + σn1+1,n2−1 + · · ·+ σd−2,n1+n2−d+2.
We wish to express the product of this class with σn3 in the Schubert cycle basis and
extract the coefficient of σd−2,d−2−n4 . By the Pieri rule, each product σn1+i,n2−iσn3 will
be a sum of Schubert cycles with multiplicity 1, and σd−2,d−2−n4 appears if and only if
d− 2− n4 ≤ n1 + i. If d− 2− n4 − n1 ≤ 0, or equivalently d− n1 − 1 ≤ n4 + 1, then
this is true for all of the terms above, and we conclude that
∫Gr(2,d)
σn1σn2σn3σn4 = d− n1 − 1.
Otherwise, the number of terms for which d− 2− n4 ≤ n1 + i is n4 + 1, and
∫Gr(2,d)
σn1σn2σn3σn4 = n4 + 1.
This establishes the lemma.
Lemma 2.4.12. Let n1 ≥ n2 ≥ n3 ≥ n4 ≥ 0 be integers satisfying n1 + n2 + n3 + n4 =
2d− 4.
∫Gr(2,d+1)
σn1σn2σn3σn4(8σ11 − 2σ21) =
6 n1 = n2 = n3 = n4
4 n1 = n2 6= n3 = n4
2 n1 + n4 = n2 + n3 and n1 6= n2
−2 n1 = n2 + n3 + n4 + 2
0 otherwise
Proof. Without loss of generality, suppose that n1 ≥ n2 ≥ n3 ≥ n4. First, if n1 > d−1,
then σn1 = 0, and it is clear that none of the first four conditions on the right hand
48
side can be satisfied. If n1 = d − 1, then we are in the fourth case on the right hand
side, as n2 + n3 + n4 = (2d− 4)− (d− 1) = d− 3. In this case, the Pieri rule implies
that σd−1σ11 = 0 and ∫Gr(2,d+1)
σd−1σn2σn3σn4σ21 = 1
so again the Lemma holds.
We next dispose of the case n1 ≤ 1: the possibilities are
(n1, n2, n3, n4) = (0, 0, 0, 0), (1, 1, 0, 0), (1, 1, 1, 1),
and one easily checks that the Lemma holds here.
Thus, we assume that the d − 2 ≥ n1 ≥ n2 ≥ n3 ≥ n4 ≥ 0 and n1 ≥ 2. Applying
the Pieri rule and Lemma 2.4.11,
∫Gr(2,d+1)
σn1σn2σn3σn4(8σ11 − 2σ21)
=
∫Gr(2,d+1)
σn1σn2σn3σn4(6σ11 − 2σ11)
= 6
∫Gr(2,d)
σn1σn2σn3σn4 − 2
∫Gr(2,d+1)
(σn1+2 + σn1+1,1 + σn1,2)σn2σn3σn4
= 4
∫Gr(2,d)
σn1σn2σn3σn4 − 2
∫Gr(2,d+1)
σn1+2σn2σn3σn4 − 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4
= 4 min(d− n1 − 1, n4 + 1)− 2 min(d− n1 − 2, n4 + 1)− 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4 .
We now consider the first, second, third, and fifth cases separately: as n1 ≤ d− 2,
we cannot have n1 = n2 + n3 + n4 + 2. Suppose first that n1 = n2 = n3 = n4 = n, and
49
d = 2n+ 2. We then have
4 min(d− n1 − 1, n4 + 1)− 2 min(d− n1 − 2, n4 + 1)− 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4
= 4 min(n, n+ 1)− 2 min(n+ 1, n+ 1)− 2
∫Gr(2,2n+1)
σ3nσn−2
= 4n− 2(n+ 1)− 2 min(n, n− 1)
= 6.
Next, consider the case n1 = n2 6= n3 = n4, so that d = n1 + n3 + 2. We have
4 min(d− n1 − 1, n4 + 1)− 2 min(d− n1 − 2, n4 + 1)− 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4
= 4 min(n3 + 1, n4 + 1)− 2 min(n3, n4 + 1)− 2
∫Gr(2,n1+n3+1)
σn1σn1−2σ2n3.
To evaluate the last term, we consider two sub-cases: if n1 − 2 ≥ n3, then by Lemma
2.4.11, we have
∫Gr(2,n1+n3+1)
σn1σn1−2σ2n3
= min(n3, n3 + 1) = n3.
On the other hand, if n1 − 2 < n3, we must have n1 − n3 = 1, as n1 > n3. Thus,
∫Gr(2,n1+n3+1)
σn1σn1−2σ2n3
= min(n1, n1 − 1) = n1 − 1 = n3.
Therefore, in both sub-cases, we have
∫Gr(2,d+1)
σn1σn2σn3σn4(8σ11 − 2σ21) = 4(n3 + 1)− 2n3 − 2n3 = 4.
Next, consider the case n1 + n4 = n2 + n3 and n1 6= n2. Then, d − n1 = n4 + 2.
50
Thus,
4 min(d− n1 − 1, n4 + 1)− 2 min(d− n1 − 2, n4 + 1)− 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4
= 4(n4 + 1)− 2n4 − 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4 .
If n1− n2 ≥ 2, then the last integral is equal to min(d− n1, n4 + 1) = (n4 + 1), and we
immediately deduce the lemma. If, on the other hand, n1 − n2 = 1, we first note that
n4 ≤ n1 − 2, or else n2 = n3 = n4, an impossibility. Then, Lemma 2.4.11 implies that
the last term is again equal to min(d− n2 − 1, n4 + 1) = min(d− n1, n4 + 1) = n4 + 1,
so we are done in this case.
Finally, suppose n1 +n4 6= n2 +n3. In particular, we have either d−n1−2 ≥ n4 +1
or d− n1 ≤ n4 + 1. First, assume that n1 − n2 ≥ 2. Then,
∫Gr(2,d−1)
σn1−2σn2σn3σn4 = min(d− n1, n4 + 1).
Thus, the expression
4 min(d−n1− 1, n4 + 1)− 2 min(d−n1− 2, n4 + 1)− 2
∫Gr(2,d−1)
σn1−2σn2σn3σn4 (2.8)
is equal to either
4(n4 + 1)− 2(n4 + 1)− 2(n4 + 1) = 0
or
4(d− n1 − 1)− 2(d− n1 − 2)− 2(d− n1) = 0,
so we have the lemma if n1 − n2 ≥ 2.
Suppose instead that n1−n2 = 0 or n1−n2 = 1. Then, we can check as before that
n1−2 ≥ n4. Furthermore, we claim that we must have d−n1−2 ≥ n4+1. If not, then we
51
have instead d−n1 ≤ n4 + 1, so n2 +n4 ≥ (n1−1) +n4 ≥ d−2 = 12(n1 +n2 +n3 +n4),
which is impossible unless n1 = n2, n3 = n4. From here, one easily evaluates the
expression (2.8) as in the previous cases, so we are done.
Proposition 2.4.13. Nd1,d2,d3,d4 is the constant term of the Laurent polynomial
Pd1−1Pd2−1Pd3−1Pd4−1,
where
Pr = rqr + (r − 2)qr−2 + · · ·+ (−r + 2)q−r+2 + (−r)q−r.
Proof. First, observe that, by the Pieri rule,
τdi−2 :=∑
ai+bi=di−2
σaiσbi =∑
a′i+b′i=di−2
(a′i − b′i + 1)σa′i,b′i
By Lemma 2.4.12, the positive contributions to
∫Gr(2,d+1)
τd1−2τd2−2τd3−2τd4−2(8σ11 − 2σ21)
correspond to terms σa′1,b′1σa′2,b′2σa′3,b′3σa′4,b′4 with
(a′i − b′i) + (a′j − b′j) = (a′k − b′k) + (a′` − b′`)
for some permutation (i, j, k, `) of (1, 2, 3, 4), Moreover, if we fix i = 1, the contribution
to the integral is
2m4∏i=1
(a′i − b′i + 1)
where m is the number of such permutations. Similarly, the negative contributions to
52
the integral correspond to terms where
(a′i − b′i + 1) = (a′j − b′j + 1) + (a′k − b′k + 1) + (a′` − b′` + 1),
and the contribution to the integral is
−24∏i=1
(a′i − b′i + 1)
We match these contributions exactly with the contributions to the constant term
in Pd1−1Pd2−1Pd3−1Pd4−1: the positive contributions come from terms
4∏i=1
miqmi
with exactly two of the mi positive, and the negative contributions come from terms
in which one or three of the mi are positive.
One can easily deduce the following, which is also a consequence of Proposition
2.3.17.
Corollary 2.4.14. Suppose that d1 = d. Then,
Nd1,d2,d3,d4 = 2(d+ 1)(d2 − 1)(d3 − 1)(d4 − 1).
In particular, we recover [Har84, Theorem 2.1(f)], as well as the fact that the
number of degree d covers f : E → P1 totally ramified at d and one other point is
equal to #E[d2]− 1 = d2 − 1.
2.4.4 Explicit formula, and proof of Theorem 2.1.4
Using the Laurent polynomial formula of the previous section, we now complete the
proofs of Theorems 2.1.3 and 2.1.4.
53
Let m,n be integers with m ≥ n. We first compute PmPn. When 0 ≤ k ≤ n, the
coefficients of qm+n−2k and q−m−n+2k in PmPn are
∑r+s=k
(m− 2r)(n− 2s) = (k + 1)mn− k(k + 1)(m+ n) + 4∑k
i=0 i(k − i)
= (k + 1)mn− k(k + 1)(m+ n) + 4
(k2(k + 1)
2− k(k + 1)(2k + 1)
6
)= (k + 1)
[mn− k(m+ n) +
2
3k(k − 1)
]
When 0 ≤ k ≤ m− n, the coefficients of qm−n−2k are
(m− 2k)(−n) + (m− 2k − 2)(−n+ 2) + · · ·+ (m− 2k − 2n+ 2)(n− 2) + (m− 2k − 2n)(n)
=1
2[(2n)(−n) + (2n− 2)(−n+ 2) + · · ·+ (−2n+ 2)(n− 2) + (−2n)(n)],
where we have paired summands from the outside inward. In particular, the value of
this coefficient does not depend on k, so we may take k = 0, in which case we have
already computed the qm−n coefficient to be
(n+ 1)
[mn− n(m+ n) +
2
3n(n− 1)
]=(n+ 1)
[−1
3n2 − 2
3n
]=− 1
3n(n+ 1)(n+ 2).
Also, the coefficients of qr and q−r are equal for all r.
We now evaluate the constant term of (Pd1−1Pd2−1) · (Pd3−1Pd4−1) by matching
coefficients in the two factors. Without loss of generality, assume that d ≥ d1 ≥ d2 ≥
d3 ≥ d4 ≥ 0. We consider the coefficients q` in the first term and q−` in the second
with −d3 − d4 + 2 ≤ ` ≤ d3 + d4 − 2.
First, suppose that d1 − d2 ≥ d3 − d4. Then, there are five intervals over which we
vary `: [−d3 − d4 + 2, d2 − d1], (d2 − d1, d4 − d3], (d4 − d3, d3 − d4), [d3 − d4, d1 − d2),
54
and [d1 − d2, d3 + d4 − 2].
The contribution from the interval (d4 − d3, d3 − d4) to Nd1,d2,d3,d4 is
(−1
3d2(d2 + 1)(d2 − 1)
)·(−1
3d4(d4 + 1)(d4 − 1)
)· (d3 − d4 − 1)
The contribution from (d2 − d1, d4 − d3] and [d3 − d4, d1 − d2) is
2
(−1
3d2(d2 + 1)(d2 − 1)
)
·(d1−d2)−(d3−d4)
2∑j=1
(d4 − j + 1)
[(d3 − 1)(d4 − 1)− (d4 − j)(d3 + d4 − 2) +
2
3(d4 − j)(d4 − j − 1)
]
The contribution from [−d3 − d4 + 2, d2 − d1] and [d1 − d2, d3 + d4 − 2] is
2
d2+d3+d4−d12
−1∑k=0
(k + 1)
[(d3 − 1)(d4 − 1)− k(d3 + d4 − 2) +
2
3k(k − 1)
]· (k′ + 1)
[(d1 − 1)(d2 − 1)− k′(d1 + d2 − 2) +
2
3k′(k′ − 1)
]
where k′ = k + (d1+d2)−(d3+d4)2
.
Summing these contributions yields a formula for Nd1,d2,d3,d4 in the case d1 − d2 ≥
d3− d4: we may expand each summand and apply standard formulas for sums of m-th
55
powers of integers for m ≤ 6. This is implemented with the help of SAGE, and yields:
Nd1,d2,d3,d4 = − 1
3360d7
1 +1
240d5
1d22 −
1
96d4
1d32 +
1
96d3
1d42 −
1
240d2
1d52 +
1
3360d7
2 +1
240d5
1d23
− 1
48d3
1d22d
23 +
1
48d2
1d32d
23 −
1
240d5
2d23 −
1
96d4
1d33 +
1
48d2
1d22d
33 −
1
96d4
2d33 +
1
96d3
1d43
− 1
96d3
2d43 −
1
240d2
1d53 −
1
240d2
2d53 +
1
3360d7
3 +1
240d5
1d24 −
1
48d3
1d22d
24 +
1
48d2
1d32d
24
− 1
240d5
2d24 −
1
48d3
1d23d
24 +
1
48d3
2d23d
24 +
1
48d2
1d33d
24 +
1
48d2
2d33d
24 −
1
240d5
3d24
− 1
96d4
1d34 +
1
48d2
1d22d
34 −
1
96d4
2d34 +
1
48d2
1d23d
34 +
1
48d2
2d23d
34 −
1
96d4
3d34 +
1
96d3
1d44
− 1
96d3
2d44 −
1
96d3
3d44 −
1
240d2
1d54 −
1
240d2
2d54 −
1
240d2
3d54 +
1
3360d7
4 −1
480d5
1
+1
96d4
1d2 −1
48d3
1d22 +
1
48d2
1d32 −
1
96d1d
42 +
1
480d5
2 +1
96d4
1d3 −1
48d2
1d22d3
+1
96d4
2d3 −1
48d3
1d23 −
1
48d2
1d2d23 +
1
48d1d
22d
23 +
1
48d3
2d23 +
1
48d2
1d33 +
1
48d2
2d33
− 1
96d1d
43 +
1
96d2d
43 +
1
480d5
3 +1
96d4
1d4 −1
48d2
1d22d4 +
1
96d4
2d4 −1
48d2
1d23d4
− 1
48d2
2d23d4 +
1
96d4
3d4 −1
48d3
1d24 −
1
48d2
1d2d24 +
1
48d1d
22d
24 +
1
48d3
2d24
− 1
48d2
1d3d24 −
1
48d2
2d3d24 +
1
48d1d
23d
24 −
1
48d2d
23d
24 +
1
48d3
3d24 +
1
48d2
1d34 +
1
48d2
2d34
+1
48d2
3d34 −
1
96d1d
44 +
1
96d2d
44 +
1
96d3d
44 +
1
480d5
4 +1
60d3
1 −1
60d2
1d2 +1
60d1d
22
− 1
60d3
2 −1
60d2
1d3 −1
60d2
2d3 +1
60d1d
23 −
1
60d2d
23 −
1
60d3
3 −1
60d2
1d4 −1
60d2
2d4
− 1
60d2
3d4 +1
60d1d
24 −
1
60d2d
24 −
1
60d3d
24 −
1
60d3
4 −1
70d1 +
1
70d2 +
1
70d3 +
1
70d4
(2.9)
Now, consider the case d1−d2 ≤ d3−d4. Similarly to the first case, the contribution
from the interval (d2 − d1, d1 − d2) to Nd1,d2,d3,d4 is
(−1
3d2(d2 + 1)(d2 − 1)
)·(−1
3d4(d4 + 1)(d4 − 1)
)· (d1 − d2 − 1)
56
The contribution from (d4 − d3, d2 − d1] and [d1 − d2, d3 − d4) is
2
(−1
3d4(d4 + 1)(d4 − 1)
)
·(d3−d4)−(d1−d2)
2∑j=1
(d2 − j + 1)
[(d1 − 1)(d2 − 1)− (d2 − j)(d1 + d2 − 2) +
2
3(d2 − j)(d2 − j − 1)
]
Finally, the contribution from [−d3 − d4 + 2, d4 − d3] and [d3 − d4, d3 + d4 − 2] is
d4−1∑k=0
(k + 1)
[(d3 − 1)(d4 − 1)− k(d3 + d4 − 2) +
2
3k(k − 1)
]· (k′ + 1)
[(d1 − 1)(d2 − 1)− k′(d1 + d2 − 2) +
2
3k′(k′ − 1)
],
where k′ = k + (d1+d2)−(d3+d4)2
.
