THE LOCUS OF PLANE CURVES IN THE MODULI STACKOF CURVES

AARON LANDESMAN

ABSTRACT. Let d ≥ 4 and let Ud denote the locus of smoothcurves in the Hilbert scheme of degree d plane curves. If themembers of Ud have genus g, let Mg denote the moduli stackof genus g curves. We show that the natural map [Ud/ PGL3] →Mg is a locally closed embedding. Along the way, we show that[Ud/ PGL3] represents a functor parameterizing families of planecurves.

1. INTRODUCTION

One of the most fundamental objects of study in algebraic geom-etry is the study of the moduli curves. Centuries ago, when peoplewere first exploring curves, they first considered plane curves: thosecut out from P2 by a single equation.

The goal of this note is to define a stack parameterizing familiesof plane curves. Perhaps the most natural candidate definition forsuch a stack would be families of curves so that all geometric fibersare isomorphic to plane curves. While it is not to difficult to see thisis a substack of Mg, it is not at all clear whether this is algebraic. Inorder to fix this issue, we add the extra condition that the first de-rived pushforward of the family is locally free of the appropriate de-gree, or equivalently that the family commutes with base change (seeLemma 3.5). We define the functor in Definition 3.1. Recall that Mg

is a smooth Deligne-Mumford stack for g ≥ 2. Since plane curves ofdegree< 4 have genus≤ 1, for the purposes of examining these sub-stacks of Mg parameterizing plane curves, it is natural to restrict tothe case d ≥ 4. Given this definition, we have two natural questions:

Question 1.1. For d ≥ 4, is the stack of plane curves as defined inDefinition 3.1 a locally closed algebraic substack of Mg (i.e., is thenatural map to Mg representable by locally closed embeddings)?

Question 1.2. For d ≥ 1, is the substack of degree d plane curvessmooth?

1

2 AARON LANDESMAN

We answer both these questions in the affirmative. To do so, wefirst recall classical facts line bundles on plane curves in section 2.Next, we show in section 3 (specifically in Theorem 3.7) that the stackof plane curves defined in Definition 3.1 is isomorphic to the quotientstack [Ud/ PGL3]. Here, Ud is the locus in the Hilbert scheme ofsmooth degree d plane curves and PGL3 acts on Ud via its action onthe universal family Cd ⊂ Ud × P2 by automorphisms of P2 . Thisimplies Pd is smooth, as mentioned in Corollary 3.11. We show thatthe natural map Pd → Mg is a locally closed embedding in section 4(specifically in Theorem 4.5).

Indeed, it is claimed in many places that [Ud/ PGL3] is the locusof plane curves. For example, it is done in [SB88, p. 51], [H+04, p.1], and (implicitly in) [BGvB10]. The main goal of this document itto provide a rigorous stack-theoretic proof of this claim. All stacks,unless otherwise specified are defined over Spec Z.

We work with stacks in the etale topology and in general followthe conventions used in [Ols16].

2. CLASSICAL FACTS ABOUT PLANE CURVES

In this section, we recall some classical facts regarding plane curvesover an algebraically closed field. The main result of this sectionis Proposition 2.11, which states that a smooth plane curve has aunique g2d, and that g2d corresponds to a reduced point in the schemeparameterizing g2d’s. We will need this later to test a certain map ofstacks is an isomorphism, by testing it on geometric points.

Many of the results of this section can be found in the exercises[ACGH85, Appendix A, Exercises 17 and 18], and we include proofsfor completeness. We note that the results of this section hold overfields of arbitrary characteristic (as we prove) even though [ACGH85,Appendix A, Exercises 17 and 18] typically has the hypothesis thatthe field has characteristic 0. This independence of characteristic iscrucial for defining our stacks over Spec Z (instead of over a field ofcharacteristic 0).

We begin with some standard definitions.

Definition 2.1. Let k be an algebraically closed field. A 0-dimensionalsubscheme S ⊂ P2k is said to impose independent conditions oncurves of degree n if

h0(P2k, IS(n)) = h0(P2k,OP2k(n)) − d,

where IS ⊂ OP2kis the ideal sheaf of S.

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 3

Definition 2.2. A grd on a smooth curveC is a line bundle L of degreed on Cwith h0(C,L) ≥ r+ 1.

2.1. Showing there is a single g2d. In this section, we show that asmooth plane curve has only one g2d in Proposition 2.6.

We start with a lemma characterizing when finite reduced sub-schemes of P2 supported on at most n+ 2 points impose indepen-dent conditions on curves of degree n.

Lemma 2.3. Let S be any reduced subscheme of P2 whose support consistsofn+1 points. Then, S imposes independent conditions on curves of degreen. Further, if S ⊂ P2 is a reduced subscheme supported on n+ 2 points,then S fails to impose independent conditions on curves of degree n if andonly if S is contained in some line.

Proof. Since S has degree d, it follows from the exact sequence

(2.1) IS(n) OP2k(n) OS

that

h0(P2k, IS(n)) ≥ h0(P2k,OP2k(n)) − d.

So, we only need verify the reverse inequality. By induction on thedegree of S, it suffices to show that for any d ≤ n+ 1 we can findsome plane curve passing of degree n through all but one point of S,but not passing through the last point of S. Further in the case d =n+ 2, it suffices to show we can find such a curve passing throughall but one point of S, provided the n+ 2 points do not lie on a line.

In the case d ≤ n+ 1, to see this, let pd denote a particular point ofS and for each point pi ∈ S,pi 6= pd, choose a line `i passing throughpi but not through pd. In the case d ≤ n+ 1, taking C to be the unionof the lines ∪i 6=d`i provides a curve of degree ≤ n passing throughall but one point of S. Taking the union of this with a curve of de-gree n− d− 1 not passing through pd provides the desired curve ofdegree n.

To conclude, we only need verify that if S is supported on n+ 2non collinear points, there is some curve passing through all but oneof these points. Choose three noncollinear points p1,p2, and p3 inthe support of S. Upon reordering points, it suffices to show there isa curve passing through all points except p3. Then, let `1 be the linejoining p1 and p2. For 2 ≤ i ≤ n, let `i be a line passing through pi+2but not p3. Then, ∪ni=1`i provides the desired curve of degree n notpassing through p3. �

4 AARON LANDESMAN

Using Lemma 2.3 we can compute the cohomology of invertiblesheaves of low degree on smooth plane curves.

Lemma 2.4. LetC be a smooth plane curve of degree d over an algebraicallyclosed field k. Let p1, . . . ,pm be distinct points and L := OC(p1, . . . ,pm).Then, if m ≤ d − 2, we have H0(C,L) = 1. If m = d − 1, thenh0(C,L) = 1 unless p1, . . . ,pm lie in a line `, in which case h0(C,L) = 2and h0(C,OC(`∩C)) = 3.

