Plane Curves as Pfaffians

Pfaffian RepresentationsElementary Transformations

Plane QuarticGeneralisations to HyperPfaffians

Plane Curves as Pfaffians

Anita Buckley1

1Department of MathematicsFaculty of Mathematics and Physics

University of LjubljanaSlovenia

Workshop GeoLMI, Toulouse, FranceNovember 19-20, 2009

Partly joint work with Tomaž Košir1

A. Buckley Plane Curves as Pfaffians



Outline1 Pfaffian Representations

Determinantal RepresentationsModuli SpaceExplicit Construction

2 Elementary Transformations- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations

3 Plane QuarticTheta CharacteristicAronhold Bundles

4 Generalisations to HyperPfaffiansPfaffiansHyperPfaffians





Notation

k algebraically closed fieldF (x0, x1, x2) homogeneous polynomial of degree dC a smooth curve in P2 defined by F

Find a 2d × 2d skew-symmetric matrix

A =

0 L1 2 L1 3 · · · L1 2d

−L1 2 0 L2 3 · · · L2 2d−L1 3 −L2 3 0

......

. . ....

−L1 2d −L2 2d · · · 0

with linear forms Lij = a0

ij x0 + a1ij x1 + a2

ij x2 such that

Pf A(x0, x1, x2) = c F (x0, x1, x2) for some c ∈ k , c 6= 0.





Definition: Pfaffian representation

Matrix A is called linear pfaffian representation of C.Two pfaffian representations A and A′ are equivalent if thereexists X ∈ GL2d(k) such that

A′ = XAX t .

Its cokernel is a rank 2 vector bundle on C. Throughout thepaper we identify vector bundles with locally free sheaves.

A locally free sheaf E of rank 2 is stable if for every invertiblesheaf E → F → 0 holds

degF >12

deg E .

Replacing > by ≥ defines semistable.A. Buckley Plane Curves as Pfaffians






A′ = XAX t .



degF >12

deg E .







A′ = XAX t .



degF >12

deg E .





Definition: Determinantal representation

Study of pfaffian representations is strongly related to andmotivated by determinantal representations. A lineardeterminantal representation of C is a d × d matrix of linearforms

M = x0M0 + x1M1 + x2M2, M0,M1,M2 ∈ Md(k)

satisfyingdet M = c F , c ∈ k , c 6= 0.

Two determinantal representations M and M ′ are equivalent ifthere exist X ,Y ∈ GLd(k) such that

M ′ = XMY .





Determinantal representation ↔ Cokernel line bundle

Theorem (Beauville, 2000)Let C be a plane curve defined by a polynomial F of degree dand let L be a line bundle of degree 1

2d(d − 1) on C withH0(C,L(−1)) = 0. Then there exists a d × d linear matrix Mwith det M = F and an exact sequence

0 →d⊕

i=1

OP2(−1)M−→

d⊕i=1

OP2 → L → 0. (1)

Conversely, let M be a linear d × d matrix with det M = F. Thenits cokernel is a line bundle of degree 1

2d(d − 1) andH0(C,Coker M(−1)) = 0.





Jacobian Variety

Corollary (Vinnikov, 1989)All linear determinantal representations of F (up toequivalence) can be parametrised by the nonexceptional pointson the Jacobian variety of C.

Analogy: parametrise all linear pfaffian representations by points inan open subset of the moduli space MC(2,KC).





Jacobian Variety

Corollary (Vinnikov, 1989)All linear determinantal representations of F (up toequivalence) can be parametrised by the nonexceptional pointson the Jacobian variety of C.

Analogy: parametrise all linear pfaffian representations by points inan open subset of the moduli space MC(2,KC).





Definition: Moduli Space

DefinitionThe moduli space MC(2,KC) consists of semistable rank 2vector bundles on C with canonical determinant.

It is an irreducible, normal projective variety and for C ofgenus g ≥ 2 it has dimension 3(g − 1).





Definition: Moduli Space

DefinitionThe moduli space MC(2,KC) consists of semistable rank 2vector bundles on C with canonical determinant.

It is an irreducible, normal projective variety and for C ofgenus g ≥ 2 it has dimension 3(g − 1).