Summing as in the previous case, we get that Nd1,d2,d3,d4 is equal to:
Nd1,d2,d3,d4 =− 1
48d4
1d34 +
1
24d2
1d22d
34 −
1
48d4
2d34 +
1
24d2
1d23d
34 +
1
24d2
2d23d
34 −
1
48d4
3d34
− 1
120d2
1d54 −
1
120d2
2d54 −
1
120d2
3d54 +
1
1680d7
4 +1
48d4
1d4 −1
24d2
1d22d4
+1
48d4
2d4 −1
24d2
1d23d4 −
1
24d2
2d23d4 +
1
48d4
3d4 +1
24d2
1d34 +
1
24d2
2d34
+1
24d2
3d34 +
1
240d5
4 −1
30d2
1d4 −1
30d2
2d4 −1
30d2
3d4 −1
30d3
4 +1
35d4 (2.10)
Proof of Theorem 2.1.3. Combine the above with Propositions 2.4.10 and 2.4.13.
Proof of Theorem 2.1.4. One checks by direct computation (carried out in SAGE) that
the right hand sides of (2.9) and (2.10) are sent to each other under the involution
di 7→ 12(d1 + d2 + d3 + d4)− di.
57
2.5 The general case via limit linear series
In this section, we use the theory of limit linear series to give a more precise version
of Theorem 2.1.5, which yields explicit answers to Question 1 for any given values of
g, d, di. A similar degeneration technique is used in [Log03, Oss03, FMNP19].
2.5.1 The degeneration formula
We adopt the notation of Question 1 and assume that (2.1) holds. In addition, we take
m = 3g, following Proposition 2.2.3 and the ensuing discussion. Then, let N gd1,...,dn+3g
be the answer to Question 1, counting covers f : C → P1 with ramification index di at
fixed points p1, . . . , pn and moving points pn+1, . . . , pn+3m.
Definition 2.5.1. Fix general elliptic curves (Ej, qj), j = 1, 2, . . . , g, and fix a general
(n + g)-pointed rational curve (P1, p1, . . . , pn, r1, . . . , rg). Then, let (X0, p1, . . . , pn) be
the nodal curve obtained by attaching the Ej to P1, gluing the point rj to rj for
j = 1, 2, . . . , g.
Lemma 2.5.2. Consider the moduli space GX0 of tuples (V0, pn+1, . . . , pn+3g), where
V0 is a limit linear series of degree d on X0, and p1, . . . , p3g ∈ X are pairwise distinct
smooth points of X0 such that V0 has vanishing at least (0, di) at pi. Then, we have:
(a) Given any [(V0, pn+1, . . . , pn+3g)] ∈ GX0, V0 is refined (in the sense of [EH86]),
and exactly three of the moving points pi, i = n + 1, . . . , n + 3g lie on each
Ej. Moreover, the vanishing sequence of V0 at pi is exactly (0, di) for all i =
1, 2, . . . , n+ 3g. In particular, none of pn+1, . . . , pn+3g lie on P1.
(b) GX0 is reduced of dimension 0.
(c) Any [(V0, pn+1, . . . , pn+3g)] ∈ GX0 smooths to a linear series on the general fiber
of the versal deformation of (X0, p1, . . . , pn+3g), preserving the ramification con-
ditions at the pi.
58
Proof. By condition (2.1), we always have ρ(V0, {p1, . . . , pn+3g}) = −3g. By sub-
additivity of the Brill-Noether number (2.2) and Proposition 2.2.1, we have that
ρ(V0, {pi})Ej= −3 for all j, and ρ(V0.{pi})P1 = 0. Thus, the Brill-Noether num-
ber is in fact additive, so V0 is a refined limit linear series. Moreover, it follows that we
need three moving points on each Ej, and that V0 cannot have higher-than-expected
ramification at any of the pi; this establishes (a).
Part (b) follows from the same statements for the moduli of linear series on the
individual components; on the rational spine, this is Theorem 2.1.1, and on the elliptic
components, this is a consequence of the transversality argument given in Lemma
2.3.15. Finally, part (c) follows immediately from [EH86, Corollary 3.7], as V0 is refined,
and dimensionally proper with respect to the pi.
Lemma 2.5.3. Let R be a discrete valuation ring, and let B = SpecR. Let π : X →
B, σi : B → X, i = 1, 2, . . . , n be a 1-family of pointed n-pointed, genus g curves with
special fiber isomorphic to (X0, p1, . . . , pn) and smooth total space X. Let p′i denote
the restriction of σi to the geometric generic fiber Xη for i = 1, 2, . . . , n. Suppose that
(V ′, p′n+1, . . . , p′n+3g) is a tuple where V ′ is a linear series on Xη and p′1, . . . , p
′n+3g ∈ Xη
are pairwise distinct points such that V ′ has ramification sequence (0, di) at p′i. Then,
(V ′, p′n+1, . . . , p′n+3g) specializes to a tuple (V0, pn+1, . . . , pn+3g) as in Lemma 2.5.2.
Proof. The content of the lemma is that the p′i, i = n + 1, . . . , n + 3g specialize to
distinct smooth points of the special fiber. Suppose that this is not the case: then,
after a combination of blow-ups and base-changes, the p′i specialize to distinct smooth
points on a compact-type curve Y0 with a non-trivial map c : Y0 → X0 contracting
rational tails and bridges. Moreover, Y0 is equipped with a limit linear series W0 with
ramification conditions as above at distinct smooth points p′i. As before, we have
ρ(W0, {pi}) = −3g. Let E ′j denote the unique component of Y0 mapping to Ej ⊂ X0,
and q′j ∈ E ′j denote the unique point of E ′j such that c(q′j) = qj.
We claim that if such aW0 exists, then in fact Y0 = X0. First, note that ρ(W0, {pi})R ≥
59
0 for any rational component R ⊂ Y0, by Proposition 2.2.3. Also, the elliptic compo-
nents of Y0 are general, so by Propositions 2.2.1 and 2.2.3, the ρ(W0, {pi})E′j ≥ −3 for
all j. Thus, by sub-additivity of the Brill-Noether number (2.2), equality must hold
everywhere, and moreover W0 is a refined limit.
For each j, let αj be the number of moving points on E ′j, and let βj be the number of
trees of rational curves attached to E ′j away from q′j. Then, ρ(W0, {pi})E′j = −(αj+βj).
Thus,
3g = −ρ(W0) =∑j
(αj + βj).
On the other hand, each such tree of rational curves attached to an E ′j away from q′j
contains at least two of the p′i, so we have
∑j
(αj + 2βj) ≤ 3g.
Therefore, βj = 0 for all j, from which it follows that c is an isomorphism. This
completes the proof.
Proposition 2.5.4. The answer N gd1,...,dn+3g
to Question 1 is computed in the following
way. Consider all distributions
S = ({p′1, p′2, p′3}, {p′4, p′5, p′6}, . . . , {p′3g−2, p′3g−1, p
′3g})
of the points pn+1, . . . , pn+3g onto the Ej such that each elliptic component contains
exactly three of the pi. For each Ej, containing the points r3j−2, r3j−1, r3j, consider all
possible vanishing sequences (aj, bj) such that
(aj + bj) + (d3j−2 + d3j−1 + d3j) = 2d+ 4.
60
Then, take the product
[∫Gr(2,d+1)
(g∏j=1
σd−aj−1,d−bj ·n∏i=1
σi
)]·
g∏j=1
Nbj−aj ,d3j−2−aj ,d3j−1−aj ,d3j−aj .
Finally, sum the resulting products over all choices of S, (aj, bj).
Proof. By Lemmas 2.5.2 and 2.5.3, N gd1,...,dn+3g
is the equal to the number of number of
(V0, pn+1, . . . , pn+3g) as described in Lemma 2.5.2(a). To enumerate such limit linear
series, we consider all possible S as above, then all possible combinations of vanishing
sequences (aj, bj) at the nodes qj ∈ Ej. Then, as V0 is a refined series, the vanishing
sequence at rj ∈ P1 must be (d− bj, d− aj). After twisting away base-points at the qj,
the terms in the product then count the number of linear series on the components of
X0, by Theorems 2.1.1 and 2.1.3.
Proof of Thoerem 2.1.5. Immediate from Proposition 2.5.4.
2.5.2 Weighted counts via degeneration
Simplifying the degeneration formula of Proposition 2.5.4 seems to be a difficult com-
binatorial problem. It seems natural to guess that in higher genus, weighted counts of
pencils are better behaved than unweighted counts of branched covers. We remark here
that in this setting, one gets a degeneration formula for the weighted number of pencils
N gd1,...,dn+3g
on C by replacingNbj−aj ,d3j−2−aj ,d3j−1−aj ,d3j−aj with N◦bj−aj ,d3j−2−aj ,d3j−1−aj ,d3j−aj
in Proposition 2.5.4, see Proposition 2.3.17.
We make one final observation, that in the weighted setting, it suffices to consider
the case n = 1, that is, the case in which there is only one fixed ramification condition.
Proposition 2.5.5. Adopt the notation of Question 1 and condition (2.1). Then,
the weighted number of tuples (V, pn+1, . . . , pn+m) is equal to the same weighted count
when p1, . . . , pn are replaced by a single general point p1 ∈ C, at which we impose the
61
condition of total vanishing at least (d1 + · · ·+ dn)− n+ 1.
Proof. We degenerate C to the nodal curve C0∼= C ∪ P1 so that that the pi specialize
to general points on P1, and count limit linear series on C0. The details are left to the
reader.
62
2.A Brill-Noether curves inM1,4 via admissible cov-
ers and Hurwitz numbers
Loci of pointed curves admitting special linear series with ramification conditions im-
posed at the marked points provide important examples of cycles on moduli spaces of
curves, especially from the point of view of birational geometry, see the discussion of
§1.3. Here, we consider the example arising from the main computation of this chapter.
Fix integers d, d1, d2, d3, d4 as in §2.3, so that 2 ≤ di ≤ d and d1+d2+d3+d4 = 2d+4.
Let Admd1,d2,d3,d41/0,d be the stack parametrizing the following data:
• A stable curve (Y, y1, y2, y3, y4) ∈M0,4,
• A connected nodal curve X of arithmetic genus 1,
• Distinct smooth points x1, x2, x3, x4 ∈ X, and
• An admissible cover (the reader may refer to §3.2.3 for definitions) f : X → Y
with f(xi) = yi of degree d ramified at xi to order di.
We have a map πd1,d2,d3,d41/0,d : Admd1,d2,d3,d41/0,d → M1,4 remembering the marked nodal
curve (X, x1, x2, x3, x4), possibly after contracting non-stable components. We thus
get a cycle [(πd1,d2,d3,d41/0,d )∗(1)] ∈ A1(M1,4), which we denote [πd1,d2,d3,d41/0,d ] for short.
To compute the (rational) Chow class of [πd1,d2,d3,d41/0,d ], it suffices to compute its
intersection with boundary divisors of M1,4, by the results of [Bel98]. We have the
following natural maps:
• ξ0 : M0,6 → M1,4, where the map glues together the last two marked points.
The image of ∆0 is the closure of the locus of irreducible curves.
• ξ1,S : M1,5−|S| × M0,|S|+1 → M1,4 for S ⊂ {1, 2, 3, 4}, where the map glues
together one point each of the components coming from each factor. The image
63
of ∆1 is the closure of the locus of reducible curves formed by the union elliptic
curve and a copy of P1, where the marked points corresponding to the elements
of S lie on the rational component. We require |S| ≥ 2.
Let ∆0,∆1,S denote the classes of ξ0, ξ1,S in A1(M1,4). By [AC87], these form a
basis of A1(M1,4) = Pic(M1,4):
We will express the intersection of [πd1,d2,d3,d41/0,d ] and the boundary divisors in terms
of certain 3-point Hurwitz numbers. Fix integers d ≥ 1 and g ≥ 0, and let λ1, λ2, λ3
be ordered partitions of d with a total of d− 2g + 2 parts. Then, let Hdg (λ1, λ2, λ3) be
the number, weighted by automorphisms, of degree d covers f : C → P1, where C is a
connected curve of genus g, f is unramified over P1 − {0,∞, 1}, and the ramification
profiles above 0,∞, 1 are λ1, λ2, λ3, respectively. For ease of notation, we will drop the
superscript d, as well as some components equal to 1 from the partitions λi.
Let Hd
g(λ1, λ2, λ3) be the pointed Hurwitz number counting covers f : C → P1 as
before, but where the point of C corresponding to the first component of each λi is
marked.
Lemma 2.A.1. Suppose d1, d2, d3 satisfy d1 + d2 + d3 = 2d+ 1 and 2 ≤ di ≤ d. Then,
we have H0((d1), (d2), (d3)) = 1.
Proof. This amounts to the fact that
∫Gr(2,d+1)
σd1−1σd2−1σd3−1 = 1,
which is straightforward to check using the Pieri Rule.
Proposition 2.A.2. We have:
∫M1,4
[πd1,d2,d3,d41/0,d ] ·∆1,S =
0 if |S| > 2∑u
H1((di), (dj), (u)) if {1, 2, 3, 4} − S = {i, j}
64
In the second case, if S = {k, `}, then the sum is over integers u satisfying the following
conditions:
• u ≤ di + dj − 3, dk + d` − 1
• u ≥ |di − dj|+ 3, |dk − d`|+ 1
• u ≡ di + dj + 1 (mod 2)
Proof. If f : X → Y is an admissible cover in the intersection of πd1,d2,d3,d41/0,d and any
boundary divisor, then Y must be singular, consisting of two rational components
Y ′, Y ′′ with two marked points each, meeting at y ∈ Y . If X contains a component
X1 ⊂ X of genus 1, then X1 must be smooth; assume without loss of generality that
X1 maps to Y ′. It is straightforward to check that X1 is ramified at three points over
Y ′, to orders di, dj over the marked points of Y ′ and to order u over y. Then, X1 must
be attached to a component X0 ⊂ X mapping to Y ′′ and ramified at three points, to
orders dk, d` over the marked points and to order u over y. The rest of the components
of X are rational, and map isomorphically to one of the components of the target.
The first part of the Proposition, where |S| > 2, is now clear. For the second
part, the conditions above on u correspond to constraints on the degrees of X1 over
Y ′ and X0 over Y ′′: they must be integers, and greater than or equal to the specified
ramification indices. It is easy to check that πd1,d2,d3,d41/0,d and ∆1,S intersect transversely
at f : X → Y : indeed, [f : X → Y ] is a smooth point of Admd1,d2,d3,d41/0,d whose non-
trivial tangent direction corresponds to the smoothing of the node at which X0 and
X1 meet, see [HM82] or Proposition 3.2.4. Finally, there is a unique cover X0 → Y ′′
satisfying the needed properties, by Lemma 2.A.1. The Proposition now follows.
Remark 2.A.3. Liu [Liu06] has announced an explicit formula for H1((d1), (d2), (d3)),
but to our knowledge a proof has not appeared in the literature.
65
Proposition 2.A.4. We have:
∫M1,4
[πd1,d2,d3,d41/0,d ] ·∆0 = Nd1,d2,d3,d4
= 2
(4∑j=2
(∑u,v
v ·H0((d1), (dj), (u, v)) ·H0((dk), (d`), (u, v))
))
Above, k, ` are chosen (without regard to order) so that {j, k, l} = {2, 3, 4}. The sum
is over integers u, v satisfying the following conditions:
• u+ v ≤ d1 + dj − 2, dk + d` − 2
• u+ v ≥ |d1 − dj|+ 2, |dk − d`|+ 2
• u+ v ≡ d1 + dj (mod 2)
Proof. The first part, that the intersection number in question is equal to Nd1,d2,d3,d4 ,
follows from the fact that we may replace ∆0 with the locus inM1,4 of pointed curves
with fixed underlying elliptic curve, that is, a general geometric fiber of the forgetful
morphism g :M1,4 →M1,1.1
Now, the intersection in question consists of admissible covers f : X → Y con-
structed in the following way. The target curve Y must be as in Proposition 2.A.2.
X must then contain two rational components X ′, X ′′ mapping to Y ′, Y ′′, respectively,
attached at two points over y. Let u, v denote the ramification indices of f at these two
points, both of which are marked. We take the first of these to correspond to the node
formed in the gluing map ξ0 : M0,6 →M1,4. Away from y, the components X ′, X ′′ are
ramified over the marked points of Y to the required orders, and all other components
of X are rational, mapping isomorphically to the target.
The numbers of covers X ′ → Y ′, X ′′ → Y ′′ are given by the marked Hurwitz
numbers in the Proposition; we sum over the three possible marked targets Y . We get
1The pullback of g by the geometric point SpecC →M1,1 corrresponding to the boundary pointis birational to, but not isomorphic to, M0,6.