Proof. Let S := ∪mi=1V(pi). By Lemma 2.3 applied in the case n =d− 3, observe that we have an exact sequence

0 H0(P2k,OP2k(d− 3)⊗ IS) H0(P2k,OP2k

(d− 3)) H0(P2k,OS) 0.

We obtain a corresponding map of exact sequences(2.2)

0 H0(P2k,OP2k(d− 3)⊗ IS) H0(P2k,OP2k

(d− 3)) H0(P2k,OS)

0 H0(C,OC(d− 3)⊗L∨) H0(C,OC(d− 3)) H0(C,OS)

coming from the natural restriction of sheaves to C and using thatIS|C ' L∨.

Observe that the latter two maps of (2.2) are isomorphisms, as fol-lows from the exact sequence on cohomology associated to(2.3)

0 OP2k(α− d) OP2k

(α) OC(α) 0

applied in the cases α = 0 and α = d− 3. Therefore, the first verticalmap of (2.2) is also an isomorphism by the five lemma.

We next claim that h0(C,L) = 1 if S is not contained in a line.Indeed, if S is not contained in a line, by Lemma 2.3 and the isomor-phisms from (2.2), h0(C,OC(d− 3)⊗L) = h0(OC(d− 3)) −m. Usinggeometric Riemann-Roch and the fact that the canonical bundle of Cis OC(d− 3), (since OC(d− 3) is a degree 2g− 2 bundle with g global

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 5

sections,) we have

h0(C,L) = h0(C,KC ⊗L∨) +m− g+ 1

= h0(C,OC(d− 3)⊗L∨) +m− g+ 1

= h0(C,OC(d− 3)) −m+m− g+ 1

=

(d− 1

2

)− g+ 1

= g− g+ 1

= 1.

To conclude, we prove the second statement of the lemma. Ifm = d− 1 and p1, . . . ,pm are collinear, we knowH0(C,OC(p1+ · · ·+pm−1)) = 1 by the above. Hence, H0(C,OC(p1 + · · ·+ pm)) ≤ 2. But,if the points lie on a line `, then we know H0(C,C ∩ `) ≥ 3. SinceC ∩ `− (p1 + · · ·+ pm) is an effective divisor of degree 1, we obtainthe two inequalities must be equalities, soH0(C,OC(p1+ · · ·+pm) =2 and H0(C,C∩ `) = 3. �

Using the prior cohomological calculations, in preparation for prov-ing Proposition 2.6, we show smooth plane curves have no g1d−2’sand characterize the g1d−1’s.

Lemma 2.5. Let C ⊂ P2k be a smooth plane curve of degree d ≥ 4, with kan algebraically closed field. Then,

(1) C has no g1m form ≤ d− 2 and(2) any g1d−1 is of the formD−p for p ∈ C andD in the linear system

H0(C,OC(1)).

Proof. We first show that C has no g1m for m ≤ d− 2. Suppose thatC has a g1m for m ≤ d − 2. Such a line bundle determines a mapC→ P1 of degree at most m (after possibly removing basepoints bytwisting the line bundle down). Therefore, it suffices to show thatC has no dominant degree m maps to P1k for m ≤ d− 2. That is, itsuffices to show C has no basepoint free line bundles of degreem form ≤ d− 2.

So, suppose C has some basepoint free line bundle of degree m ≤d− 2 corresponding to a dominant map C → P1k. We next reduceto showing that C has no line bundles corresponding to genericallyseparable dominant maps C → P1k. This is automatic if k has char-acteristic 0. If k has characteristic p, and C → P1k is generically in-separable and dominant, then C → P1k factors as the composition of

6 AARON LANDESMAN

some generically separable dominant mapC→ P1k with some powerof Frobenius. Therefore, it suffices to show C has no generically sep-arable dominant map to P1k.

So, we now show that there are no generically separable mapsC → P1k. Such a map has general fiber given by a collection of mdistinct points on C. Thus, it remains to show there are no line bun-dles L = OC(p1+ · · ·+pn) with h0(C,L) ≥ 2. (The reason for the re-duction to generically separable morphisms is that we may assumethe points defining the line bundle are distinct, so we may applyLemma 2.4.) Therefore, by Lemma 2.4, C has no g1m’s.

We next verify the second claim that only g1d−1’s on C are givenby divisors of the form D− p with p ∈ C and D in the linear sys-temH0(C,OC(1)). The proof is quite similar to the previous case. LetL be some g1d−1. Note that L must be basepoint free, as otherwise,twisting down by the basepoints, we would obtain some g1m form ≤d − 2, contradicting the previous part. Thus L determines a mapC → P1k. This map is necessarily separable, as otherwise it wouldfactor as the composition of a generically separable map of lower de-gree with Frobenius. But, there are no generically separable maps toP1k of lower degree because there are no g1m’s for m ≤ d− 2. Hence,we know L determines a generically separable dominant morphismC → P1k. Therefore, we may assume that L ∼= OC(p1 + · · · + pd−1)for p1, . . . ,pd−1 distinct. Then, by Lemma 2.5, we have h0(C,L) = 1unless the points lie on a line. In the case that the points p1, . . . ,pd−1lie on a line, taking D to be the intersection of C with that line inP2k, we obtain that p1 + · · ·+ pd−1 = D− q, where q is by definitionD− (p1 + · · ·+ pd−1). �

Using Lemma 2.5, we can now deduce the main result of this sec-tion.

Proposition 2.6. Let C ⊂ P2k be a smooth plane curve of degree d ≥ 4,with k an algebraically closed field. Then, C curve has at most one g2d andthat g2d is a complete linear series.

Proof. First, suppose C is a smooth curve with a g2d. That is, C hasan invertible sheaf L with h0(C,L) ≥ 3. We claim h0(C,L) = 3. Tosee this, let D be an effective divisor in the linear system H0(C,L).Let p1,p2 be two points. If h0(C,L) > 3, then D− p1 − p2 is a g1d−2,which does not exist by Lemma 2.5.

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 7

To complete the proof, we only need to check C has a uniqueg2d. For this, let M be any g2d. Then, for any D in the linear sys-tem H0(C,M) and p ∈ Supp(D), we have OC(D− p) is a g1d−1. ByLemma 2.5, D− p consists of d− 1 collinear points. Therefore, thereis some point q so that M(−p) ∼= OC(1)(−q). Consider the invertiblesheaf,

L := M⊗OC(1)∨ ∼= OC(−q+ p).