66
an additional factor of 2 coming from the ways to distribute the fifth and sixth marked
points on X ′ and X ′′ after normalizing the first marked node. Finally, the intersection
multiplicity of ∆0 and πd1,d2,d3,d41/0,d at [f ] is equal to v. Indeed the complete local ring at
[f ] of Admd1,d2,d3,d41/0,d is isomorphic to C[[x1, x2]]/(xu1 = xv2) (see [HM82] or Proposition
3.2.4), and the map ξ0 : M0,6 →M1,4 kills x1 on the level of complete local rings.
Finally, the conditions on u+ v are tantamount to the fact that the degrees of the
covers X ′ → Y ′, X ′′ → Y ′′ must be integers, and at least equal to the ramification
indices over the marked points, and the sum u+ v over y. Combining all of the above,
we get the conclusion.
Remark 2.A.5. Proposition 2.A.4 gives a new formula for Nd1,d2,d3,d4 , which, while
not explicit, is non-negative, that is, it is the sum of visibly non-negative terms. A
comparison to the formulas of Theorem 2.1.3 is in order; we will pursue this in future
work.
2.B Positivity of enumerative counts
In this section, we show that Question 1 always has a non-zero answer.
Proposition 2.B.1. Let d, d1 . . . , dn be integers satisfying 2 ≤ di ≤ d and∑
i(di−1) =
2d − 2. Let x1, . . . , xn ∈ P1 be general points. Then, there are a non-zero number of
maps f : P1 → P1 ramified to order di at xi.
Proof. This is a special case of [EH83, Theorem 2.3], that the space of linear series
on P1 with imposed ramification conditions at fixed general marked points has the
expected dimension, and in particular is non-empty when the expected dimension is
non-negative. Alternatively, one can apply the Pieri Rule directly to Theorem 2.1.1.
Proposition 2.B.2. Fix integers d, d1, d2, d3, d4, so that 2 ≤ di ≤ d and d1 + d2 + d3 +
d4 = 2d+ 4. Then, we have Nd1,d2,d3,d4 > 0.
67
Proof. It suffices to find one positive term in the formula of Theorem 2.A.4. Without
loss of generality, assume that d1 ≥ d2 ≥ d3 ≥ d4. By Theorem 2.1.4, we may
additionally assume that d1 − d2 ≥ d3 − d4. We will take j = 2 and v = 1. Then, we
need u to satisfy
d1 − d2 + 1 ≤ u ≤ d3 + d4 − 3,
and
u+ 1 ≡ d1 + d2 (mod 2).
Such a u must exist, because
d3 + d4 − 3− (d1 − d2 + 1) = 2d− 2d1 ≥ 1,
unless d1 = d, in which case we may take u = d1 − d2 + 1.
To finish, it follows from Lemma 2.A.1 that
H0((d1), (d2), (u, 1)) = H0((d3), (d4), (u, 1)) = 1,
so we have found a positive contribution to Nd1,d2,d3,d4 .
Theorem 2.B.3. Suppose g is arbitrary. Then, the answer to Question 1 is non-zero,
provided we have (2.1).
Proof. We may assume that m = 3g. We proceed by induction on g; Propositions
2.B.1 and 2.B.2 give the cases g = 0, 1.
Let C0 = D1∪Dg−1 be a reducible curve formed by attaching general smooth curves
Di of genus i at a node p. Let p1, . . . , pn be general marked points on Dg−1. By a similar
argument as in §2.5, it suffices to show that on (C0, p1, . . . , pn), there are a non-zero
number of limit linear series V with the desired moving ramification conditions. We
68
may impose ramification conditions at the moving points pn+1, . . . , pn+3g−3 ∈ Dg−1 and
pn+3g−2, pn+3g−1, pn+3g ∈ D1.
We will show that, for some choice of vanishing sequences at p, there exists a (fine)
limit linear series V with the needed ramification properties. Let (a, b) be the vanishing
sequence at p on the D1-aspect of V : we need 0 ≤ a < b ≤ d and
a+ b+ dn+3g−2 + dn+3g−1 + dn+3g = 2d+ 4.
By the g = 1 case, there exists a D1-aspect with the needed ramification properties if
and only if we have the inequalities
2 ≤ b− a
dn+3g−2, dn+3g−1, dn+3g ≥ d− a,
as we must twist away the order a base-point at p to obtain a base-point-free pencil as
counted in in the g = 1 case.
The vanishing sequence at p of the Dg−1-aspect of V must be (d − b, d − a), and
by the inductive hypothesis, there exists such an aspect if and only if we have the
inqualities
2 ≤ b− a
d1, d2, . . . , dn+3g−3 ≤ b.
Eliminating redundancies, we need to find (a, b) satisfying the following list of
properties:
• a+ b = 2d+ 4− (dn+3g−2 + dn+3g−1 + dn+3g)
• a < b+ 1
69
• 0 ≤ a ≤ d−max(dn+3g−2, dn+3g−1, dn+3g)
• max(d1, . . . , dn+3g−3) ≤ b ≤ d
The last three conditions define a connected region R ⊂ R2. Thus, it is enough
to check that among the lattice points in R, the required value of a + b lies in be-
tween the minimum and maximum possible. Indeed, we see that the minimum is
achieved when (a, b) = (0,max(d1, . . . , dn+3g−3)) and the maximum when (a, b) =
(d−max(dn+3g−2, dn+3g−1, dn+3g), d). Now, the inequality
max(d1, . . . , dn+3g−3) ≤ 2d+ 4− (dn+3g−2 + dn+3g−1 + dn+3g)
is immediate from (2.1), and the inequality
2d+ 4− (dn+3g−2 + dn+3g−1 + dn+3g) ≤ 2d−max(dn+3g−2, dn+3g−1, dn+3g)
is clear from the fact that di ≥ 2. This completes the proof.
Theorem 2.B.3 may be rephrased in the following way. Let H be the Hurwitz space
parametrizing branched covers f : C → P1 as enumerated in Question 1. We have a
diagram
H π //
ψ
��
Mg,n
M0,n+3g
where π, ψ are the maps remembering the source and target, respectively, of a cover.
As usual, ψ is etale, so H has the expected dimension of 3g − 3 + n. Then, Theorem
2.B.3 asserts that the map π, between spaces of the same dimension, is dominant, in
the sense that at least one component of H dominates Mg,n.
A question which seems to be much more subtle is whether every component of H
dominates Mg,n. We plan to pursue this question in future work.
70
Chapter 3
d-elliptic loci in genus 2 and 3
“Eins, zwei, drei, vier, funf, sechs...
Sieb’n, acht, neun, zehn, elf, zwolf,
Hopp, hopp, hopp, hopp, das geht
im Galopp:
Sechshundertundneun!
...Ein halbe Million, ja eine halbe
Million!
Da mag der Teufel richtig zahlen!”
Johann Strauss II/Karl
Haffner/Richard Genee, Die
Fledermaus (1874)
3.1 Introduction
Let Hg/1,d denote the moduli space of degree d covers f : C → E, where C is a smooth
curve of genus g, E is a smooth curve of genus 1, and f is simply branched at marked
points x1, . . . , x2g−2 ∈ C mapping to distinct points y1, . . . , y2g−2 ∈ E. We then have a
71
diagram
Hg/1,d
πg/1,d //
ψg/1,d
��
Mg
M1,2g−2
where the map πg/1,d remembers the source curve, and ψg/1,d remembers the target
curve with the branch points. The degree of ψg/1,d is given by a Hurwitz number,
which counts monodromy actions of π1(E − {y1, . . . , y2g−2}, y) on the d-element set
f−1(y). We have the following groundbreaking result of Dijkgraaf:
Theorem 3.1.1 ([Dij95]). For g ≥ 2, the generating series
∑d≥1
deg(ψg/1,d)qd
is a quasimodular form of weight 6g − 6.
Okounkov-Pandharipande [OP06] give a substantial generalization to the Gromov-
Witten theory of an elliptic curve with arbitrary insertions.
In this paper, we consider instead the enumerative properties of the map πg/1,d.
Here, the geometry is more subtle, as we no longer have a combinatorial model as in
Hurwitz theory. For enumerative applications, one needs to compactify the moduli
spaces involved; we pass to the Harris-Mumford stack Admg/1,d of admissible covers,
see [HM82] or §3.2.3. We have the diagram:
Admg/1,d
πg/1,d //
ψg/1,d
��
Mg
M1,2g−2
We prove:
72
Theorem 3.1.2. For g = 2, 3, we have:
∑d≥1
[(πg/1,d)∗(1)]qd ∈ Ag−1(Mg)⊗Qmod,
where Qmod is the ring of quasimodular forms.
More precisely, we have the following formulas (here σk(d) is the sum of the k-th
powers of the divisors of d):
Theorem 3.1.3. The class of (π2/1,d)∗(1) in A1(M2) is:
(2σ3(d)− 2dσ1(d)) δ0 + (4σ3(d)− 4σ1(d)) δ1.
Theorem 3.1.4. The class of (π3/1,d)∗(1) in A2(M3) is:
((−6264d2 + 6780d− 960)σ1(d) + (5592d− 5400)σ3(d) + 252σ5(d)
)λ2
+((1224d2 − 1068d+ 156)σ1(d) + (−1152d+ 840)σ3(d)
)λδ0
+((2160d2 − 696d+ 216)σ1(d) + (−1920d+ 240)σ3(d)
)λδ1
+((−54d2 + 39d− 6)σ1(d) + (51d− 30)σ3(d)
)δ2
0
+((−216d2 + 36d− 12)σ1(d) + (192d)σ3(d)
)δ0δ1
+((−216d2 − 132d+ 36)σ1(d) + (192d+ 120)σ3(d)
)δ2
1
+((216d2 − 444d+ 60)σ1(d) + (−192d+ 360)σ3(d)
)κ2.
As a check, both formulas become zero after substituting d = 1. When d = 2,
we recover the main results of Faber-Pagani [FP15]. Indeed, the morphism π2/1,2 is
generically 4-to-1, where one factor of 2 comes from the complement C → E2 to any
bielliptic map C → E1 (see [Kuh88, §2] or §3.B.2), and another comes from the labelling
of the two ramification points. Thus, our answer differs from [FP15, Proposition 2] by
a factor of 4. In genus 3, the morphism π3/1,2 is has degree 4!, coming from the ways of
73
labelling the ramification points of a bielliptic cover, and our answer differs from the
correction to [FP15, Theorem 1] given by van Zelm [vZ18a, (3.5)] by a factor of 24.
We are then led to conjecture:
Conjecture 1. The statement of Theorem 3.1.2 holds for all g ≥ 2.
In fact, our method, which we now outline, suggests a number of possible refine-
ments of Conjecture 1. For g = 2, 3, it suffices to intersect πg/1,d with test classes of
complementary dimension, all of which lie in the boundary of Mg. The test classes
may then be moved to general cycles in a boundary divisor ofMg. The intersection of
general boundary cycles with the admissible locus can be expressed in terms of contri-
butions from admissible covers of a small number of topological types, and we compute
these contributions in terms of branched cover loci in lower genus. This leads naturally
to a number of auxiliary situations in which quasimodularity phenomena also occur,
for example:
• Loci of d-elliptic curves with marked points having equal image under the d-
elliptic map, see §3.3.2
• Loci of curves covering a fixed elliptic curve, see §3.5.1 and also [OP06]
• Correspondence maps (πg/1,d)∗ ◦ ψ∗g/1,d, see §3.5.2
• Loci of d-elliptic curves with marked ramification points, see §3.5.3 and Appendix
3.A
Let us mention one consequence of Conjecture 1. Building on work of Graber-
Pandharipande [GP03], van Zelm [vZ18b] has shown that the class (πg/1,2)∗(1) ∈
H2g−2(Mg) is non-tautological for g ≥ 12, and that (πg/1,2)∗(1) ∈ H2g−2(Mg) is non-
tautological for g = 12. Assuming Conjecture 1, we have that for these values of g, the
generating functions ∑d≥1
(πg/1,d)∗(1)qd
74
in the quotients of Ag−1(Mg) and Ag−1(Mg) by their tautological subgroups are non-
zero quasimodular forms (note that the tautological part of Ag−1(Mg) is zero by Looi-
jenga’s result [Loo95]). In particular, we would get infinitely many non-tautological
classes from d-elliptic loci on Mg for g ≥ 12, and on M12.
The structure of this paper is as follows. We collect preliminaries in §3.2, recording
the needed facts about intersection theory on Mg,n and recalling the definitions of
admissible covers and quasimodular forms. In §3.3, we carry out some enumerative
calculations for branched covers that we will need when considering d-elliptic loci. In
§3.4, we prove Theorem 3.1.3 on the d-elliptic loci in genus 2; it is here where we
explain our method in the most detail. In §3.5, we establish variants of Theorem 3.1.3
suggesting possible variants of Conjecture 1. Finally, we put together all of the previous
results to prove Theorem 3.1.4 on the d-elliptic loci in genus 3 in §3.6.
We remark in Appendix 3.A that we have quasimodularity for d-elliptic loci on
M2,2, where the ramification points of a d-elliptic cover are marked. However, we
explain a new feature: not all contributions to the classes of the d-elliptic loci from
admissible covers of individual topological types are themselves quasimodular.
3.2 Preliminaries
3.2.1 Conventions
We work over C. Fiber products are over Spec(C) unless otherwise stated. All curves,
unless otherwise stated, are assumed projective and connected with only nodes as
singularities. The genus of a curve X refers to its arithmetic genus and is denoted
pa(X). A rational curve is an irreducible curve of geometric genus 0. All moduli
spaces are understood to be moduli stacks, rather than coarse spaces. In all figures,
unlabelled irreducible components of curves are rational, and all other components are
labelled with their geometric genus.
75
If X is a nodal curve, its stabilization, obtained by contracting rational tails and
bridges (that is, non-stable components), is denoted Xs. We use similar notation for
pointed nodal curves.
All Chow rings are taken with rational coefficients and are denoted A∗(X), where
X is a variety or Deligne-Mumford stack over C. When referring to Chow groups, we
use subscripts (recording the dimensions of cycles) and superscripts (recording their
codimensions) interchangeably when X is smooth. We will frequently refer to the
Chow class of a proper and generically finite morphism f : Y → X, by which we mean
f∗([Y ]). The class of f in A∗(X) is denoted [f ]. When there is no opportunity for
confusion, we sometimes refer to the same class by “the class of Y ” or [Y ] ∈ A∗(X).”
If X is proper and f : X → Spec(C) is the structure morphism, we denote the proper
pushforward map f∗ by∫X
.
We deal throughout this paper with boundary classes on moduli spaces of curves.
We will find it more convenient to carry out intersection-theoretic calculations using
classes obtained as pushforwards of fundamental classes from (products of) moduli
spaces of curves of lower genus, and label these classes using upper case Greek letters.
For example, when g ≥ 2, we denote by ∆0 ∈ A∗(Mg) the class of the morphism
Mg−1,2 → Mg that glues together the two marked points. We reserve lower-case
letters for substack classes (also known as Q-classes): for example, δ0 ∈ A∗(Mg) is the
class of the substack of curves with a non-separating node. We have
δ0 =1
2∆0.
In general, the denominator is the order of the automorphism group of the stable
graph associated to the boundary stratum, which in this case is the graph consisting
of a single vertex and a self-loop.
76
Figure 1: Boundary classes in A1(M1,2)
3.2.2 Intersection numbers on moduli spaces of curves
Here, we collect notation for various classes on moduli spaces of curves, and intersection
numbers of these classes. We will frequently abuse notation: for instance, ∆0 will
always denote the class of the locus of irreducible nodal curves on any Mg,n, but it
will be clear in context the spaces on which these classes are defined. The intersection
numbers given here can be verified using the admcycles.sage package, [DSvZ20].
3.2.2.1 M1,2
The rational Picard group A1(M1,n) is freely generated by boundary divisors, see
[AC87]. When n = 2, we have the boundary divisors ∆0, parametrizing irreducible
nodal curves, and ∆1, parametrizing reducible curves, see Figure 1. The intersection
pairing is as follows:
∆0 ∆1
∆0 0 1
∆1 1 − 124
3.2.2.2 M1,3
In A1(M1,3), we have the boundary divisor ∆0 parametrizing irreducible nodal curves,
and the boundary divisors ∆1,S, where S ⊂ {1, 2, 3}, parametrizing reducible nodal
curves with the marked points corresponding to elements of S lying on the rational
component, see Figure 2. (We require |S| ≥ 2.)
77
For the same S, let ∆01,S ∈ A2(M1,3) be the class of curves in the boundary divisor
∆1,S whose genus 1 component is nodal; if |S| = 2, let ∆11,S ∈ A2(M1,3) be the class
of curves consisting of a chain of three components, where the rational tail contains
the two marked points corresponding to the elements of S, see Figure 3.
Figure 2: Some boundary classes in A1(M2)
Figure 3: Some boundary classes in A2(M2)
We will not need the intersection numbers of the boundary divisors with all curve
classes in A2(M1,3), but we record the intersections of the boundary divisors with those
defined above:
∆0 ∆1,{1,2,3} ∆1,{2,3} ∆1,{1,3} ∆1.{1,2}
∆01,{1,2} 0 12
0 0 −12
∆01,{1,3} 0 12
0 −12
0
∆11,{1,2}12
− 124
0 0 0
∆11,{1,3}12
− 124
0 0 0
3.2.2.3 M2
Mumford has computed A∗(M2) in [Mum83]. A1(M2) is generated by the boundary
classes ∆0,∆1, parametrizing irreducible nodal curves and reducible curves, respec-
78
tively, see Figure 4. A2(M2) is generated by the boundary classes ∆00,∆01, parametriz-
ing irreducible binodal curves and reducible curves where one component is a rational
nodal curve, respectively, see Figure 5.
Figure 4: Boundary classes in A1(M2)
Figure 5: Boundary classes in A2(M2)
The intersection pairing is as follows:
∆00 ∆01
∆0 −4 1
∆1 2 − 112
3.2.2.4 M2,1
It follows from [Fab90a] that A2(M2,1) has dimension 5, with a basis given by the
boundary classes ∆00,∆01a,∆01b,Ξ1,∆11, shown in Figure 6.
The intersection pairing is as follows:
79
Figure 6: Boundary classes in A2(M2,1)
Figure 7: Some boundary classes in A1(M2,1) and A3(M2,1)
∆00 ∆01a ∆01b Ξ1 ∆11
∆00 0 0 0 −4 2
∆01a 0 1 −1 1 0
∆01b 0 −1 1 0 − 112
Ξ1 −4 1 0 112
0
∆11 2 0 − 112
0 1288
We will also need the classes ∆1 ∈ A1(M2,1) and Γ(5),Γ(6),Γ(11) ∈ A3(M2,1), shown
in Figure 7.
The subscripts in the classes Γ(i) ∈ A3(M2,1) are chosen in such a way that Γ(i) ×
M1,1 = ∆(i) ∈ A2(M3), see §3.2.2.5. We have the following intersection numbers:
Γ(5) Γ(6) Γ(11)
∆1 1 0 − 124
80
Figure 8: Boundary classes in A4(M3)
3.2.2.5 M3
Faber has computed A∗(M3) in [Fab90a]. We have that A2(M3) and A4(M3) both
have dimension 7 and pair perfectly. To describe bases of these groups, we first recall
the definitions of the λ and κ classes. Let u : Cg → Mg be the universal curve, let
ωCg/Mgbe the relative dualizing sheaf, and let K ∈ A1(Cg) denote the divisor class of
ωCg/Mg. Then, by definition:
λi = ci(u∗ωCg/Mg) ∈ Ai(Mg)
κi = u∗(Ki+1) ∈ Ai(Mg)
We will only need the class λ1 in this paper, so we write λ = λ1 in A1(M3),
with no risk of confusion. Then, a basis for A2(M3) is given by the seven classes
λ2, λδ0, λδ1, δ20, δ0δ1, δ
21, κ2.
A basis for A4(M3) is given by surface classes ∆[i] for i ∈ {1, 4, 5, 6, 8, 10, 11},
retaining the indexing from [Fab90a], see Figure 8.
We have the following intersection numbers:
81
λ2 λδ0 λδ1 δ20 δ0δ1 δ2
1 κ2
∆[1] 0 0 0 0 4 −3 1
∆[4] 0 0 0 8 −4 2 0
∆[5] 0 − 112
124
−2 712
− 112
0
∆[6] 0 0 − 124
0 −12
112
0
∆[8] 0 − 112
124
−116
12
− 124
124
∆[10] 0 0 − 124
0 −12
18
124
∆[11]1
288124
− 1288
12
− 124
1288
0
3.2.3 Admissible covers
We recall the definition of [HM82]:
Definition 3.2.1. Let X, Y be curves. Let b = (2pa(X) − 2) − d(2pa(Y ) − 2), and
let y1, . . . , yb ∈ Y be such that (Y, y1, . . . , yb) is stable. Then, an admissible cover
consists of the data of the stable marked curve (Y, y1, . . . , yb) and a finite morphism
f : X → Y such that:
• f(x) is a smooth point of Y if and only if x is a smooth point of X,
• f is simply branched over the yi and etale over the rest of the smooth locus of
Y , and
• at each node of X, the ramification indices of f restricted to the two branches
are equal.
Remark 3.2.2. It is clear that non-separating nodes of X must map to non-separating
nodes of Y . Hence, the preimage of a smooth component of Y must be a disjoint union
of smooth components of X.
Admissible covers of degree d from a genus g curve to a genus h curve are parametrized
by a proper Deligne-Mumford stack Admg/h,d, see [HM82, Moc95, ACV03]. Admg/h,d
82
contains the Hurwitz space Hg/h,d parametrizing simply branched covers of smooth
curves as a dense open substack.
Let b = 2g − 2 − d(2h − 2). Then, we have a forgetful map ψg/h,d : Admg/h,d →
Mh,b remembering the target, and another πg/h,d : Admg/h,d →Mg,b remembering the
stabilization of the source. We will often abuse notation and write πg/h,d : Admg/h,d →
Mg,r for r < b, obtained by post-composing with the mapMg,b →Mg,r forgetting the
last b− r points.
Lemma 3.2.3. The morphism ψg/h,d : Admg/h,d → Mh,b is quasifinite (and hence
finite).
Proof. Over the open locus Mh,b, this is classical: the number of points in any fiber
is given by a Hurwitz number, counting isomorphism classes of monodromy actions
of the finitely generated group π1(Y − {y1, . . . , yb}) on a general fiber of f : X → Y .
Over a general point of any boundary stratum ofMh,b parametizing admissible covers
f : X → Y , there are a finite number of possible collections of ramification profiles
above the nodes of Y , each of which leads to finitely many collections of covers of
the individual components of Y , which in turn can be glued together in finitely many
ways.
We also recall from [HM82] the explicit local description of Admg/h,d. Let [f : X →
Y ] be a point ofAdmg/h,d. Let y′1, . . . , y′n be the nodes of Y , and let y1, . . . , yb ∈ Y be the
branch points of f . Let C[[t1, . . . , t3h−3+b]] be the deformation space of (Y, y1, . . . , yb),
so that t1, . . . , tn are smoothing parameters for the nodes y′1, . . . , y′n. Let xi,1, . . . , xi,ri
be the nodes of X mapping to y′i, and denote the ramification index of f at xi,j by ai,j.
Proposition 3.2.4 ([HM82]). The complete local ring of Admg/h,d at [f ] is
C[[t1, . . . , t3h−3+b, {ti,j}1≤i≤n
1≤j≤ri
]]/(t1 = t
a1,11,1 = · · · = t
a1,r11,r1
, . . . , tn = tan,1
n,1 = · · · = tan,rnn,rn
).
83
Here, the variable ti,j is the smoothing parameter for X at xi,j. In particular,
Admg/h,d is Cohen-Macaulay of pure dimension 3h − 3 + b. Moreover, if the ai,j are
all equal to 1, that is, f is unramified over the nodes of Y (or if Y is smooth to begin
with), then Admg/h,d is smooth at [f ].
One can readily extend the theory to construct stacks of admissible covers with
arbitrary ramification profiles; we use this in §3.3.3. Even in this more general setting,
we always require the target curve, marked with branch points, to be stable.
In this paper, we primarily study the case h = 1, that is, the moduli of covers
of elliptic curves. The space Admg/1,d is reducible when d is composite, due to the
existence of covers C → E1 → E2 factoring through a non-trivial isogeny. However,
the open and closed substack Admprimg/1,d parametrizing primitive covers, that is, those
that do not factor through a non-trivial isogeny (more generally, through a non-trivial
admissible cover of genus 1 curves), is irreducible. In fact, this is already true for a
fixed elliptic target, see [GK87] or [Buj15, Theorem 1.4]. For our enumerative results,
however, the individual components of Admg/1,d play no essential role: we consider the
entire moduli space, including the components parametrizing non-primitive covers.
3.2.4 Quasimodular forms
For positive integers d, k, define
σk(d) =∑a|d
ak
84
Recall that the ring Qmod of quasimodular forms is generated over C by the Eisenstein
series
E2 = 1− 24∞∑d=1
σ1(d)qd
E4 = 1 + 240∞∑d=1
σ3(d)qd
E6 = 1− 504∞∑d=1
σ5(d)qd,
where we take q to be a formal variable. The weight of Ek is k, and Qmod is a graded
C-algebra by weight.
We have the Ramanujan identities
qdE2
dq=E2
2 − E4
12
qdE4
dq=E2E4 − E6
3
qdE6
dq=E2E6 − E2
4
2,
so in particular∞∑d=1
P (d)σ2k−1(d)qd ∈ Qmod
for any P (d) ∈ C[d]. Thus, Theorem 3.1.2 will be an immediate consequence of Theo-
rems 3.1.3 and 3.1.4.
The Ramanujan identities also give the convolution formulas
∑d1+d2=d
σ1(d1)σ1(d2) =
(−1
2d+
1
12
)σ1(d) +
5
12σ3(d)
∑d1+d2=d
d1σ1(d1)σ1(d2) =
(−1
4d2 +
1
24d
)σ1(d) +
5
24dσ3(d)
∑d1+d2+d3=d
σ1(d1)σ1(d2)σ1(d3) =
(1
8d2 − 1
16d+
1
192
)σ1(d) +
(− 5
32d+
5
96
)σ3(d) +
7
192σ5(d)
85
3.3 Auxiliary computations
In this section, we record a number of enumerative results for branched covers that we
will take as inputs in the main computation.
3.3.1 Counting isogenies
Lemma 3.3.1. Let (E, p) be an elliptic curve and d be a positive integer. Then, the
number of isomorphism classes of isogenies E → F of degree d is σ1(d). Likewise, the
number of isomorphism classes of isogenies F → E of degree d is σ1(d).
Proof. We see that these two numbers are equal by taking duals, so it suffices to count
isogenies E → F of degree d, i.e., quotients of E by a subgroup of order d, which is
the number of index d sublattices of Z2. A sublattice of Z2 is determined by a Z-basis
(a, 0), (b, c), where a, c are positive and as small as possible; b is uniquely determined
modulo a. As ac = d, the number of such sublattices is exactly σ(d).
Corollary 3.3.2. The degrees of the morphisms ψ1/1,d : Adm1/1,d →M1,1 and π1/1,d :
Adm1/1,d → M1,1 remembering the target and (contracted) source, respectively, of a
cover, are both σ1(d). Moreover, both morphisms are unramified over M1,1.
Proof. The first statement is exactly the content of Lemma 3.3.1. To see that both
morphisms are unramified over M1,1, note that the open locus H1/1,d ⊂ Adm1/1,d
parametrizing covers of smooth curves is smooth, and that the set-theoretic fibers of
both morphisms over any point of M1,1 all have the same size.
3.3.2 The 2-pointed d-elliptic locus on M1,2
Lemma 3.3.3. Let (E, p) be an elliptic curve and d a positive integer. Then, the
number of pairs (up to isomorphism) (f, q) where f : E → F is an isogeny and q 6= p
is a pre-image of the origin of F is (d− 1)σ1(d).
86
Proof. We give a bijection between the set of (f, q) and the set of pairs (G, g) where
G ⊂ E[d] is a subgroup of order d and 0 6= g ∈ G. The claim then follows from Lemma
3.3.1. In one direction, given (f, q), we take G = ker(f) and g = q. In the other, let
F = E/G and f be the quotient map, and take q = g.
LetAdm1/1,d,2 be the moduli space of triples (f, x1, x2), where f : X → Y is a degree
d cover of a marked genus 1 curve (Y, y) by a genus 1 curve X, and x1, x2 ∈ X are dis-
tinct points with f(x1) = f(x2) = y. We have a finite morphism ψ1/1,d,2 : Adm1/1,d,2 →
M1,1 remembering the target curve, and a morphism π1/1,d,2 : Adm1/1,d,2 → M1,2
remembering the stabilized source curve.
Proposition 3.3.4. Adopting the notation of §3.2.2.1, we have:
∫M1,2
[π1/1,d,2] ·∆0 = (d− 1)σ1(d),∫M1,2
[π1/1,d,2] ·∆1 = 0.
Proof. Let (E, p) be a general elliptic curve. Any two geometric points of M1,1 are
equivalent, so the boundary class ∆0 is equivalent to the class of the morphism tE :
E →M1,2 sending q 7→ (E, p, q). Then, the first statement follows from Lemma 3.3.3
provided the intersection of tE and π1/1,d,2 is transverse. This is easy to see: at an
intersection point (E, p, q), a tangent vector from E fixes (E, p) but moves q to first
order, while a tangent vector from Adm1/1,d,2 moves (E, p) to first order, owing to
Corollary 3.3.2.
The second statement follows from the fact that no (pointed) admissible cover
f : X → Y in Adm1/1,d,2 has the property that X contracts to a curve in ∆1. Indeed,
Y would need to be singular, in which case X must be a cycle of m rational curves,
each of which maps to the normalization of Y via the map x 7→ xd/m, totally ramified
at the nodes. (To see this, one can follow the method of §3.4.1.2.) The contraction of
such a curve does not lie in ∆1.
87
Corollary 3.3.5. We have
[π1/1,d,2] = (d− 1)σ1(d)
(1
24∆0 + ∆1
)
in A1(M1,2).
Proof. Immediate from §3.2.2.1.
3.3.3 Doubly totally ramified covers of P1
The following is an easy special case of Theorem 2.1.3.
Lemma 3.3.6. Let (E, x1) be a general elliptic curve and d a positive integer. Then,
the number of tuples (f, x1, x2, x3, x4), where x1, . . . , x4 ∈ E are distinct and f : E → P1
is a degree d morphism (considered up to automorphism of the target) totally ramified
at x1, x2 and simply ramified at x3, x4, is 2(d2 − 1).
Proof. The linear system defining f must be a 2-dimensional subspace W of V =
H0(E,O(d · x1)). In order for W to be totally ramified at x1, we need O(d · x1) ∼=
O(d · x2), that is, x1 ∈ E[d] − {x1}. For such an x2, there are unique (up to scaling)
sections in V vanishing to maximal order at x1, x2; thus W is uniquely determined by
the d2−1 possible choices of x2. Moreover, f will be simply branched over two distinct
points x3, x4 of P1 unless it has two simple ramification points over the same point of
P1 or a triple ramification point; however, this will only happen of E admits a degree
d cover of P1 branched over 3 points, which is impossible for E general. There are two
ways to label the simple ramification points, so the conclusion follows.
Corollary 3.3.7. Let (E, x1) be a general elliptic curve and d a positive integer. Then,
the number of tuples (f, x1, x2, x3, x4), where x1, . . . , x4 ∈ E are distinct and f : E →
P1 is a degree d morphism simply ramified at x1, x2 and totally ramified at x3, x4, is
2(d2 − 1).
88
Proof. Pullback by translation by x3 − x1 defines a bijection with the objects counted
here and those in Lemma 3.3.6.
Let Admd,d,2,21/0,d be the moduli space of degree d admissible covers f : X → Y , where
X has genus 1, Y has genus 0, and f is ramified at four points x1, x2, x3, x4 to orders
d, d, 2, 2, respectively. We consider the map πd,d,2,21/0,d : Admd,d,2,21/0,d → M1,3 sending f to
the stabilization of (X, x1, x2, x3).
Proposition 3.3.8. Adopting the notation of §3.2.2.2, we have:
∫M1,3
[πd,d,2,21/0,d ] ·∆0 = 2(d2 − 1),∫M1,3
[πd,d,2,21/0,d ] ·∆1,S = 0,
for all S ⊂ {1, 2, 3} with |S| ≥ 2.
Proof. Similarly to the proof of Proposition 3.3.4, we replace ∆0 with the class of ME,3,
as defined by the Cartesian diagram
ME,3//
��
M1,3
��
[E] //M1,1
where [E] is the geometric point corresponding to a general elliptic curve E. Then, to
check that [ME,3] and πd,d,2,21/0,d intersect transversely, note that Admd,d,2,21/0,d is unramified
over a general point ofM1,1, so a tangent vector from Admd,d,2,21/0,d will deform E to first
order, whereas a tangent vector from ME,3 will fix E and deform the marked points to
first order. The first statement then follows from Lemma 3.3.6.
On the other hand, it is straightforward to check that the image of πd,d,2,21/0,d is disjoint
from every ∆1,S: if Y is singular and [f : X → Y ] ∈ Admd,d,2,21/0,d , then X is either a
union of an elliptic curve and rational tails, all of which will be contracted, or a union
89
of smooth rational curves. The second statement follows.