To conclude the proof, it suffices to show p = q. We know M⊗(d−3) ∼=KC ∼= L⊗(d−3), since M⊗(d−3) and L⊗(d−3) are degree 2g− 2 bundleswith a g dimensional space of global sections. Therefore, L⊗(d−3) ∼=OC. This implies OC((d−3)p)⊗OC(−(d−3)q) ∼= OC. Using Lemma 2.5,we see H0(C,OC((d− 3)p)) = 1. This implies that the only section ofOC((d− 3)p) is the trivial section, and so OC((d− 3)p)⊗ OC(−(d−3)q) has no sections unless p = q. Therefore, M ∼= OC(1), as de-sired. �

2.2. Showing that g2d is reduced. In this section, we show that theunique g2d on a smooth plane curves (whose uniqueness was estab-lished in Proposition 2.6) corresponds to a reduced point of a param-eter space for g2d’s (which we shall define in Definition 2.9). We do soin Proposition 2.11. In order to do so, we first recall a standard factthat plane curves are projectively normal.

Lemma 2.7. Let k be a field. Any smooth plane curve C ⊂ P2k is projec-tively normal, meaning that for all n > 0, the map H0(P2k,OP2k

(n)) →H0(C,OC(n)) is surjective.

Proof. By the long exact sequence associated to

(2.4) 0 IC OP2kOC 0

to verify projective normality, we only need verify H1(P2, IC(n)) =0 for all n ≥ 0. Say C has degree d. Then, IC(n) ∼= OP2k

(n − d).

Therefore, H1(P2, IC(n)) = H1(P2,OP2k(n− d)) = 0. �

We next define the scheme Grd(p) for p : C→ S a family of smoothgenus g curves. For the moment, we will only need it in the case S isa field, in which case the functor parameterizes invertible sheaves onC and a space of global sections of dimension at most r+1. However,later in section 4 we will need this functor for arbitrary families p :C→ S, so we define it in general now.

8 AARON LANDESMAN

Definition 2.8. We say a morphism C → S is a family of smoothgenus g curves if it is a projective flat morphism so that each fiber isa geometrically connected smooth curve of genus g.

Definition 2.9 ( [ACG11, Chapter XXI, Definition 3.12]). Suppose p :C → S is a family of smooth genus g curves. Define the fiberedcategory Grd(p) sending a map f : T → S to the set of equivalenceclasses of pairs (L,H)/ ∼ defined as follows: Let ι : t→ T be a pointand define the corresponding fiber square

(2.5)

Ct CT

t T .

ιC

pt pT

ι

A pair (L,H) consists of a line bundle L on C×S T whose restrictionto each fiber of pT : CT → T has degree d and a locally free sheafH which is a subsheaf of pT∗L of rank r + 1 so that for each fiberι : t→ T , the natural composition

ι∗H → ι∗pT∗L → pt∗ι∗CL(2.6)

is injective. The equivalence relation ∼ defined on pairs (L,H) dic-tates that two pairs (L,H) and (L ′,H ′) are equivalent if there is aninvertible sheaf Q on T and an isomorphism L ′ ∼= L ⊗ p∗TQ whichinduces an isomorphism H ′ ' H⊗ Q.

Theorem 2.10 ( [ACG11, Chapter XXI, Theorem 3.13]). For p : C →S a family of smooth genus g curves admitting a section, the functor Grddefined in Definition 2.9 is represented by an S scheme.

We are now ready to state and prove the main result of this section.

Proposition 2.11. Let C be a degree d ≥ 4 smooth plane curve over analgebraically closed field k. Then, G2d(p : C → Spec k) is isomorphic to areduced point.

Proof. Since the underlying set of G2d(p) is the set of g2d’s on C, to-gether with a 3-dimensional space of global sections, by Proposi-tion 2.6, G2d(p) is supported on a point. Here we are using that theunique g2d is not a g3d, as was shown in Proposition 2.6.

It only remains to show this point is reduced. Indeed, using [ACGH85,Chapter IV, Proposition 4.1(iii)], (whose proof holds equally well in

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 9

positive characteristic,) the tangent spaces will be 0-dimensional ifthe multiplication map

H0(C,OC(1))⊗H0(C,KC ⊗OC(−1)) → H0(C,KC)

is surjective. Under the identification KC ∼= OC(d− 3), we want toshow the map

H0(C,OC(1))⊗H0(C,OC(d− 4)) → H0(C,OC(d− 3))

is surjective.To verify this, note that we have a commutative square

(2.7)

H0(P2k,OP2k(1))⊗H0(P2k,OP2k

(d− 4)) H0(P2k,OP2k(d− 3))

H0(C,OC(1))⊗H0(C,OC(d− 4)) H0(C,OC(d− 3)).

We know that the vertical maps are surjective by projective normal-ity of C, as shown in Lemma 2.7. Further, the top horizontal mapis surjective, since every degree d − 3 polynomial is a linear com-bination of products of degree 1 and degree d− 3 polynomials in 3variables. Therefore, the bottom horizontal map is also surjective, asdesired. �

3. THE MODULI STACK OF PLANE CURVES

In subsection 3.1, we define the moduli stack of plane curves of de-gree d, which we will denote Pd, and verify it is an algebraic stack.Then, in subsection 3.2, we review a standard application of coho-mology and base change. Finally, in subsection 3.3, we show thatPd ' [Ud/ PGL3], where Ud denotes the open subscheme of theHilbert scheme parameterizing smooth degree d plane curves. Inparticular, this implies that Pd is a smooth algebraic stack of finitetype.

3.1. Defining the stack of plane curves. To begin, we define thestack of plane curves.

Definition 3.1. Let d ≥ 1 be an integer. Let g :=(d−12

). Define

the moduli stack of degree d plane curves denoted Pd to be thefibered category of pairs (f,L) where f : C → S are projective flatmorphisms such that for every geometric point s ∈ S, the fiber Cs isa geometrically proper smooth curve of genus g, (i.e., f is a family

10 AARON LANDESMAN

as defined in Definition 2.8,) and L is a degree d invertible sheaf onC so that R1f∗L is locally free of rank 2− d+ g such that for everygeometric point s→ S, L|s is a very ample invertible sheaf.

A morphism of two families (f ′ : C ′ → S ′,L ′) → (f : C → S,L) isa fiber square

(3.1)C ′ C

S ′ S

g ′

f ′ f

g

so that there is some line bundle Q on S ′ with g ′∗L ⊗ f ′∗Q ∼= L ′.This makes Pd into a fibered category over the category of schemesby sending a family C→ S to S.

We note that Pd is a stack.

Lemma 3.2. The functor Pd is a stack in the etale topology.