Proposition 3.3.9. Let C be a general curve of genus 2. Then, up to automorphisms
of the target, there are 48(d4 − 1) tuples (f, x1, x2, x3, x4, x5, x6), where x1, . . . , x6 ∈ C
are distinct points, and f : C → P1 is a degree d morphism totally ramified at x1, x2
and simply ramified at x3, x4, x5, x6.
Proof. The fact that such an f is simplify ramified at four other points follows from a
dimension count: the dimension of the space of covers C → P1 branched over 5 points
or fewer is 2, so a general point of M2 admits no such covers.
We appeal to the degeneration technique described in §2.5: it suffices to count limit
linear series V on the reducible curve C0 formed by attaching general elliptic curves
E1, E2 to a copy of P1, where each elliptic component contains three of the xi. Of the(63
)= 20 possible distributions of the xi onto the elliptic components, there are 12 ways
for x1, x2 to lie on different components, and 8 for them to lie on the same component.
In the first case, it follows from Lemma 3.3.6 that there are 2(d2 − 1) possible aspects
of V on each Ei, and the aspect of V on P1 must be the unique pencil with vanishing
sequence (0, d) at both marked points. In the second, we have, by Corollary 3.3.7,
2(d2− 1) possible aspects on the component containing x1, x2 and 2(22− 1) = 6 on the
other, and the aspect of V on P1 must be the unique pencil with vanishing sequences
(0, 2), (d− 2, d) at the two marked points.
Thus, our answer is
12 · (2(d2 − 1))2 + 8 · 6 · 2(d2 − 1) = 48(d4 − 1),
as desired.
Remark 3.3.10. One can also recover Proposition 3.3.9 using Tarasca’s formula for
the closure of the locus of [(C, p, q)] ∈ M2 such that C admits a cover of P1 totally
ramified at p and q, see [Tar15].
90
3.4 The d-elliptic locus on M2
In this section, we prove Theorem 3.1.3. It suffices to compute the intersection of
the morphism π2/1,d : Adm2/1,d → M2 with an arbitrary curve class in A1(M2). It
is possible to simplify the computation by specializing to particular test curves and
computing these intersection numbers with these classes directly, but instead we explain
a more abstract approach that we will employ later for pointed genus 2 curves and in
genus 3.
It follows from [Mum83] that A1(M2) = 0, and hence that any curve class on
M2 comes from the boundary. In particular, any curve class may be represented as
a rational linear combination of morphisms from a smooth, connected scheme C of
dimension 1 to one of the two boundary divisors:
(∆0) C →M1,2 →M2
(∆1) C →M1,1 ×M1,1 →M2
We may furthermore take C to be general, in the sense that C intersects any
given finite collection of subvarieties of its associated boundary divisor as generically
as possible. Now, consider the intersection of such a C with the admissible locus
π2/1,d : Adm2/1,d → M2. Note that we get a stratification of Adm2/1,d by pulling
back the stratification of M1,2 by boundary strata under the finite morphism ψ2/1,d,
see Lemma 3.2.3. Then, C may be chosen to avoid the zero-dimensional strata of
Adm2/1,d.
Thus, when intersecting C with the admissible locus, we need only consider ad-
missible covers f : X → Y where Y has at most one node; in fact, because X must
be singular, Y must have exactly one node. We classify such covers in §3.4.1, then in
§3.4.2 and §3.4.3 compute the contributions of the covers of each topological type to
the intersection numbers.
91
3.4.1 Classification of Admissible Covers
Let f : X → Y be a point of Adm2/1,d where Y has exactly one single node, and the
stabilization Xs of X lies in one of the boundary divisors ∆i. We consider the cases
i = 0, 1 separately.
3.4.1.1 [Y ] ∈ ∆1
Let Yi be the component of Y of genus i, and let y = Y0 ∩ Y1. By assumption, both
Yi are smooth. The pre-image of Yi is then a union of smooth curves; we may ignore
the case where one of the components has genus 2, by the assumption on Xs. Thus,
f−1(Y1) either consists of a single genus 1 curve X1, or two disjoint elliptic curves
X1, X′1.
In both cases, the f must be unramified over y, and simply ramified over two points
of Y0. Thus, the pre-image of Y0 consists of smooth rational curves attached to the Y1
at the pre-images of y, all of which map isomorphically to Y0 except one, which has
degree 2 over Y0. In the case that f−1(Y1) has two components, the degree 2 component
must be a bridge X1 and X ′1 in order for X to be connected.
We thus get covers of two topological types, which we denote by (∆0,∆1) (Figure
9) and (∆1,∆1) (Figure 10); the coordinates are the boundary strata in which Xs and
Y lie, respectively.
Figure 9: Cover of type (∆0,∆1) Figure 10: Cover of type (∆1,∆1)
92
3.4.1.2 [Y ] ∈ ∆0
Let [f : X → Y ] ∈ Adm2/1,d be an admissible cover, where Y is an irreducible nodal
curve of genus 1. We have a diagram
XνX //
f��
X
f
��P1 νY // Y
where the maps νX , νY are normalizations. Let y0 ∈ Y denote the node, and let
y′, y′′ ∈ P1 be its pre-images under ν. Let y1, y2,∈ Y be the branch points of f ; by
abuse of notation, we also let y1, y2 ∈ P1 denote their preimages under νY . Then, f is
simply branched over y1, y2, possibly branched over y′, y′′, and unramified everywhere
else.
Let X1, . . . , Xn be the components of X, and let di be the degree of Xi over P1.
Let si be the total number of points of Xi lying over y′ and y′′. Then, the number of
points of f−1(y) is t = 12
∑si. We have pa(X) ≥ 1 − n. On the other hand, X is is
the blowup of X at t nodes, so pa(X) = 2− t ≥ 1− n, hence t ≤ n+ 1. On the other
hand, si ≥ 2 for each i, so t ≥ n.
Because t is an integer, the three possibilities for the si (up to re-indexing) are:
si = 2 for all i (type (∆0,∆0), Figure 11), s1 = 4 and si = 2 for all i ≥ 2 (type
(∆00,∆0), Figure ??), and s1 = s2 = 3 and si = 2 for all i ≥ 3. In the last case, it is
easy to check that Xs will lie in one of the zero-dimensional boundary strata of M2,
so we may disregard covers of this type.
Suppose f is a cover of type (∆0,∆0). Then, X must consist of a smooth genus
1 component X1 attached at two points to a chain of m − 1 rational curves, each of
which maps to Y0 via x 7→ xa. (Note that a ≥ 2, as X1 has degree a over P1.) The
map f |X1 : X1 → P1 is totally ramified at the two nodes on X1, and is simply ramified
at two other points on X1. We may also have m = 1, in which case X is irreducible
93
with a single node, and its normalization X = X1 maps to P1 as above.
Finally, suppose f is a cover of type (∆00,∆0). We have a single component X0 ⊂ X
with four points mapping to y0 ∈ Y ; all other components of X have two points
mapping to y0. As X0 is connected, we see that X must consist of two disjoint chains
of curves X1, . . . , Xm−1 and X ′1, . . . , X′n−1 attached at two points to X0. We allow one
or both of m,n to be equal to 1; in this case, X0 has a non-separating node. We first
assume m,n > 1, in which case all components of X have genus 0.
Each of the components of X other than X0 is unramified over P1 away from the
two nodes; thus, each Xi → P1 is of the form x 7→ xa for some a (independent of
i), branched over y′, y′′. Similarly each X ′j → P1 is of the form x 7→ xb for some b
(independent of j), branched over y′, y′′.
Now, X0 has degree a+ b over P1, and each of y′, y′′ has two points in its pre-image,
of ramification indices a and b. By Riemann-Hurwitz, there are two additional simple
ramification points on X0 mapping to y1, y2 ∈ Y .
The situation is similar when at least one of m,n is equal to 1: the chains of smooth
rational curves attached to X0 are replaced with a non-separating node on X0, and the
normalization of X0 maps to P1 as before.
Figure 11: Cover of type (∆0,∆0)
Figure 12: Cover of type (∆00,∆0)
94
3.4.2 Intersection numbers: the case [C] ∈ ∆1
Suppose that we have a general curve class C →M1,1×M1,1. In the Cartesian diagram
AC //
��
Adm2/1,d
π2/1,d��
C //M1,1 ×M1,1//M2
we wish to compute the degree of AC has a 0-cycle on M2.
Suppose d1, d2 are positive integers satisfying d1+d2 = d. We also form the diagram
Ad1,d2C//
��
Adm1/1,d1 ×∆ Adm1/1,d2//
��
M1,1
∆��
Adm1/1,d1 ×Adm1/1,d2
ψ1/1,d1×ψ1/1,d2 //
π1/1,d1×π1/1,d2��
M1,1 ×M1,1
C //M1,1 ×M1,1
where both squares are Cartesian.
Lemma 3.4.1. We have a bijection of sets
AC(Spec(C)) ∼=∐
d1+d2=d
Ad1,d2C (Spec(C))
In particular, the groupoids of geometric points of AC and Ad1,d2C are in fact sets, i.e.,
have no non-trivial automorphisms.
Proof. By §3.4.1, a geometric point of AC consists of a point x ∈ C and a cover
f : X → Y of type (∆1,∆1), along with the data of an isomorphism of Xs with
the curve corresponding to the image of x in M2. It is clear that this data has no
non-trivial automorphisms.
From f , we may associate an ordered pair of covers of the same elliptic curve, whose
degrees are integers a, b satisfying d1 +d2 = d. We thus get a geometric point of Ad1,d2C ,
95
and it is again easy to see that such points have no non-trivial automorphisms.
The construction of the inverse map is clear: we need only note that C may be
chosen to avoid the point ([E0], [E0]) ∈ M1,1 × M1,1, where E0 denotes a singular
curve of genus 1; thus, all covers in Ad1,d2C are covers of (smooth) elliptic curves.
Lemma 3.4.2. The intersection multiplicity at all points of Ad1,d2C is 1, and the inter-
section multiplicity at all points of AC is 2.
Proof. Analytically locally near a point of Ad1,d2C , the map Adm1/1,d1 ×∆ Adm1/1,d2 →
M1,1×M1,1 is the inclusion of a smooth curve in a smooth surface, by Corollary 3.3.2
and the smoothness of M1,1. Thus, C ⊂ M1,1 ×M1,1 may be chosen to intersect
Adm1/1,(d1,d2) transversely.
Now, let f : X → Y be an admissible cover of type (∆1,∆1). The complete local
ring of Adm2/1,d at [f ] is isomorphic to C[[s, t]], where t is a smoothing parameter
for the node of Y , and the quotient C[[s, t]] → C[[s]] corresponds to the universal
deformation of the elliptic component of Y . Let C[[x, y, z]] be the complete local ring
ofM2 at [Xs], where z is a smoothing parameter for the node of Xs, and the variables
x, y are deformation parameters for the two elliptic components.
Consider the induced map T : C[[x, y, z]]→ C[[s, t]]. We have the following, up to
harmless renormalizations of the coordinates:
• T (z) ≡ t2 mod (s2, st, t3). To see this, consider any 1-parameter deformation
of Y that smooths the node to first order. The corresponding deformation of
f smooths the nodes of X to first order, and the induced deformation of Xs is
obtained by contracting the rational bridge of X in the total space, introducing
an ordinary double point. It follows that the node of Xs is smoothed to order 2.
• T (x) ≡ T (y) ≡ s mod (t, s2). Indeed, varying the elliptic component of Y to
first order varies the elliptic components of X to first order.
96
The map on complete local rings at [Xs] induced by C 7→ M1,1×M1,1 →M2 is of
the form C[[x, y, z]] 7→ C[[x, y]] 7→ C[[u]], where x, y map to power series with generic
linear leading terms. It is straightforward to check that the complete local ring of AC
at ([f ], [Xs]) is isomorphic to C[t]/(t2).
Proposition 3.4.3. The degree of AC as a 0-cycle on M2 is:
2
( ∑d1+d2=d
σ1(d1)σ1(d2)
)∫M1,1×M1,1
[C] · [∆]
where [∆] = [p ×M1,1] + [M1,1 × p] is the class of the diagonal in M1,1 ×M1,1, and
p is the class of a geometric point in M1,1.
Proof. By the previous two lemmas, it suffices to show that the degree of Ad1,d2C is:
σ1(d1)σ1(d2)
∫M1,1×M1,1
[C] · [∆]
Let E be any elliptic curve. Writing [∆] = [p×M1,1]+[M1,1×p], where we take p to be
the class of [E] ∈M1,1, the contribution of the first summand toAdm1/1,d1×∆Adm1/1,d2
is Adm1/1,d1(E) × Adm1/1,d2 , where the first factor is the moduli space of degree d1
covers of the fixed curve E. By §3.3.1, this class pushes forward to σ(d1)σ(d2)[p×M1,1]
on M1,1 ×M1,1. Similarly, the second summand gives the class σ(d1)σ(d2)[M1,1 × p],
so adding these contributions and intersecting with C yields the result.
3.4.3 Intersection numbers: the case [C] ∈ ∆0
Given C →M1,2 general, we wish to compute the degree of AC , as defined below:
AC //
��
Adm2/1,d
π2/1,d��
C //M1,2//M2
97
By §3.4.1, we have three topological types of covers contributing to AC : type
(∆0,∆1), type (∆0,∆0), and type (∆00,∆0). We now consider each contribution sep-
arately.
3.4.3.1 Contribution from type (∆0,∆1)
Consider the Cartesian diagram
A(∆0,∆1)C
//
��
Adm1/1,d,2
π1/1,d,2
��
C //M1,2
where Adm1/1,d,2 and its forgetful map to π1/1,d,2 : Adm1/1,d,2 → M1,2 are as defined
in §3.3.2.
Proposition 3.4.4. The contribution to AC from covers of type (∆0,∆1) is:
2
∫M1,2
[C] · [π1/1,d,2].
Proof. It is easy to check that A(∆0,∆1)C (Spec(C)) is a set, and is isomorphic to the
subgroupoid of AC(Spec(C)) consisting of covers of type (∆0,∆1). If C is general,
it intersects π1/1,d,2 transversely (see, for example, the proof of Proposition 3.3.4) on
M1,2. On the other hand, an argument analogous to the proof of Lemma 3.4.2 shows
that a general C intersects π2/1,d with multiplicity 2 at any cover f : X → Y of type
(∆0,∆1), due to the contraction of the rational bridge of X after applying π2/1,d.
98
3.4.3.2 Contribution from type (∆0,∆0)
Fix positive integers a,m satisfying am = d. Consider the Cartesian diagram
A(∆0,∆0),aC
//
��
Adma,a,2,21/0,a
πa,a,2,21/0,a
��
C //M1,2
where Adma,a,2,21/0,a is as defined in §3.3.3, and the map πa,a,2,21/0,a : Adma,a,2,2
1/0,a → M1,2
remembers the (stabilized) source curve with the two total ramification points. Thus,
the points of A(∆0,∆0),aC record the main data of a cover of type (∆0,∆0), namely the
restriction to the genus 1 component, which is a cover of P1 totally ramified at two
points. For a general C, all points of A(∆0,∆0),aC correspond to covers of smooth curves.
Proposition 3.4.5. The contribution to AC from covers of type (∆0,∆0) is:
∑am=d
(m
∫M1,2
[C] · [πa,a,2,21/0,a ]
).
Proof. It is easy to check that the geometric points of AC(Spec(C)) are in bijection
with the geometric points of A(∆0,∆0),aC , where a ranges over the positive integer factors
of d. However, a cover f : X → Y of type (∆0,∆0) has automorphism group of order
am−1, as the group of a-th roots of unity acts on each rational component of X. On
the other hand, each A(∆0,∆0),aC (Spec(C)) is a set. If C is general, it intersects πa,a,2,21/0,a
transversely (see, for example, the proof of Proposition 3.3.8).
It now suffices to show that the intersection multiplicity (on the level of complete
local rings) of C with π2/1,d at a cover f : X → Y of type (∆0,∆0) is mam−1; af-
ter dividing by the order of the automorphism group, we get the factor of m in the
statement, and the conclusion follows.
99
By Proposition 3.2.4, the complete local ring at [f ] is
C[[s, t, t1, . . . , tm]]/(t = ta1 = · · · = tam),
which is canonically a C[[s, t]]-algebra via ψ2/1,d : Adm2/1,d → M1,2. Here, t is a
smoothing parameter for the node of Y and ti is a smoothing parameter for the node
xi ∈ X. The quotient C[[s]] corresponds to the deformation of the target that moves
the marked points y1, y2 ∈ Y apart.
Let C[[x, y, z]] be the complete local ring ofM2 at [Xs], where the coordinates are
chosen as follows. The coordinate z is the smoothing parameter for the node, so that
C[[x, y]] is the deformation space of the marked normalization (X1, x1, xm) of Xs (the
points x1, xm are the nodes along X1 ⊂ X). Then, y is the deformation parameter of
the elliptic curve (X1, x1), and the quotient C[[x]] corresponds to the deformation of
X moving x1 and xm apart.