Proof. First, observe that Pd is indeed a fibered category as all mor-phisms are Cartesian arrows. Full faithfulness follows from a generalfact for fppf coverings (in particular for etale coverings) as writtenin [Ols16, Corollary 4.2.13]. Now, suppose we have an etale coverg : S ′ → S and are given an object (f ′ : C ′ → S ′,L ′) together withdescent data σ. (see [Ols16, 4.2.1] for precise details of what is meantby descent data, but it is essentially an isomorphism of the two pull-backs of f S ′ ×S S ′ which satisfies the cocycle condition). By descentfor polarized schemes, see [Ols16, Proposition 4.4.12], it follows thatthe line bundle L ′ and familyC ′ → S ′ over S ′ is the pullback of someline bundle L on a family f : C → S. It only remains to verify thatR1f∗L is locally free. Since by assumption g is flat, it follows from flatbase change that the base change map is an isomorphism. Hence, ifwe consider the diagram

(3.2)

C ′ C

S ′ S,

g ′

f ′ f

g

we see R1f ′∗(L ′) = R1f ′∗(g ′∗L) ' g∗R1f∗(L). Since f ′∗L ′ is locally freeby assumption, g∗R1f∗L is locally free, which implies R1f∗L is alsolocally free by [Ols16, Exercise 4.C(a)]. �

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 11

3.2. Applications of cohomology and base change. We next aim toprove that Pd is the quotient [Ud/ PGL3]. Before doing so, we willneed some preparatory results using cohomology and base change,which we prove in this section. The main result is Proposition 3.6.To start, we introduce notation for the base change map.

Definition 3.3. Suppose we have a Cartesian diagram

(3.3)W X

Z Y.

ψ ′

π ′ π

ψ

Then, for F a sheaf on X, we denote by φpZ the natural map

ψ∗Rpπ∗(F) → Rpπ ′∗(ψ′∗F).

We next note a variant of cohomology and base change, showingit can be verified on geometric points, as opposed to points.

Lemma 3.4 (Cohomology and base change for maps from points).Suppose π : X→ Y is proper, Y is locally Noetherian, F is coherent and flatover Y and for each point q ∈ Y there is some ψ : Spec L → Y with imageq, for L a field, so that φpSpec L is surjective. Then, the following hold:

(i) For any Z→ Y, the base change map φpZ is an isomorphism.(ii) Furthermore, for any map Spec L ′ → Y with L ′ a field with image

q, we have φp−1Spec L ′ is surjective (hence an isomorphism) if and onlyif Rpπ∗F is locally free in some neighborhood of q.

Proof. We will reduce this version to the version [Vak, 28.1.6] (whichreplaces our maps Spec L → Y with inclusions of points of Y). Wehave a Cartesian diagram

(3.4)

XSpec L Xq X

Spec L q Y.

iq

t

To complete the proof, it suffices to show φpq is surjective (respec-

tively, an isomorphism) if and only if φpSpec L is surjective (respec-tively, an isomorphism). We just show the surjectivity statement, asthe proof of the isomorphism is nearly the same, mutatis mutandis.

12 AARON LANDESMAN

By flat base change ([Vak, Theorem 24.2.8]) the base change mapfor i∗qF applied to the left square is an isomorphism. Therefore, thebase change map applied to F for the outer rectangle is surjectiveif and only if the map φpq pulled back along t is surjective. Sincepulling back along t is merely a base change of a map of finite di-mensional vector spaces along a field extension, it follows that thepullback of φpq along t is surjective if and only if φpq is surjective. �

We next verify the equivalence of R1f∗(L) being locally free and fcommuting with base change.

Lemma 3.5. Suppose f : C → S is a family of curves of genus g over S,L is a locally free sheaf of degree d on C, with g =

(d−12

), and S is locally

Noetherian. Then, R1f∗L is locally free if and only if φ0q is an isomorphismfor all points q. Further, if R1f∗L is locally free of rank 2− d+ g, then forany map g : S ′ → S with corresponding fiber square

(3.5)

C ′ C

S ′ S.

g ′

f ′ f

g ′

the base change maps φpS ′ are isomorphisms and f∗L is locally free of rank

3.

Proof. First, let us show that R1f∗L is locally free if and only if φ0q isan isomorphism for all points q. Observe that for all p ≥ 2, φpq is anisomorphism for all points q in S because all cohomologies in degree≥ 2 vanish for relative curves. Further, for p ≥ 2, we have Rpf∗L =0, hence it is locally free. Applying these two statements above inthe case p = 2, by cohomology and base change, we obtain that φ1qis an isomorphism for all q. Then, since φ1q is an isomorphism for allpoints q, cohomology and base change implies R1f∗L is locally freeof rank 1 if and only if φ0q is an isomorphism for all q ∈ S.

It remains to prove the second statement. Indeed, note that sinceφ0q is an isomorphism for all points q, and we automatically haveφ−1q is an isomorphism (as negative cohomology groups vanish), we

obtain f∗L is locally free. By cohomology and base change, we knowthat for all maps S ′ → S the base change mapsφp

S ′ are isomorphisms.It remains only to show that the rank of f∗L is 3. For this it suf-

fices to show that for every point q that (f∗L)|q has rank 3. Becausethe map φ0q is an isomorphism, this is equivalent to showing that

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 13

f|q∗(L|q) has rank 3. That is, we want to show h0(C|q,L|q) = 3. ByRiemann-Roch, we know h0(C|q,L|q) − h1(C|q,L|q) = d− g+ 1. So,showing h0(C|q,L|q) = 3 is equivalent to showing h1(C|q,L|q) =

2−d+ g. This is equivalent to showing R1f|q∗L|q has rank 2−d+ g.By cohomology and base change, we know φ1q is an isomorphism,so this is equivalent to showing (R1f∗L)|q has rank 2− d+ g, whichfollows from the assumption that R1f∗L is locally free of rank 2−d+g. �

Using Lemma 3.5, we can immediately deduce the first two partsof the following proposition.

Proposition 3.6. Let (f : S→ C,L) ∈ Pd be an object. Then,(1) For any map g : S ′ → S we may construct the fiber square

(3.6)

C ′ C

S ′ S

g ′

f ′ f

g ′

and corresponding map of objects of Pd (f : C → S,L) → (f ′ :C ′ → S ′,g∗L). Then, the base change map

g∗f∗L → f ′∗g′∗L

is an isomorphism.(2) We have that f∗L is locally free sheaf of rank 3 on S.(3) The map f∗f∗L → L is surjective and the resulting map C →

P(f∗L) is a closed embedding.

Proof. Note that since f is flat and L is locally free, hence flat on C,it follows that L is flat over S. By spreading out, writing S as a limitof its finite type Z-algebras, it suffices to verify the special case thatS is Noetherian. The first two parts then follow immediately fromLemma 3.5.

To conclude, we verify the third part. We start by reducing to thecase that S is the spectrum of an algebraically closed field. As above,by spreading out, we may assume S and S ′ are Noetherian. To checksurjectivity of f∗f∗L → L, it suffices to check surjectivity at all stalks.Further, since L and f∗f∗L are finite type, it suffices to check sur-jectivity at all geometric fibers. This explains why we it suffices tocheck the surjectivity statement when S = Spec k.