Consider the induced map on complete local rings
T : C[[x, y, z]]→ C[[s, t, t1, . . . , tm]]/(t = ta1 = · · · = tam),
We have the following, up to harmless renormalizations of the coordinates:
• T (z) ≡ tm1 mod (s, t1 − t2, . . . , t1 − tn). Indeed, in the 1-parameter deformation
of f smoothing the nodes of X to first order, consider the total space of the
associated deformation of X. Contracting the rational components of X produces
an Am-singularity, so the node of Xs is smoothed to order m in its induced
deformation.
• T (z) ≡ 0 mod (ti), as a deformation that keeps the node xi in X also keeps the
node in Xs.
• T (y) ≡ s mod (t1, . . . , tn). The content here is that first-order deformation of [f ]
100
that moves y1.y2 apart varies the elliptic curve (X1, x1) to first order. Indeed, the
map πa,a,2,21/0,a is unramified over a general point of of M0,4, and ψa,a,2,21/0,a is unramified
over M1,1, so we have the claim for C general.
The first two claims imply that T (z) = (t1 · · · tm)u, where u is a unit. Then, it is
straightforward to check that the complete local ring of AC at ([f ], [Xs]) is
C[t, t1, . . . , tm]/(t− tai , t1 · · · tm),
which has dimension mam−1 as a C-vector space (for example, a basis is given by
monomials tete11 · · · temm , where 0 ≤ e ≤ m− 1 and 0 ≤ ei ≤ a− 1). This completes the
proof.
3.4.3.3 Contribution from type (∆00,∆0)
Lemma 3.4.6. Let x1, x2, x3, x4 ∈ P1 be four distinct points, and let a, b ≥ 1 be
integers. Then, up to scaling on the target, there is a unique cover g : P1 → P1 of
degree a + b such that f has zeroes of orders a, b at x1, x2, respectively, and poles of
orders a, b at x3, x4, respectively. If the xi are general, then g is simply ramified over
two other distinct points.
Proof. The unique such map is the meromorphic function
g(x) =(x− x1)a(x− x2)b
(x− x3)a(x− x4)b.
The second half of the statement follows from Riemann-Hurwitz and a dimension count,
as there are finitely many covers of P1 branched over 3 points.
101
Consider the Cartesian diagram
A(∆00,∆0)C
//
��
M0,4
��
C //M1,2
where the map M0,4 → M1,2 glues the third and fourth marked points; its class is
∆0 ∈ A1(M1,2). For generic C, the points of A(∆00,∆0)C correspond to the points on C
whose images in M1,2 are irreducible nodal curves.
Proposition 3.4.7. The contribution to AC from covers of type (∆00,∆0) is:
2
( ∑d1+d2=d
σ1(d1)σ1(d2)
)∫M1,2
[C] ·∆0
Proof. It is clear that A(∆00,∆0)C (Spec(C)) is a set. Given positive integers a, b,m, n
satisfying am+ bn = d, a geometric point of A(∆00,∆0)C gives rise to a unique cover g as
in Lemma 3.4.6, which, for general C, will be simply ramified over two points distinct
from each other and from 0,∞. Then, g gives rise to two admissible covers of type
(∆00,∆0), distinguished by the labelling of the two simple ramification points; all such
covers are obtained (uniquely) in this way.
Each cover f : X → Y of type (∆00,∆0) has automorphism group of order am−1bn−1,
coming from the actions of roots of unity on the rational components of X. Now,
consider the complete local rings of AC at ([f ], Xs).
As in the proof of Proposition 3.4.5, consider the map on complete local rings
T : C[[x, y, z]]→ C[[s, t, t1, . . . , tm, u1, . . . , un]]/(t− tai , t− ubj),
induced by π2/1,d. We take y to be the smoothing parameter of the node obtained from
the gluing map M1,2 → M2, z to be that coming from the map M0,4 → M1,2, and
102
x such that the quotient C[[x]] corresponds to the deformation of Xs along M0,4. We
verify that, up to renormalizing coordinates,
• T (x) = s mod (t1, . . . , tm, u1, . . . , un),
• T (y) = (t1 . . . tm)v, and
• T (z) = (u1 . . . un)v′,
where v, v′ are units. The complete local ring of A(∆00,∆0)C at [f ] is thus
C[t, t1, . . . , tm, u1, . . . , un]/(t− tai , t− ubj, u1 · · ·un),
which has a C-basis given by monomials te(te11 · · · tem−1
m−1 )(uf11 · · ·ufnn ), where 0 ≤ e ≤
m− 1, 0 ≤ ei ≤ a− 1, and 0 ≤ fj ≤ b− 1. The total contribution to AC of each f is
therefore (mam−1bn)/(am−1bn−1) = mb.
Summing over all a,m, b, n, we obtain that each point of intersection of C and ∆0
contributes
2
( ∑am+bn=d
mb
)= 2
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
to AC .
3.4.4 The class of the admissible locus
We are now ready to compute the class of the d-elliptic locus in genus 2, that of
π2/1,d : Adm2/1,d →M2.
Proposition 3.4.8. We have:
∫M2
[π2/1,d] ·∆00 = 4(d− 1)σ1(d)∫M2
[π2/1,d] ·∆01 = 2
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
103
Proof. Note first that the formulas of Propositions 3.4.3, 3.4.4, 3.4.5, and 3.4.7 hold
for any curve class defined on the relevant boundary divisor, as such a class may be
written as a linear combination of general curves, and the formulas are all linear in [C].
We first take ∆01 as the pushforward of [p×M1,1] ∈ A1(M1,1×M1,1) toM2. We
have∫M1,1×M1,1
∆01 · [∆] = 1, so the formula for∫M2
[π2/1,d] ·∆01 follows immediately
from Proposition 3.4.3.
As a check, ∆01 may also be expressed as the pushforward of ∆1 ∈ A1(M1,2)
to M2, so we can apply Propositions 3.4.4, 3.4.5, and 3.4.7. Combining these with
Proposition 3.3.4, Proposition 3.3.8, and §3.2.2.1, respectively, we find that the first
two contributions are zero, and the third is
2
( ∑d1+d2=d
σ1(d1)σ1(d2)
),
so we obtain the same result.
Finally, ∆00 is the pushforward of ∆0 ∈ A1(M1,2) to M2. Applying the same
formulas as above, we get a contribution of 2(d−1)σ1(d) in type (∆0,∆1), a contribution
of ∑am=d
2(a2 − 1)m = 2(d− 1)σ1(d)
in type (∆0,∆0) (where we have applied the projection formula for the forgetful mor-
phismM1,3 →M1,2, under which ∆0 pulls back to ∆0), and zero in type (∆00,∆0).
Proof of Theorem 3.1.3. Immediate from of §3.2.2.3 and the fact that δi = 12∆i for
i = 0, 1, along with the identity
∑d1+d2=d
σ1(d1)σ1(d2) =
(−1
2d+
1
12
)σ1(d) +
5
12σ3(d),
see §3.2.4.
104
3.5 Variants in genus 2
3.5.1 Covers of a fixed elliptic curve
Fix a general elliptic curve E; we consider genus 2 curves covering E. Define the space
of such covers Adm2/1,d(E) by the Cartesian diagram
Adm2/1,d(E) //
��
Adm2/1,d
��
[E] //M1,1,
where the map Adm2/1,d →M1,1 is the composition of ψ2/1,d : Adm2/1,d →M1,2 with
the map forgetting the second marked point.
By post-composing with π2/1,d, we get a map π2/1,d(E) : Adm2/1,d(E) → M2; we
wish to compute its class in A2(M2). We do so by intersecting with the boundary
divisors ∆0 and ∆1. If f : X → Y is an admissible cover appearing in one of these
intersections, then Y = E∪P1, where both components are attached at a node and both
marked points are on the rational component, and f is of type (∆0,∆1) or (∆1,∆1).
Proposition 3.5.1. We have
∫M2
π2/1,d(E) ·∆0 = (d− 1)σ1(d)∫M2
π2/1,d(E) ·∆1 = 2
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
Proof. The points of intersection of π2/1,d(E) and the gluing morphism M1,2 → M2
consist of admissible covers of type (∆0,∆1) with target Y , which correspond to iso-
genies E ′ → E with a second marked point in the kernel. However, note that each
intersection is counted twice in this way, as (E ′, 0, x) and (E ′, 0,−x) correspond to
the same point of M1,2 (if x is 2-torsion, then multiplication by −1 defines an auto-
105
morphism of the cover of order 2). As in Proposition 3.4.4, each cover appears with
multiplicity 2, so the first claim follows.
The points of intersection of π2/1,d(E) and the gluing morphismM1,1×M1,1 →M2
consist of admissible covers of type (∆1,∆1) with target Y , which correspond to ordered
pairs of isogenies E1 → E,E2 → E. As in Lemma 3.4.2, each such admissible cover
appears with multiplicity 2, and it is easy to check that the geometric points have no
non-trivial automorphisms. We are now done by Lemma 3.3.1.
Using the fact that δ00 = 18∆00 and δ01 = 1
2∆01, and applying again the identity
∑d1+d2=d
σ1(d1)σ1(d2) =
(−1
2d+
1
12
)σ1(d) +
5
12σ3(d)
we conclude:
Theorem 3.5.2. The class of π2/1,d(E) in A2(M2) is:
((−22
5d+
2
5
)σ1(d) + 4σ3(d)
)δ00 +
((−12
5d− 8
5
)σ1(d) + 4σ3(d)
)δ01.
3.5.2 Interlude: quasimodularity for correspondences
Theorem 3.5.3. Consider the correspondence
(π2/1,d)∗ ◦ ψ∗2/1,d : A∗(M1,2)→ A∗(M2).
Then, for a fixed α ∈ A∗(M1,2), we have
(π2/1,d)∗ ◦ ψ∗2/1,d(α) ∈ A∗(M2)⊗Qmod .
Proof. It suffices to check the claim on a basis of A∗(M1,2). Note that all classes of ge-
ometric points onM1,2 orM2 are rationally equivalent, as both spaces are unirational.
106
When α is the class of a point, we have Theorem 3.1.1. When α is the fundamental
class, the claim follows from Theorem 3.1.2 (in genus 2). When α = ∆0, we may
replace the locus of covers of a nodal curve of genus 1 with that of covers of a fixed
smooth curve, in which case we are done by Theorem 3.5.2.
It remains to consider α = ∆1. Consider the intersection of a general divisor
D → M2 with the cycle (π2/1,d)∗ ◦ ψ∗2/1,d∆1. The contribution from covers of type
(∆0,∆1) to the intersection of D with the admissible locus is the intersection of D with
the pushforward of π1/1,d,2 to M2, which is quasimodular by Proposition 3.3.4. The
contribution from type (∆1,∆1) is the intersection of D with Adm1/1,d1 ×∆Adm1/1,d2 ,
where d1, d2 range over integers satisfying d1 + d2 = d; this is also quasimodular by
Corollary 3.3.2.
3.5.3 The d-elliptic locus on M2,1
Here, we compute the class in A2(M2,1) of the morphism π2/1,d : Adm2/1,d → M2,1,
whose image is the closure of the locus of pointed curves (C, p) admitting a degree d
cover of an elliptic curve, ramified at p. We do so by intersecting with test surfaces,
following the same method as in §3.4. Because A2(M2,1) = 0 (see §3.2.2.4), it suffices
to consider test surfaces in the boundary of M2,1, which is the union of the boundary
divisors
(∆0) M1,3 →M2,1
(∆1) M1,2 ×M1,1 →M2,1
Here, the map M1,3 →M2,1 glues together the second and third marked points, and
the mapM1,2×M1,1 →M2,1 glues the second marked point on the first component to
the marked point of the second. As in the unpointed case, a general surface S mapping
to one of these boundary classes will intersect the admissible locus at covers of one of
the four types described in §3.4.1.
107
3.5.3.1 The case [S] ∈ ∆1
All admissible covers f : X → Y in the intersection of S → M1,2 ×M1,1 → M2,1
and π2/1,d : Adm2/1,d → M2,1 have type (∆1,∆1). Let s : M1,1 → M1,2 be the map
attaching a 2-pointed rational curve to an elliptic curve at its origin. Then, the 1-
pointed curve X is obtained by gluing a point of M1,2 in the image of s, at its first
marked point, to a point of M1,1, at its origin.
For integers d1, d2 satisfying d1 + d2 = d, we have a diagram
Ad1,d2S//
��
Adm1/1,d1 ×∆ Adm1/1,d2//
��
M1,1
∆��
Adm1/1,d1 ×Adm1/1,d2
π1/1,d1×π1/1,d2 //M1,1 ×M1,1
s×id��
S //M1,2 ×M1,1
where both squares are Cartesian. Taking the union over all possible (d1, d2), the
geometric points of Ad1,d2S are in bijection with those of the intersection of Adm2/1,d
and S, and there are no non-trivial automorphisms on either side. In the map π2/1,d :
Adm2/1,d → M2,1, the rational bridge of X in a cover f : X → Y of type (∆1,∆1)
does not get contracted, so in fact the argument of Lemma 3.4.2 shows that both
intersections are transverse for general S.
Using the fact that ∆ = [p ×M1,1] + [M1,1 × p] and applying Corollary 3.3.2, we
find, as in Proposition 3.4.3:
Proposition 3.5.4. For S →M1,2 ×M1,1, we have:
∫M2,1
[S] · [π2/1,d] =
(∫M1,2×M1,1
([p×M1,1] + [∆1 × p]) · [S]
)·∑
d1+d2=d
σ1(d1)σ1(d2).
108
3.5.3.2 The case [S] ∈ ∆0.
In the intersection
AS //
��
Adm2/1,d
π2/1,d��
S //M2,1
we have contributions from covers of types (∆0,∆1), (∆0,∆0), and (∆00,∆0).
First, consider a cover f : X → Y of type (∆0,∆1). The 1-pointed curve X may
be obtained by identifying the points x2, x3 of [(X1, x1, x2, x3)] ∈ M1,3 lying in either
of the boundary divisors ∆1,{1,2},∆1,{1,3}. Let s{1,2}, s{1,3} :M1,2 →M1,3 be the maps
defining these two boundary divisors, we have the Cartesian diagrams
A(∆1,∆0),{1,2}S
//
��
Adm1/1,d,2
π1/1,d,2
��
M1,2
s{1,2}��
S //M1,3
A(∆1,∆0),{1,3}S
//
��
Adm1/1,d,2
π1/1,d,2
��
M1,2
s{1,3}��
S //M1,3
Following the proof of Proposition 3.4.4, both intersections above are seen to be trans-
verse (note that we no longer contract the rational bridge of X) and have no non-trivial
automorphisms. We conclude:
Proposition 3.5.5. The contribution to AS from covers of type (∆0,∆1) is:
∫M1,3
[S] ·([s{1,2} ◦ π1/1,d,2] + [s{1,3} ◦ π1/1,d,2]
)The analysis of covers of type (∆0,∆0) is essentially identical to that of Proposition
3.4.5. We find:
109
Proposition 3.5.6. The contribution to AS from covers of type (∆0,∆0) is:
∑am=d
(m
∫M1,3
[S] · [πa,a,2,21/0,a ]
).
Finally, consider covers of type (∆00,∆0). Fix a, b,m, n with am + bn = d, and
let Adm(a,b),(a,b),2,20/0,a+b be the space of tuples (f, x1, . . . , x6), where f : X → Y is a de-
gree a + b admissible cover of genus 0 curves, with six marked points x1, . . . , x6 ∈ X
such that f(x1) = f(x2), f(x3) = f(x4), and the ramification indices at x1, . . . , x6
are a, b, a, b, 2, 2, respectively. As usual, there is a canonical morphism π(a,b),(a,b),2,20/0,a+b :
Adm(a,b),(a,b),2,20/0,a+b → M0,6 remembering the pointed source curve. Let r : M0,6 → M1,3
be the map sending
[(X, x1, . . . , x6)] 7→ [(X/(x2 ∼ x4), x5, x1, x3)s]
We have a Cartesian diagram
A(∆00,∆0)S
//
��
Adm(a,b),(a,b),2,20/0,a+b
π(a,b),(a,b),2,20/0,a+b��
M0,6
r��
S //M1,3
It is routine to check, following the proof of Proposition 3.4.7:
Proposition 3.5.7. The contribution to AS from covers of type (∆00,∆0) is
∑am+bn=d
(mb
∫M1,3
[S] · [r ◦ π(a,b),(a,b),2,20/0,a+b ]
).
In order to complete the calculation, we will need to compute the class [r◦π(a,b),(a,b),2,20/0,a+b ] ∈
A1(M1,3).
110
Proposition 3.5.8. We have:
∫M1,3
[r ◦ π(a,b),(a,b),2,20/0,a+b ] ·∆0 = 0∫
M1,3
[r ◦ π(a,b),(a,b),2,20/0,a+b ] ·∆1,S = 1 for S = {2, 3}, {1, 2, 3}∫
M1,3
[r ◦ π(a,b),(a,b),2,20/0,a+b ] ·∆1,S = 0 for S = {1, 2}, {1, 3}
Proof. For the first statement, we may replace ∆0 with the locus of pointed curves
with a fixed underlying elliptic curve (E, x1), which clearly has empty intersection
with [r ◦ π(a,b),(a,b),2,20/0,a+b ].
Now, consider a cover in Adm(a,b),(a,b),2,20/0,a+b whose image inM1,3 has a non-separating
node. We claim that the only such cover, up to isomorphism, is constructed as follows.
Let Y be the union of two copies Y1, Y2 of P1, attached at a node, with two marked
points on each component. Then, X contains two copies of P1 mapping to Y1 via the
maps x 7→ xa and x 7→ xb, respectively. These two components are connected by a
copy of P1 mapping to Y2 via x 7→ x2, and the rest of the components of X map to Y2
isomorphically.
The cover f : X → Y constructed above gives a single point of intersection of
the admissible locus with ∆1,{2,3} and ∆1,{1,2,3}; it is now standard to check that the
multiplicity is 1.
3.5.3.3 Final computation
We are now ready to intersect [π2/1,d] ∈ A2(M2,1) with the boundary test surfaces.
111
Proposition 3.5.9. We have:
∫M2,1
[π2/1,d] ·∆00 = 4(d− 1)σ1(d)∫M2,1
[π2/1,d] ·∆01a =∑
d1+d2=d
σ1(d1)σ1(d2)∫M2,1
[π2/1,d] ·∆01b =∑
d1+d2=d
σ1(d1)σ1(d2)∫M2,1
[π2/1,d] · Ξ1 = − 1
24(d− 1)σ1(d)∫
M2,1
[π2/1,d] ·∆11 = − 1
24
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
Proof. First, we consider the classes ∆01a,∆01b, and ∆11 contained inM1,2×M1,1; these
are the push-forwards of the boundary divisors M1,2 × p, ∆0 ×M1,1, and ∆1 ×M1,1,
respectively. Thus,
∫M1,2×M1,1
([p×M1,1] + [∆1 × p]) · [∆01a] = 1∫M1,2×M1,1
([p×M1,1] + [∆1 × p]) · [∆01b] = 1∫M1,2×M1,1
([p×M1,1] + [∆1 × p]) · [∆11] = − 1
24,
applying §3.2.2.1. The formulas for the intersections of π2/1,d with these three classes
now follow from Proposition 3.5.4.
Now, we consider the classes ∆00,∆01a,∆01b,Ξ1 contained in M1,3; these are the
push-forwards of the boundary divisors ∆0, ∆1,{2,3}, ∆1,{1,2,3}, and ∆1,{1,3}, respectively.
(We have included the middle two classes, which arose earlier, as a check.)
By Propositions 3.3.8 and 3.5.6, the only class for which covers of type (∆0,∆0)
contribute is ∆00, in which we get a contribution of 2(d − 1)σ1(d). By Propositions
3.5.7 and 3.5.8, covers of type (∆00,∆0) contribute∑
d1+d2=d σ1(d1)σ1(d2) to each of
∆01a,∆01b and nothing to the others.
112
Finally, consider covers of type (∆0,∆1). By Corollary 3.3.5, we have
[s{2,3} ◦ π1/1,d,2] = (d− 1)σ1(d)
(1
24∆01,{1,2} + ∆11,{1,2}
)[s{1,3} ◦ π1/1,d,2] = (d− 1)σ1(d)
(1
24∆01,{1,3} + ∆11,{1,3}
)
Applying Proposition 3.5.5 and §3.2.2.2, we get no contributions to ∆01a and ∆01b,
a contribution of 2(d − 1)σ1(d) to ∆00, and a contribution of − 124
(d − 1)σ1(d) to Ξ1.
Combining all of the above yields the needed intersection numbers.
Theorem 3.5.10. The class of π2/1,d in A2(M2,1) is:
(− 1
12dσ1(d) +
1
12σ3(d)
)δ00 +
(1
12σ1(d)− 1
12σ3(d)
)δ01a
+
((−d− 1
12
)σ1(d) +
13
12σ3(d)
)δ01b + (2σ3(d)− 2dσ1(d)) ξ1 + (4σ3(d)− 4σ1(d)) δ11.
In particular, ∑d≥1
[π2/1,d]qd ∈ Qmod⊗A2(M2,1).
Proof. Immediate from §3.2.2.4; note that the dual graphs associated to all five bound-
ary classes have automorphism group of order 2, except ∆00, which has automorphism
group of order 8.
The classes δ00, δ01a, δ01b push forward to zero on M2, and the classes ξ1, δ11 push
forward to δ0, δ1, respectively. Thus, Theorem 3.5.10 recovers Theorem 3.1.3. In addi-
tion, taking d = 2 in Theorem 3.5.10 recovers [vZ18a, Proposition 3.2.9].
3.6 The d-elliptic locus on M3
We now carry out the methods developed earlier and use the results above to compute
the (unpointed) d-elliptic locus in genus 3, [π3/1,d] ∈ A2(M3). As A2(M3) = 0 [Fab90a,
113
Theorem 1.9], any test surface lies in one of the two boundary divisors:
(∆0) M2,2 →M3
(∆1) M2,1 ×M1,1 →M3
3.6.1 Classification of Admissible Covers
We will compute the intersection of the admissible locus with a general test surface S
in one of the two boundary divisors. The same arguments from before show that we
need only consider the codimension 1 strata in Adm3/1,d, parametrizing covers whose
targets have exactly one node. Moreover, the same dimension count shows that we
may disregard the strata whose images inM3 have dimension 2 or less, or equivalently
whose general fiber under the map π3/1,d is positive-dimensional.
A similar analysis as in genus 2 yields seven topological types of covers in Adm3/1,d
that that give non-zero contributions to general test surfaces, shown in Figures 13, 14,
15, 16, 17, 18, 19.
Figure 13: Cover of type (∆0,∆1,2) Figure 14: Cover of type (∆1,∆1,2)
114
Figure 15: Cover of type (∆1,∆1,3) Figure 16: Cover of type (∆11,∆1,4)
Figure 17: Cover of type (∆0,∆0) Figure 18: Cover of type (∆00,∆0)
Figure 19: Cover of type (∆000,∆0)
115
3.6.2 The case [S] ∈ ∆1
Define the intersection AS by the Cartesian diagram
AS //
��
Adm3/1,d
π3/1,d��
S //M2,1 ×M1,1//M3
We consider the contributions to AS from covers of the three possible types: (∆1,∆1,2),
(∆1,∆1,3), and (∆11,∆1,4).
3.6.2.1 Type (∆1,∆1,2)
For any d ≥ 1, we define the space Adm12/1,d as a functor by the Cartesian diagram
Adm12/1,d
//
π12/1,d
��
Adm2/1,d
π2/1,d��
M2,1u //M2
Here, u : M2,1 →M2 is the map forgetting the marked point. Generically, Adm12/1,d
parametrizes covers f : X → Y along with an arbitrary point of X. Define the map
ψ12/1,d by the composition
Adm12/1,d → Adm2/1,d →M1,2 →M1,1
where the middle map is ψ2/1,d and the last map forgets the second point.
As in §3.4.2, we have a diagram
116
A(∆1,∆1,2),(d1,d2)S
//
��
Adm12/1,d1
×∆ Adm1/1,d2//
��
M1,1
∆
��
Adm12/1,d1
×Adm1/1,d2
ψ12/1,d1
×ψ1/1,d2 //
π12/1,d1
×π1/1,d2��
M1,1 ×M1,1
S //M2,1 ×M1,1
where both squares are Cartesian. A geometric point of AS corresponding to a cover
f : X → Y of type gives rise to a geometric point of A(∆1,∆1,2),(d1,d2)S in an obvious way.
Note, however, that the image of [f ] in M1,1 ×M1,1 is of the form ([Y1, q], [Y1, y]),
where Y1 ⊂ Y is the elliptic component, q ∈ Y1 is one of the branch points of f , and
y ∈ Y1 is the node. While q 6= y, [(Y1, q)] and [(Y1, y)] are isomorphic via translation.
Owing to the contraction of the rational bridge of any admissible cover of type
(∆1,∆1,2), each cover f of type (∆1,∆1,2) appears with multiplicity 2 in AS. In addi-
tion, each point A(∆1,∆1,2),(d1,d2)S comes from
(42
)= 6 points of AS, due to the possible
labelings of the branch points of the d1-elliptic map.
Using the fact that ∆ = [p×M1,1]+[M1,1×p], and applying the projection formula,
we find:
Proposition 3.6.1. The contribution to AS from covers of type (∆1,∆1,2) is:
12
(∫M2×M1,1
u∗([S]) ·
( ∑d1+d2=d
σ1(d2)([[π2/1,d1(E)]×M1,1] + [[π2/1,d1 ]× p]
)))
117
3.6.2.2 Type (∆1,∆1,3)
We have a Cartesian diagram
A(∆1,∆1,3)S
//
��
Adm2/1,d
π2/1,d
��
S //M2,1 ×M1,1pr1 //M2,1
where pr1 :M2,1 ×M1,1 →M2,1 is the projection. Given a point ([(C, q)], [(E, p)]) of
M2,1×M1,1, the curve C ∪p∼q E is the image under π3/1,d of a cover of type (∆1,∆1,3)
if and only if there exists a d-elliptic map g : C → E ramified at q, in which we may
glue g to the unique double cover E → P1 ramified at the origin (and attach additional
rational tails) to form an admissible cover whose source contracts to C ∪p∼q E. The
transversality is straightforward, and we find that:
Proposition 3.6.2. The contribution to AS from covers of type (∆1,∆1,3) is:
24
(∫M2,1
pr1∗([S]) · [π2/1,d]
)
The factor of 24 comes from the 4! ways to label the ramification points.
3.6.2.3 Type (∆11,∆1,4)
For d1, d2 with d1 + d2 = d, we have a diagram
A(∆11,∆1,4)S
//
��
M1,2 ×Adm1/1,d1 ×∆ Adm1/1,d2
��
//M1,2 ×Adm1/1,d1 ×Adm1/1,d2
id×π1/1,d1×π1/1,d2��
M1,2 ×M1,1id×∆ //M1,2 ×M1,1 ×M1,1
ξ1×id��
S //M2,1 ×M1,1
118
where ξ1 :M1,2 ×M1,1 →M2,1 is the map defining the boundary divisor ∆1 inM2,1,
and both squares above are Cartesian.
Proposition 3.6.3. The contribution to AS from covers of type (∆11,∆1,4) is:
24
( ∑d1+d2=d
σ1(d1)σ1(d2)
)·
(∫M2,1×M1,1
[S] · ([∆01a ×M1,1] + [∆1 × p])
)
Proof. Given covers E1 → E and E2 → E of degrees d1, d2, respectively, and a 2-
pointed curve (E ′, p1, p2) of genus 1, we construct an admissible cover of type (∆11,∆1,4)
by attaching E ′ the Ei at their origins along the pi, mapping E ′ → P1 via the complete
linear series |O(p1 + p2)|, and labelling the ramification points in one of 4! = 24 ways.
The transversality is straightforward. Decomposing the class of ∆ as usual, we get the
desired result; here the class ∆01a ∈ A2(M2,1) arises as the pushforward of M1,2 × p
under ξ1.
3.6.3 The case [S] ∈ ∆0
The only such S we will need is defined as follows. Let C be a general curve of genus
2, and take S = C × C. The map S 7→ M2,2 is defined by (x, y) 7→ [(C, x, y)], where
if x = y, we interpret the image as the reducible curve with a 2-pointed rational curve
attached at x. Clearly, covers of types (∆00,∆0) and (∆000,∆0) do not appear along
S, so we do not give a general formula for contributions from such covers. Moreover,
if C is general, then it is not d-elliptic, so we also do not see covers of type (∆0,∆1,2).
Therefore, the only contributions to the intersection of S with the admissible locus
come from covers of type (∆0,∆0).
Proposition 3.6.4. We have
∫M3
[C × C] · [π3/1,d] = 48(dσ3(d)− σ1(d)).
119
Proof. This amounts to enumerating covers of type (∆0,∆0) where the genus 2 com-
ponent in the source is isomorphic to C; the result is then immediate from Proposition
3.3.9 and a local computation identical to that of Proposition 3.4.5. Indeed, we have
∑am=d
48(a4 − 1)m = 48d
(∑am=d
a3
)− 48
(∑am=d
m
)= 48(dσ3(d)− σ1(d)).
In the final computation, we will use the following:
Proposition 3.6.5. We have [C × C] = 2(∆(1) + ∆(4)) in A2(M3), where the classes
on the right hand side are defined as in §3.2.2.5.
Proof. Because any two geometric points of M2 are rationally equivalent, we may
replace the general genus 2 curve C by the reducible genus 2 curve C0 obtained by
gluing two nodal curves of arithmetic genus 1 together at a separating node. Then, the
space MC0,2 parametrizing two points on C0 (that is, the fiber over [C0] of the forgetful
map M2,2 → M2) has four components, corresponding to the choices of components
of C0 on which the marked points can lie. The two components of MC0,2 for which the
marked points lie on the same component of C0 each contribute ∆(1) to the class of
MC0,2, and the two components for which the marked points lie on opposite components
each contribute ∆(4).
120
3.6.4 The class of the admissible locus
Proposition 3.6.6. We have:
∫M3
[π3/1,d] ·∆(1) = 96(d− 1)σ1(d)∫M3
[π3/1,d] ·∆(4) = 24(dσ3(d)− σ1(d))− 96(d− 1)σ1(d)∫M3
[π3/1,d] ·∆(5) = 24
( ∑d1+d2=d
(2d1 − 1)σ1(d1)σ1(d2)
)∫M3
[π3/1,d] ·∆(6) = 0∫M3
[π3/1,d] ·∆(8) = 12
( ∑d1+d2=d
(d1 + 1)σ1(d1)σ1(d2)
)− (d− 1)σ1(d)
∫M3
[π3/1,d] ·∆(10) = 48
( ∑d1+d2=d
σ1(d1)σ1(d2)
)∫M3
[π3/1,d] ·∆(11) = 24
( ∑d1+d2+d3=d
σ1(d1)σ1(d2)σ1(d3)
)−
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
Proof. We first deal with the surface classes ∆(i) for i = 1, 5, 6, 8, 10, 11, which factor
through M2,1 ×M1,1. These are pushed forward from the following classes:
(∆(1)) ∆00 × p
(∆(5)) Γ(5) ×M1,1
(∆(6)) Γ(6) ×M1,1
(∆(8)) Ξ1 × p
(∆(10)) ∆01a × p
(∆(11)) (a) ∆(11) × p or (b) Γ(11) ×M1,1
We summarize the contributions from covers of the three possible types in the
table below, where we have applied Propositions 3.6.1, 3.6.2, and 3.6.3, along with the
121
intersection numbers of §3.2.2.4 and the intersection numbers with d-elliptic loci in
genus 2 from Propositions 3.4.8, 3.5.1, and 3.5.9. The last two rows correspond to the
computation of the intersection of the admissible locus with ∆(11), computed as the
pushforwards of the two classes labelled (a) and (b) above.
Type (∆1,∆1,2) Type (∆1,∆1,3) Type (∆11,∆1,4)
∆(1) 0 96(d− 1)σ1(d) 0
∆(5) 48
( ∑d1+d2=d
(d1 − 1)σ1(d1)σ1(d2)
)0 24
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
∆(6) 0 0 0
∆(8) 12
( ∑d1+d2=d
(d1 − 1)σ1(d1)σ1(d2)
)−(d− 1)σ1(d) 24
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
∆(10) 0 24
( ∑d1+d2=d
σ1(d1)σ1(d2)
)24
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
∆(11)(a) 24
( ∑d1+d2+d3=d
σ1(d1)σ1(d2)σ1(d3)
)−
( ∑d1+d2=d
σ1(d1)σ1(d2)
)0
∆(11)(b) 24
( ∑d1+d2+d3=d
σ1(d1)σ1(d2)σ1(d3)
)0 −
( ∑d1+d2=d
σ1(d1)σ1(d2)
)
Combining the above yields six of the intersection numbers claimed; the seventh, of
[π3/1,d] with ∆(4), now follows from Propositions 3.6.4 and 3.6.5.