Next, we explain why it suffices to check the closed embeddingstatement for S the spectrum of an algebraically closed field. Since a

14 AARON LANDESMAN

morphism is a closed embedding if and only if it is a proper monomor-phism (and being a monomorphism is equivalent to the fiber overeach point having degree 0 or 1, by [Gro67, Proposition 17.2.6]) itsuffices to check the map is a closed embedding over each point.

Summing up what we have just shown, taking the map g : q → Sto be the inclusion of a point, we only need verify g ′∗f∗f∗L → g ′∗Lis surjective and determines a closed embedding. We can write thepreceding map as the composition

g ′∗f∗f∗L ' f ′∗g∗f∗L ' f ′∗f ′∗g ′∗L → g ′∗L,

where the middle map is an isomorphism from cohomology andbase change and the last map is the natural adjunction map. Hence,it suffices to verify that the adjunction map is surjective and deter-mines a closed embedding, which completes the reduction to thecase that S = Spec k, for k an algebraically closed field.

So, we now assume that S = Spec k, and complete the proof. Inthis case, by the second part, f∗L is locally free of rank 3. Since weare assuming C is a plane curve by definition of Pd, it follows formProposition 2.6 that since h0(C,L) ≥ 3, L must be the unique in-vertible sheaf on C which determines a closed embedding C → P2.Further, the resulting map f∗f∗L → L is then a surjection because themap C → P2 is base point free. (In more detail, since f∗f∗L at somepoint p can be identified with non-projectivized local coordinates forp in P2 while L can be identified with the non-projectivized point p.The statement that the map is surjective corresponds to not all coor-dinate functions vanishing at p, which means the map is basepointfree.) �

3.3. The stack of plane curves as a quotient stack. In this section,we prove the map [Ud/ PGL3] → Pd is an isomorphism. The proofconsists of constructing an inverse map and doing a fairly routineverification that the two maps are mutually inverse. However, theproof is fairly lengthy as there are a number of details to verify.

Theorem 3.7. For all d ≥ 1, we have an isomorphism

Pd → [Ud/ PGL3] .

Proof. We prove this in four steps: In subsubsection 3.3.1 we definethe map above. In subsubsection 3.3.2 we define an inverse map.In subsubsection 3.3.3 we show one composition of these maps isnaturally isomorphic to the identity. In subsubsection 3.3.4 we showthe other composition is naturally isomorphic to the identity.

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 15

3.3.1. Defining a map from Pd. We now define a map Pd → [Ud/ PGL3].We define this map on T points, for T a scheme. Say we have somefamily of degree d plane curves (f : C → T ,L) in Pd(T). We want toobtain a corresponding point [Ud/ PGL3]. By definition of the quo-tient stack, this corresponds to the datum

(3.7)

E Ud

T

h

g

with g : E → T a PGL3 torsor and E → U a PGL3 equivariant map.We construct such an E and show it is a PGL3 torsor in Lemma 3.8.We then construct the map h and show it is PGL3 equivariant inLemma 3.9.

Lemma 3.8. The scheme E := IsomT (Pf∗L, PO3T )g−→ T is a principal

PGL3-torsor, where g is the natural projection.

Proof. By From the definition for principal PGL3-torsor given in [Ols16,Definition 4.5.4], it is apparent that conditions (i) and (ii) are satisfied,so we only need check condition (iii), that the map

(ρ,π2) : PGL3×TE→ E×T E(g,p) 7→ (gp,p)

is an isomorphism. To verify this, we may do so Zariski locally.Choose a cover {Ti}i∈I of T so that f∗L is trivial on each Ti. This ispossible since f∗L is locally free of rank 3 by Proposition 3.6, henceisomorphic to O3Ti . Since the construction of proj commutes with basechange, we see that E|Ti ∼= IsomTi(P(f∗L|Ti), PO3Ti). Therefore, uponfixing an isomorphism f∗L|Ti ' O3Ti the above E|Ti can be identi-fied with IsomTi(PO3Ti , PO3Ti)

∼= PGL3 in which case it is clear thatthe action map (ρ,π2) is an isomorphism. Therefore, we obtain thatg : E→ T is a PGL3 torsor. �

Next, we define a map h : E → T , which we will verify is PGL3equivariant in Lemma 3.9 Define the fiber square

(3.8)

CE C

E T

g ′

f ′ f

g

16 AARON LANDESMAN

Define L ′ := g ′∗L. We claim f ′∗L′ is a locally free rank 3 sheaf

on E. To see this, observe that by Proposition 3.6(1), we know themap g∗f∗L → f ′∗L

′ is an isomorphism. Since f∗L is locally free byProposition 3.6, it follows that f ′∗gL ′ is as well. Therefore, the fam-ily (f ′ : CE → E,L ′) defines an element of Pd(E). Further, applyingProposition 3.6(3) to the element (f ′ : CE → E,L ′) ∈ Pd(E), we ob-tain a resulting closed embedding C→ P(f ′∗L

′). This is a flat familywhose geometric fibers are smooth plane curves of degree d by con-struction. Therefore, the universal property of the Hilbert schemedetermines an map from E to the Hilbert scheme of plane curves,which factors through Ud as all geometric fibers are smooth. Callthis map h : E→ Ud.

Lemma 3.9. Further, the resulting map defined above h : E→ Ud is PGL3equivariant.

Proof. We want to check that the diagram

(3.9)

PGL3×Spec ZE PGL3×Spec ZUd

E Ud

id×ρ

h

commutes, where the vertical maps are the multiplication maps. Tosee this, observe that we have a universal family over Ud, call it fd :Cd → Ud. This comes with a universal line bundle Ld on Cd and an

embedding Cd → P(d+22 )−1Ud

. Let h ′ : CE → Cd be the resulting map.Observe that from the definition of the Hilbert scheme, L ′ (definedas g ′∗L, with g ′ as in (3.8),) is isomorphic to h ′∗Ld.

We next claim that cohomology and base change commutes forLd, as we prove in the following sublemma.

Lemma 3.10. Let Ud denote the open subscheme of the Hilbert scheme ofdegree d plane curves corresponding to smooth curves, let Cd → Ud bethe universal family and let Ld denote the universal line bundle. Then,cohomology and base change commutes for Ld.

Proof. Observe that for any point q ∈ P(d+22 )−1Ud

with fiber Cq, weknow Ld|Cq is a locally free sheaf of degree d on Cq and so by Propo-sition 2.6 we know that h0(Cq,Ld|Cq) = 3. By Riemann-Roch, we ob-tain h0(Cq,Ld|Cq) = 2− d+ g for all points q. Since we know Ud is

explicitly an open subscheme of P(d+22 )−1, hence reduced, it follows

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 17

from Grauert’s theorem that R1h∗(Ld) is locally free of rank 2−d+g.Therefore, by Proposition 3.6, cohomology and base change com-mutes for Ld. �

By construction of E, we know E = IsomT (PL ′, PO3T ). The multi-plication maps PGL3×Spec ZUd → Ud and PGL3×Spec ZE → E aregiven by the respective actions of PGL3 on IsomUd(P(fd∗Ld), PO3Ud)

and

IsomE(Pf′∗L′, PO3E) ' IsomE(P(h∗fd∗Ld), P(h∗O3Ud))(3.10)

' E×Ud IsomUd(P(fd∗Ld), PO3Ud).(3.11)

where the first isomorphism is obtained via Lemma 3.10. Using theseidentifications, we claim the diagram (in which the top and bottomsquares are separately Cartesian)

(3.12)

PGL3×Spec ZP2E P2E

PGL3×Spec ZCE CE

PGL3×E E

is the pullback along h : E→ Ud of the diagram

(3.13)

PGL3×Spec ZP2UdP2Ud

PGL3×Spec ZCd Cd

PGL3×Ud Ud.

In more detail, our identifications in (3.10) show that the top row of(3.12) is the pullback along h of the top row of (3.13). Restrictingthis to CE ⊂ P2E and Cd ⊂ P2Ud

tells us the middle row of (3.12) isthe pullback along h of the middle row of (3.13). This finally impliesfrom the universal property of the Hilbert scheme that the bottomrow of (3.12) is the pullback along h of the bottom row of (3.13). It

18 AARON LANDESMAN

follows that that (3.9) is in fact a fiber square. In particular, it com-mutes, as desired. �

By definition of the quotient [Ud/ PGL3], the PGL3 bundle g : E→T and PGL3 equivariant map h : E → Ud determines a map T →[Ud/ PGL3]. All in all, we have determined a map

α : Pd → [Ud/ PGL3]

defined by sending an element of Pd(T) to the element of [Ud/ PGL3] (T)obtained from the PGL3 bundle E = IsomT (Pf∗L, PO3T ).

3.3.2. Defining an inverse map to Pd. We next define a map β inverseto α,

β : [Ud/ PGL3] → Pd.

First, as shown in Lemma 3.10, the universal line bundle Ld on theuniversal family Cd over Ud satisfies cohomology and base change,and therefore determines an element (fd : Cd → Ud,Ld) ∈ Pd(Ud).

Now, for any scheme T , a T point of [Ud/ PGL3] is a principalPGL3-bundle ν : P → T together with a PGL3-equivariant mapε : P → Ud and corresponding fiber square

(3.14)

CP Cd

P Ud.

εC

fP fd

ε

To construct our map β, we will describe how to obtain an element(fT : CT → T ,LT ) ∈ Pd(T)). (Here, we are using that PGL3 is affine,so principal PGL3-bundles are the same as PGL3 torsors, by [Ols16,Proposition 4.5.6].) Since Ld satisfies cohomology and base change,the bundle LP := Ld|CP has R1fP∗LP locally free of rank 2 − d + g,being the pullback along ε of R1fd∗Ld. Note that the map P → T isfppf by definition of principal PGL3-bundle, and the two pullbacksof LP to P ×T P ∼= PGL3(P) (the isomorphism holding because P isa PGL3 torsor over T ) are isomorphic, as follows from PGL3 equiv-ariance of the map ε. Further this descent data satisfies the cocyclecondition. Hence, by fppf descent for polarized families, we knowthat the family (CP → P,LP) descends to a polarized family of curves

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 19

(CT → T ,LT ). We have a fiber square

(3.15)

CP CT

P T

ν ′

fP fT

ν

To complete our definition of β, we only need verify that (fT :CT → T ,LT ) ∈ Pd(T). To see this, we must show that (R1fT )∗LT is lo-cally free of rank 2−d+g. Indeed, this follows from flat base changeapplied to the map ν: Flat base change implies that R1ν∗(fT )∗LT ∼=R1fP∗(ν

′∗LT ) is an isomorphism. Therefore, R1fT∗LT has pullbackalong νwhich is locally free, and hence it is locally free.

3.3.3. Showing β ◦ α ' id. Finally, we note that the two maps wehave constructed in both directions between [Ud/ PGL3] and Pd aremutually inverse.

First, we claim β ◦ α ' id. To verify this, we check it on T points.If we start with a T point of Pd, this is a family C→ T . Following theconstruction of α, we obtain a particular bundle E→ T , and pullbackCE → E, which determines a map E → Ud. Now, we wish to showthat β recovers the family C → T . However, by construction of themap E → Ud, we realized CE → E as the pullback of the universalfamily Cd → Ud, and so we indeed recover CE → E. Then, we alsorecover C → T from the full faithfulness of descent (as the resultingobject CT → T was constructed via descent in the map β).

3.3.4. Showing α ◦ β ' id. To finish the proof of Theorem 3.7, weonly need verify α ◦ β ' id. This will follow if we show the mapβ is fully faithful. To check this, by [Ols16, Proposition 3.1.10], wecan restrict to the case that both morphisms are from the same testscheme T . That is, for x,y ∈ Pd(T), it suffices to show that

homPd(T)(x,y) ' hom[Ud/ PGL3](T)(β(x),β(y)).

We first show β is faithful. To this end, suppose we have twodistinct maps f and g in homPd(T)(x,y). We want to show β(f) 6=β(g). We can think of x,y as the datum of (fx : Cx → T ,Lx) and(fy : Cy → T ,Ly) with a map T → T so that Cy pulls back to Cxand Ly pulls back to some line bundle of the form Lx ⊗ (fx)∗Q forQ an invertible sheaf on T . If the two maps fx, fy are distinct, theinduced map on PGL3 bundles will also be distinct. Therefore, wemay assume the two maps fx, fy agree. So, we may assume T → Tis the identity. Then, two such maps simply correspond to a choice

20 AARON LANDESMAN

of isomorphism between Ly ∼= Lx ⊗ fx∗Q, in which case the twobundles maps were already deemed equivalent in Pd(T). So, β isfaithful.

We next verify β is full. To this end, suppose we have two objectsx,y given by (fx : Cx → T ,Lx) and (fy : Cy → T ,Ly) so that β(x)agrees with β(y). We want to show x agrees with y. If β(x) agreeswith β(y), this means the resulting map T → T is the identity. In thiscase, we suppose that the two PGL3 bundles associated to Lx and Ly

are isomorphic, meaning

IsomT (Pfx∗L

x, PO3T )∼= IsomT (Pf

x∗L

x, PO3T ).

We want to show that Ly ' fx∗(Q)⊗Lx for Q some invertible sheafon T . For this, it suffices to show P(fx∗L

x) ∼= P(fy∗Lx). To show

this, taking a Zariski cover {Ui} → T which simultaneously triv-ializes both line bundles, we see that on each such Zariski open,upon choosing trivializations for the two line bundles, we obtain anisomorphism Pfx∗L

x ∼= Pfy∗L

y given by some gi in PGL3 with re-spect to the chosen trivializations. These gi restrict compatibly andhence determine some element of PGL3 defining an isomorphismP(fx∗L

x) ∼= P(fy∗Ly), as desired.