Remark 3.6.7. One can also implement the following check: the class ∆(7) ∈ A2(M3)
is rationally equivalent to ∆(6), so its intersection with the admissible locus should be
zero. Using the fact that ∆(7) is the pushforward of ∆01b×p fromM2,1×M1,1, we indeed
find a contribution of 0 from type (∆1,∆1,2), a contribution of∑
d1+d2=d σ1(d1)σ1(d2)
from type (∆1,∆1,3), and a contribution of−∑
d1+d2=d σ1(d1)σ1(d2) from type (∆11,∆1,4).
Proof of Theorem 3.1.4. The result now follows from Proposition 3.6.6, along with the
intersection numbers of §3.2.2.5 and the convolution formulas of §3.2.4.
122
3.A Quasi-modularity on M2,2
The quasimodularity for d-elliptic loci in genus 2 found in Theorems 3.1.3 (forgetting
marked branch points) and 3.5.10 (with one branch point) can in fact be upgraded to
M2,2, remembering both branch points of a d-elliptic cover. That is:
Theorem 3.A.1. We have
∑d≥1
[π2/1,d]qd ∈ Qmod⊗A3(M2,2).
This result will propagate to quasimodular contributions to the d-elliptic locus on
M4, providing further evidence for Conjecture 1. The method of proof is the same
as above, using the fact that A2(M2,2) = 0 [Fab90a, Lemma 1.14]; we do not carry
out the full calculation. However, we point out one new aspect, that the contributions
from admissible covers of certain topological types are not individually quasimodular,
but the non-quasimodular contributions cancel in the sum.
Let T → M1,4 be a general boundary cycle of dimension 3. Consider the contri-
butions to the intersection of T with π2/1,d : Adm2/1,d → M2,2 from covers of types
(∆0,∆0) and (∆00,∆0).
Lemma 3.A.2. Let Hdg (λ1, λ2, λ3) denote the Hurwitz number counting covers (weighted
by automorphisms) f : C → P1 branched over 3 points with ramification profiles
(λ1, λ2, λ3), where we require C to be a connected curve of genus g. We have:
(a)
Hd1 ((d), (d), (3, 1d−3)) =
(d− 1)(d− 2)
6
(b)
Hd0 ((a, b), (a, b), (3, 1d−3)) =
1 if a 6= b
0 if a = b
123
Proof. Let α ∈ Sd denote the cycle (12 · · · d), and let β = (1jk), for j, k ∈ {2, 3, . . . , d}
distinct. One readily checks that the product βα is a d-cycle if and only if j < k. There
are(d−1
2
)choices of such β, but each Hurwitz factorization is then triple-counted owing
to the simultaneous conjugations by powers of α sending 1 to j, k. The first formula
follows.
For the second, let α be the permutation (12 · · · a)(a+ 1 · · · d). We seek a 3-cycle β
for which βα also has cycle type (a, b). In order for the corresponding branched cover
to be a map of connected curves, β cannot act trivially on either orbit of α, so we may
assume that β acts nontrivially on 1, a+1. If a > b, then we find β = (1(a−b+1)(a+1)),
but if a = b, then no such β exists.
First, consider covers of type (∆0,∆0). As in Propositions 3.4.5 and 3.5.6, we have:
Proposition 3.A.3. The contribution to the intersection of T and π2/1,d from covers
of type (∆0,∆0) is ∑am=d
(m
∫M1,4
[T ] · [πa,a,2,21/0,a ]
).
We now apply Proposition 3.A.3 with T = ∆3,4. The intersection [T ] · [πa,a,2,21/0,a ]
includes admissible covers formed by gluing a degree dmap E → P1 branched over three
points with ramification indices d, d, 3 to a degree 3 map P1 → P1 with ramification
indices 3, 2, 2, at the triple points in the source and target (see also Proposition 2.A.2).
Applying Lemma 3.A.2(a) and Proposition 3.A.3, we find a contribution to∫M2,2
[T ] ·
[π2/1,d] of ∑am=d
(a− 1)(a− 2)
6·m =
(1
6d+
1
3
)σ1(d)− 1
2dτ(d),
where τ(d) denotes the number of divisors of d; the generating function for dτ(d) is
not quasimodular.
On the other hand, consider contributions along T from covers of type (∆00,∆0).
We get a quasimodular contribution analogous to that of Proposition 3.5.7, but we get
a new contribution from admissible covers formed by gluing a degree d map P1 → P1
124
branched over three points with ramification profiles (a, b), (a, b), 3 to a degree 3 map
P1 → P1 with ramification indices 3, 2, 2, at the triple points in the source and target.
By Lemma 3.A.2(b) and the usual local computation, we get an additional contribution
of
∑am+bn=d
mb−∑
b(m+n)=d
mb
=∑
d1+d2=d
σ1(d1)σ1(d2)−∑bn′=d
n′−1∑i=1
n′b
=∑
d1+d2=d
σ1(d1)σ1(d2)−∑bn′=d
b
(n′(n′ − 1)
2
)=
∑d1+d2=d
σ1(d1)σ1(d2)− 1
2dσ1(d) +
1
2dτ(d)
In particular, the last term cancels out the non-quasimodular term from type (∆0,∆0).
3.B An enumerative application
In this section, we give an example application of Theorem 3.1.3.
Theorem 3.B.1. Let x1, . . . , x5 ∈ P1 be a very general collection of points. Let ad be
the number of points x6 ∈ P1 such that the hyperelliptic curve branched over x1, . . . , x6
is smooth and d-elliptic. Then,
ad = 5d
∑d′|d
(σ3(d′)
d′· µ(d
d′
))− d
,where µ(m) is the Mobius function.
When d = 2, we recover the classical fact that there are 15 points x6 such that the
aforementioned hyperelliptic curve is bielliptic. Indeed, there are 12
(52
)(32
)= 15 ways to
partition x1, . . . , x6 into two pairs {a1, a2}, {b1, b2}, and a fifth point c. When x1, . . . , x5
125
are general, there is a unique x6 ∈ P1 such that there is an involution of P1 swapping
a1 with a2, b1 with b2, and c with x6. The quotient by this involution has genus 1.
We may associate to x1, . . . , x5 a 1-parameter family µP , and compute its intersec-
tion with π2/1,d. We will need to analyze carefully the points of the intersection, after
which we may conclude Theorem 3.B.1.
We also remark that the method can be used in genus 3: for example, the number
of bielliptic curves in a general net of plane quartics is computed in [FP15]; the same
computation may now be carried out for general d-elliptic curves using Theorem 3.1.3.
3.B.1 The Class of µP
Let P ∈ C[x] be a square-free monic polynomial of degree 5. We define a map µP :
P1 →M2 sending t to the hyperelliptic curve branched over t2 and the roots of P , and
denote the corresponding class by [µP ] ∈ A2(M2). More precisely, the total space X
has charts
U1 = SpecC[x, y, t]/(y2 − P (x)(x− t2))
U2 = SpecC[x, y′, x]/(y′2 − P (x)(s2x− 1))
U3 = SpecC[u, v, t]/(v2 − u5P (u−1)(1− ut2))
U4 = SpecC[u, v′, s]/(v′2 − u5P (u−1)(s2 − u)).
The transition functions are as follows: between U1 and U2, we have t = 1/s and
y = ity′ (where i is a square root of −1), between U1 and U3, we have u = 1/x and
v = u3y, and between U2 and U4, we have u = 1/x and v′ = u3y′. Then, we have a
family g : X → P1 = ProjC[s, t] of stable genus 2 curves.
126
Proposition 3.B.2. We have:
∫M2
[µP ] ·∆0 = 40∫M2
[µP ] ·∆1 = 0
Proof. Set-theoretically, there are 10 nodal fibers of the family µP , corresponding to
the points where t2 becomes equal to one of the 5 roots of P . Thus, there are 20
C-points in the intersection of µP and the map M1,2 → M2 defining ∆1; there are
clearly no automorphisms. The total space X of the family µP has an ordinary double
point at each node, so each point has intersection multiplicity 2. The first statement
follows.
The second statement is immediate from the fact that every member of the family
defined by µP is irreducible.
From §3.2.2.3, we conclude:
Corollary 3.B.3. We have [µP ] = 16δ00 + 96δ01 in A2(M2).
3.B.2 Genus 2 curves with split Jacobian
In this section, all curves are assumed to be smooth, and J(X) denotes the Jacobian
of X. The main reference here is [Kuh88, §2].
Definition 3.B.4. Let f : C → E be a morphism of curves of degree d, where C
has genus 2 and E has genus 1. We say that f is primitive if it does not factor as
X → E ′ → E, where E ′ → E is an isogeny of degree greater than 1.
Let f : X → E1 be an optimal cover of degree d. Fix a Weierstrass point x0 ∈ X,
and let ι : X 7→ J(X) be the embedding sending x 7→ O(x − x0). We may regard E1
as an elliptic curve with origin f(x0). We then get an induced morphism of abelian
127
varieties φ1 : J(X)→ E1 such that φ1 ◦ ι = f . We have an exact sequence
0 // E2// J(X) // E1
// 0 (3.1)
By the optimality of f , E2 is connected. Let φ2 : J(X) → E2 be the dual map to the
embedding φ2 : E2 7→ J(X), and let f2 = φ2 ◦ ι : X → E2.
Lemma 3.B.5 ([Kuh88], §2). f2, as constructed above is an optimal cover of degree
d, and φ = φ1 ⊕ φ2 : J(X)→ E1 ⊕ E2 is an isogeny of degree d2.
Corollary 3.B.6. Let X be a d-elliptic curve of genus 2, where d is minimal, and let
f : X → E1 be an optimal cover of degree d. Let f2 : X → E2 and φ : J(X)→ E1⊕E2
be as above. Suppose that E1 and E2 are not isogenous. Then, any non-constant
morphism f0 : X → E0, where E0 is a curve of genus 1, factors uniquely through
exactly one of f1 and f2.
Proof. We may regard E0 as an elliptic curve with origin f0(x0). We then get an
induced morphism of abelian varieties φ0 : J(X)→ E0 such that f = φ0 ◦ ι, and a non-
zero dual morphism φ0 : E0 → J(X). Exactly one of the maps φ′i := pri◦φ◦φ0 must be
non-zero, because E1 and E2 are not isogenous; assume that φ′1 = 0 and φ′2 6= 0. Then,
from the exact sequence (3.1), we have that φ0 factors as φ2 ◦ g, for some non-zero
g : E0 → E2. Dualizing and pre-composing with ι shows that f0 factors through f2.
The uniqueness of f0 follows from the uniqueness of the factorization φ0 = φ2 ◦ g.
3.B.3 Intersection of µP and π2/1,d
Lemma 3.B.7. Suppose P is a general monic square-free polynomial of degree 5.
Then, µP and π2/1,d intersect in the dense open substack H2/1,d ⊂ Adm2/1,d of covers
of smooth curves.
Proof. Let Z = π2/1,d(Adm2/1,d−H2/1,d), which has dimension 1 inM2. Given a point
[C] ∈ M2 in the image of some µP , there is a 3-dimensional family of P such that µP
128
passes through [C], so there is a 4-dimensional space of P for which µP is incident to
Z. On the other hand, the space of polynomials P is 5-dimensional, so the general µP
avoides Z.
Lemma 3.B.8. Suppose P is a general monic square-free polynomial of degree 5, and
s ∈ C is not a root of P . Then, the genus 2 curve C associated to the affine equation
y2 = f(x)(x − s) has either #Aut(C) = 2 or #Aut(C) = 4. The latter occurs if and
only if C is bielliptic.
Proof. The assertion is equivalent to the following: a general choice of distinct points
x1, . . . , x5 ∈ P1 has the property that for all x6 ∈ P1 − {x1, . . . , x5}, the group of
automorphisms of P1 fixing the set {x1, . . . , x6} has order at most 2, and the order is
2 if and only the hyperelliptic curve branched over x1, . . . , x6 is bielliptic.
One checks that for x1, . . . , x5 general, there is no automorphism of P1 doing any
of the following:
(i) x1 7→ x2 7→ x3 7→ x4 7→ x5
(ii) x1 7→ x2 7→ x3 7→ x4 7→ x1
(iii) x1 7→ x2 7→ x3 and x4 7→ x5 7→ x4
(iv) x1 7→ x2 7→ x3 7→ x1 and x4 7→ x5
(v) x1 7→ x1 and x2 7→ x3 7→ x4 7→ x2
(vi) x1 7→ x1 and x2 7→ x3 7→ x4 7→ x5
(vii) x1 7→ x1, x2 7→ x3 7→ x2 and x4 7→ x5
(viii) x1 7→ x1, x2 7→ x2, and x3 7→ x4 7→ x3.
(ix) x1 7→ x1, x2 7→ x2, and x3 7→ x4 7→ x5.
129
For instance, to prove that x1 7→ x2 7→ x3 7→ x1 and x4 7→ x5 is impossible, we
may let (x1, x2, x3) = (0, 1,∞), and note that the unique automorphism of P1 sending
x1 7→ x2 7→ x3 7→ x1 will only send x4 7→ x5 if the latter two points are chosen in
special position.
We then conclude that a non-trivial permutation ρ of the xi must be a union of three
2-cycles. An automorphism of P1 inducing such a permutation must be an involution.
The quotient of the induced involution on C will have genus 1, as the involution is not
hyperelliptic.
Proposition 3.B.9. For a very general monic square-free polynomial P of degree 5,
every cover f : C → E in the intersection of µP and π2/1,d has the following properties:
(i) [f ] ∈ H2/1,d, that is, C and E are smooth.
(ii) J(C) is not isogenous to the product of an elliptic curve with itself.
(iii) Given an automorphism h of C, there exists an automorphism h′ of E and a
cover f ′ : C → E compatible with h and h′.
(iv) The intersection of µP and π2/1,d is transverse at [f ].
Proof. The first condition is Lemma 3.B.7.
The locus of smooth curves in M2 whose Jacobian is isogenous to a self-product
of an elliptic curve is a countable union of substacks of dimension 1. By the same
argument as in Lemma 3.B.7, a very general P avoids this locus.
By Lemma 3.B.8, the only possible automorphisms of C are the hyperelliptic invo-
lution, which is compatible with an involution of E, or a bielliptic involution, which
commutes with f , by Corollary 3.B.6 and condition (ii).
130
Finally, we check the transversality. Consider the Cartesian diagram
W //
��
H2/1,d
π2/1,d
��A6 −∆ //M2
where the map A6 −∆→M2 is the morphism taking six distinct points x1 . . . , x6 to
the hyperelliptic curve branched over x1, . . . , x5, x26. Consider the composition
W → A6 −∆→ A5 −∆
where the second map remembers the first five points. By generic smoothness, the
general fiber of q : W → A5 − ∆ is smooth. The previous conditions guarantee
that the very general fiber of q is precisely the intersection of µP with π2/1,d, where
P (x) = (x− x1) · · · (x− x5), so we are done.
Proof of Theorem 3.B.1. Fix a coordinate on P1, and let P be the monic polynomial
with roots x1, . . . , x5; we assume 0,∞ 6= xi. Let Cx6 denote the genus 2 curve branched
over x1, . . . , x6. For a very general collection of x1, . . . , x5, the conditions of Proposition
3.B.9 are satisfied. The C-points of the intersection of µP and π2/1,d consist of the data
of t ∈ C, a cover f : C → E (with the data of its branch points), and an isomorphism
h : Cx6∼= C. By condition (iii), we may disregard h. It is moreover clear that these
objects have no automorphisms. By condition (ii) and Corollary 3.B.6, f : C → E
factors uniquely through exactly one of two optimal covers fi : C → Ei of some degree
d′|d.
Let bd′ be the number of x6 such that Cx6 admits an optimal cover Cx6 → E of
degree d′. Then, we have
ad =∑d′|d
bd′ = (b ? 1)d,
where ? denotes Dirichlet convolution and 1d = 1. Given a d-elliptic curve Cx6 , the
131
number of points in the intersection of µP and π2/1,d factoring through an optimal cover
from C of degree d′ is 4σ1(d/d′). Indeed, we have 2 choices for the optimal cover, 2
choices for the labelling of the branch points on E, and σ1(d/d′) quotients of E of degree
d/d′, by Lemma 3.3.1. Because we may assume x6 6= 0,∞, the family µP contains Cx6
twice, so we have
[µP ] · [π2/1,d] = 8(b ? σ)d · p
Applying Theorem 3.1.3 and Proposition 3.B.3, we have
(b ? σ)d = 5(σ3(d)− dσ1(d)).
However, as σ = Id ?1, where Idd = d, and a = b ? 1, we have
(a ? Id)d = 5(σ3(d)− dσ1(d)).
Because Id−1d = µ(d)d is a Dirichlet inverse of Id, we get
ad = 5∑d′|d
(σ3(d′)− d′σ1(d′)) · µ(d
d′
)· dd′
= 5d
∑d′|d
(σ3(d′)
d′· µ(d
d′
))− d
,where we have applied Mobius Inversion. The proof is complete.
132
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