We have therefore produced an isomorphism [Pd/ PGL3] ∼= Pd,completing the proof of Theorem 3.7. �

Corollary 3.11. For all d ≥ 1, Pd is a smooth algebraic stack.

Proof. From Theorem 3.7, we have produced an isomorphism Pd '[Ud/ PGL3] whereUd is the open subscheme of P(d+22 )−1 which is thelocus of smooth curves the Hilbert scheme of plane curves of degreed. It follows that Pd is an algebraic stack. To show it is smooth, weonly need show [Ud/ PGL3] has a smooth cover by a smooth scheme.Indeed, this follows asUd → [Ud/ PGL3] is such a smooth cover. �

Remark 3.12. At this point, if we wished, we could prove that ford ≥ 3, the stack Pd is in fact Deligne-Mumford. Essentially, thiswould follow because for d ≥ 4 there are only finitely many auto-morphisms of curves and for d = 3, there are only finitely manyautomorphisms of curves which preserve a degree 3 line bundle.However, there is the delicate issue of verifying the isotropy groupsat points are actually etale. Instead, since we will show the mapPd → Mg is a locally closed embedding for d ≥ 4, it will follow thatfor d ≥ 4, Pd is Deligne-Mumford because Mg is.

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 21

4. VERIFYING THAT Pd → Mg IS A LOCALLY CLOSED EMBEDDING

Let g :=(d−12

)Our next goal is to verify that the natural map Pd →

Mg, sending a pair (f : C → T ,L) 7→ (f : C → T) is a locally closedembedding.

We do this in several steps, completing the proof in Theorem 4.5.(1) We show that in the case f : C→ T has a section, the resulting

fiber product T ×Mg Pd is in fact a scheme (in Proposition 4.2).(2) We show that the map Pd → Mg is a monomorphism (in

Proposition 4.4).(3) We show that Pd → Mg is a locally closed embedding, us-

ing the (little known!) valuative criterion for locally closedembedding (in Theorem 4.5).

Recall the definition of the stack Grd(p) for p : C → S a family ofsmooth genus g curves given in Definition 2.9 and recall that it isrepresentable when p has a section, by Theorem 2.10.

Definition 4.1. First, we define F2d(p) to be the subfunctor of G2d(p)which associates to any S scheme T pairs (L,H)/ ∼ as in Defini-tion 2.9 with the additional condition that pT∗L is locally free of rank3 and p∗TpT∗L → L is surjective.

Next, we define K2d(p) to be the subfunctor of F2d(p) which asso-ciates to any S scheme T pairs (L,H)/ ∼ as in Definition 2.9 with theadditional conditions that pT∗L is locally free of rank 3, that and thatthe resulting morphism C→ PpT∗L is a closed embedding.

Proposition 4.2. Suppose p : C→ S is a family of smooth genus g curveswith a section, corresponding to a map S→ Mg Then, we have morphismsS ×Mg Pd → K2d(p) → F2d(p) → G2d(p) with the first map being anisomorphism and the latter two maps being open embeddings. In particular,the fiber product S×Mg Pd is in fact a scheme.

Proof. We will verify that there are maps S ×Mg Pd → K2d(p) →F2d(p) → G2d(p) with the first morphism being an isomorphism andthe latter two morphisms being open embeddings. The last claimwould then follow from Theorem 2.10, since G2d(p) is representableas p has a section.

First, we show that F2d(k) is an open subfunctor of G2d(p). To seethis, we know G2d(p) is representable by some scheme X with a uni-versal invertible sheaf LX on the universal curve pX : CX → X. Itfollows that F2d(p) is then represented by the open subscheme of Xon which pX∗LX is locally free of rank 3 and the map p∗XpX∗LX → LX

22 AARON LANDESMAN

is surjective (which is open because it is the complement of the sup-port of the cokernel, and the support of a sheaf is a closed locus).

Next, we show K2d(p) → F2d(p) is an open embedding. Note thatbecause pT∗L is locally free of rank 3, we may construct the rela-tive proj P(pT∗L) which is isomorphic to P2T . The condition thatp∗TpT∗L → L is surjective implies that the sheaf L is basepoint free,and hence we obtain a resulting projective morphism C→ P(pT∗L).Now, say the functor F2d(p) is represented by some scheme Y witha universal invertible sheaf LY on the universal curve pY : CY → Y,and let φ : CY → PpY∗LY be the resulting closed embedding.

To show K2d(p) → F2d(p) is an open embedding, it suffices to showthere is an open locus over which the map C → P(pY∗L) is a closedembedding. This is intuitively clear because the locus on which itis not a closed embedding should be the one in which the degreeof the fiber jumps. We formally codify this as follows: Then, let Vdenote the cokernel of the resulting map OPpY∗LY → φ∗OCY . LetW := SuppV ⊂ P(pY∗LY) denote the support of V and let Z :=φ−1(W). Then, it follows that Z is a closed subscheme of CY andpY(CY \Z) represents the functor K2d(p).

Finally, we define a map S ×Mg Pd → K2d(p) and verify it is anisomorphism of stacks. First define the map

S×Mg Pd → G2d(p)

(f : C→ T ,L) ∈ Pd(T) 7→ (L,L)/ ∼ .

Observe this factors through K2d(p) by Proposition 3.6.We want to verify the resulting map S ×Mg Pd → K2d(p) is an

equivalence of stacks. We wish to show the map is fully faithful andessentially surjective. Full faithfulness follows tautologically fromthe definition, as it sends a pair (f : C→ T ,L) to essentially the samedata, with the same equivalence relation. That is, the equivalencerelation ∼ in Definition 2.9 is the same as the isomorphism relationgiven in Definition 3.1.

So, to conclude, we only need verify essential surjectivity. For this,we will start by showing that if we have some family pT : C → Tand an invertible sheaf L on C whose pushforward is locally freeof rank 3, then pT∗L has no subsheaves H which are locally free ofrank 3 and satisfy (2.6), other than pT∗L itself. Along the way, wewill also see that for all points q ∈ T , the base change map φ0q isan isomorphism. To verify this, suppose we have such datum. LetCt := t×T CT . Note that pt∗ι∗CH → H0(Ct,L|κ(t)) is an injective map

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 23

between vector spaces. Observe that since we are assuming pT∗L isan embedding, it follows pt∗L is an embedding. This means that infact Ct is a plane curve, as L determines an embedding Ct → P2t . ByProposition 2.6, if Ct is a plane curve, h0(Ct,L|κ(t)) ≤ 3 and so if (2.6)is an injection from a 3-dimensional vector space, it must be an iso-morphism. Now, recall the map (2.6) was defined as the compositionι∗H → ι∗pT∗L → pt∗ι∗CL. Since the composition is surjective, the lat-ter map must be surjective, hence an isomorphism. This implies, bycohomology and base change that the map H → pT∗L is an isomor-phism when restricted to any fiber, hence an isomorphism.

It only remains to show that if pT∗L is locally free then so is R1pT∗L.This will imply the map is essentially surjective. However, sincethe base change map φ0q is an isomorphism for all points q ∈ T ,it follows from Lemma 3.5 that R1pT∗L is locally free. Since the basechange maps commute by Lemma 3.5, it follows from Riemann-Rochthat R1pT∗L is locally free of rank 2− d+ g. �

Now, since Mg is Deligne-Mumford, there is an etale cover T →Mg. This corresponds to a family C → T . After choosing a furtheretale cover of T , we may assume that p : C → T has a section. Inthis case, we know from Theorem 2.10 that the functor G2d(p) is rep-resentable by a scheme. That is, the fiber product

(4.1)

T ×Mg Pd T

Pd Mg

τ

is a scheme, using Proposition 4.2. Hence, in order to verify ourmap Pd → Mg is a locally closed embedding, we only need checkτ is a locally closed embedding. For this, we will use the followingvaluative criterion for locally closed embeddings.

Lemma 4.3 (Valuative criterion for locally closed embeddings, [Moc14,Chapter 1, Corollary 2.13]). Suppose S is a Noetherian scheme and X andY are S-schemes of finite type. A morphism f : X → Y is a locally closedembedding if and only if f is a monomorphism and the following conditionholds: For all valuation rings R with fraction field K and residue field κ,

24 AARON LANDESMAN

and all maps g : Spec R→ Y with commutative diagrams

(4.2)

Spec K X Spec κ X

Spec R Y Spec R Y,

f f

g g

there exists a unique morphism h : T → X making the diagrams

(4.3)

Spec K X Spec κ X

Spec R Y Spec R Y

f f

g

h

g

h

commute.

So, we need to verify the map τ of (4.1) is a monomorphism andthat it satisfies the valuative criterion of (4.3). First, we show it is amonomorphism.

Proposition 4.4. The map τ defined in (4.1) is a monomorphism.

Proof. Using [Gro67, Proposition 17.2.6] in order to verify the nat-ural map τ is a monomorphism, it suffices to verify each fiber ei-ther has degree 0 or 1. This can be verified on geometric points.So, let Spec k → T be some geometric point. Let Ck be the corre-sponding curve over k. We wish to show the fiber over Spec k hasdegree 1. For this, we will check the fiber is reduced and is sup-ported on a single point. From the definition of the stack G2d(p) withp : Ck → Spec k, which has a section as k is algebraically closed, wesee G2d(p) parameterizes the underlying set of g2d’s on C. By Propo-sition 2.11, G2d(p) is a degree one scheme. Since P2d(p) ⊂ G2d(p) isan open subscheme by Proposition 4.2, it follows that P2d(p) also hasdegree at most 1, completing the proof. �

We can now conclude prove our main theorem, by verifying thevaluative criterion for locally closed embeddings holds.

Theorem 4.5. The map Pd → Mg is a locally closed embedding.

Proof. First, it suffices to check after pull back to an etale cover ofMg, and hence it suffices to check the map τ defined in (4.1) is a lo-cally closed embedding. For this, by Lemma 4.3, it suffices to verifythe map is a monomorphism (which follows from Proposition 4.4

THE LOCUS OF PLANE CURVES IN THE MODULI STACK OF CURVES 25

and the valuative criterion for locally closed embeddings. To applyLemma 4.3 we are using that Pd and Mg are both finite type overSpec Z. It is well known that Mg has finite type: this follows fromthe construction of Mg as a quotient of a Hilbert scheme of canoni-cally embedded curves modulo a PGL action. The statement for Pdfollows from Theorem 3.7.

It only remains to verify the valuative criterion for being a locallyclosed immersion. Retaining the notation of Lemma 4.3, the valua-tive criterion can be rephrased in the following way: Let Cκ and CKdenote the restriction of C to the closed and generic fibers of Spec R.We may suppose we have a family of curves f : CR → Spec R whoseclosed and generic fibers are plane curves with the maps to P2K andP2κ given by invertible sheaves LK and Lκ. We want to show there ex-ists a unique invertible sheaf L of degree d on CR so that L|CK = LKand LCκ = Lκ with R1f∗L locally free of rank 2−d+g. First, unique-ness follows the valuative criterion for separatedness, since Pd andMg are both separated, so the map Pd → Mg is separated.

Therefore, it suffices to show there exists a morphism Spec R →Pd, restricting to the given maps from Spec K and Spec κ, which wenow construct. Since R is regular, it follows CR is regular. Therefore,the natural map from Weil divisors to Cartier divisors is an isomor-phism, and hence LK ∼= OCk(DK) for some Weil divisor DK ⊂ CK.Let DR denote the closure of DK inside CR. Let L := OCR(DR). Byconstruction, we know L|CK

∼= LK.Next, we verify L|Cκ

∼= Lκ. Indeed, since f : CR → R is a propermorphism of Noetherian schemes and L is flat over R, we obtain thatthe map q 7→ h0(Cq,L|Cq) is upper semicontinuous. In particular,since h0(CK,L|CK) = 3, we obtain h0(Cκ,L|Cκ) ≥ 3. It follows fromLemma 3.5 that h0(Cκ,L|Cκ) = 3. Then, again using Lemma 3.5,there is a unique invertible sheaf M onCκ with h0(Cκ,M) = 3, whichimplies that both L|Cκ

∼= M and Lκ ∼= M, so L|Cκ∼= Lκ.

To conclude, we only need verify that R1f∗L is locally free of rank2− d+ g. But indeed, we have already shown that h0(CK,L|CK) =

h0(Cκ,L|Cκ) = 3, which implies h1(CK,L|CK) = h1(Cκ,L|Cκ) = 2−d + g. Since Spec R is reduced, it follows from Grauert’s theoremthat f∗L is locally free. �

5. ACKNOWLEDGEMENTS

I thank David Zureick-Brown for running the REU project whichoriginally prompted this question. I thanks Maksym Fedorchuk for

26 AARON LANDESMAN

suggesting the method of showing Pd → Mg is a locally closed em-bedding by showing Pd → G2d is an open embedding and then show-ing G2d → Mg is a locally closed embedding. I thank Anand Patel andJoe Harris for explaining why smooth plane curves have a unique g2dand suggesting other methods of approaching this question. I thankBrian Conrad for pointing out the useful valuative criterion for beinga locally closed embedding. I thank Ravi Vakil and Michael Kemenyfor listening to my argument in detail. I also thank Tony Feng, BenLim, Arpon Raksit, Zev Rosengarten, Bogdan Zavyalov, and YangZhou for helpful discussions.

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