Pfaffian representation ↔ Cokernel rank 2 bundle

Theorem (Beauville, 2000)Let C be a smooth plane curve defined by a polynomial F ofdegree d and let E be a rank 2 bundle on C with determinantOC(d − 1) and H0(C, E(−1)) = 0. Then there exists a 2d × 2dskew-symmetric linear matrix A with Pf A = F and an exactsequence

0 →2d⊕i=1

OP2(−1)A−→

2d⊕i=1

OP2 → E → 0. (2)

Conversely, let A be a linear skew-symmetric 2d × 2d matrixwith Pf A = F. Then its cokernel is a a rank 2 bundle withdet E ∼= OC(d − 1) and H0(C, E(−1)) = 0.





Moduli Space

CorollaryAll linear pfaffian representations of C (up to equivalence) canbe parametrised by the open setMC(2,KC)− {K : h0(C,K) > 0}.

An explicit construction of all representations (from theglobal sections of rank 2 vector bundles with certainproperties) yields an explicit description of the modulispace.





Moduli Space

CorollaryAll linear pfaffian representations of C (up to equivalence) canbe parametrised by the open setMC(2,KC)− {K : h0(C,K) > 0}.

An explicit construction of all representations (from theglobal sections of rank 2 vector bundles with certainproperties) yields an explicit description of the modulispace.





Determinantal representation ↔ decomposablePfaffian

There are many more pfaffian than determinantalrepresentations: every determinantal representation Minduces a decomposable pfaffian representation[

0 M−M t 0

].

Note that the equivalence relation is well defined since[0 XMY

−(XMY )t 0

]=

[X 00 Y t

] [0 M

−M t 0

] [X t 00 Y

].





Determinantal representation ↔ decomposablePfaffian

There are many more pfaffian than determinantalrepresentations: every determinantal representation Minduces a decomposable pfaffian representation[

0 M−M t 0

].

Note that the equivalence relation is well defined since[0 XMY

−(XMY )t 0

]=

[X 00 Y t

] [0 M

−M t 0

] [X t 00 Y

].





Show Picture





Rank of Pfaffian Representation

Lemma

For any x ∈ C the corank of A(x) equals 2.

Denote by Pfij A the pfaffian of the (2d − 2)× (2d − 2)skew-symmetric matrix obtained by removing the i th and j throws and columns from A. Then

∂F∂xk

(x) =1c

∑i,j

akij Pfij A(x).

If for some x ∈ C all 2d − 2 pfaffian minors vanish, then x mustbe a singular point of F . Our F is smooth, thusrank A(x) ≥ 2d − 2 for all x ∈ C. The rank of skew-symmetricmatrices is even and det A = F 2 = 0.





Rank of Pfaffian Representation

Lemma

For any x ∈ C the corank of A(x) equals 2.

Denote by Pfij A the pfaffian of the (2d − 2)× (2d − 2)skew-symmetric matrix obtained by removing the i th and j throws and columns from A. Then

∂F∂xk

(x) =1c

∑i,j

akij Pfij A(x).

If for some x ∈ C all 2d − 2 pfaffian minors vanish, then x mustbe a singular point of F . Our F is smooth, thusrank A(x) ≥ 2d − 2 for all x ∈ C. The rank of skew-symmetricmatrices is even and det A = F 2 = 0.





Pfaffian Adjoint

DefinitionThe pfaffian adjoint of A is the skew-symmetric matrix

A =

0

. . . (−1)i+j Pfij A. . .

0

.

By analogy with determinants the following holds

A · A = Pf A · Id2d .





Pfaffian Adjoint

DefinitionThe pfaffian adjoint of A is the skew-symmetric matrix

A =

0

. . . (−1)i+j Pfij A. . .

0

.

By analogy with determinants the following holds

A · A = Pf A · Id2d .





Construction of the Pfaffian Representation

TheoremLet C be a smooth plane curve of degree d. To every rank 2vector bundle E on C with properties

(i) h0(C, E) = 2d,(ii) H0(C, E(−1)) = 0,(iii) det E =

∧2 E = OC(d − 1)

we can assign a pfaffian representation A with cokernel E . Inparticular, isomorphic bundles induce equivalentrepresentations.





Construction: Proof

Choose a basis {s1, . . . , s2d} for U = H0(C, E) and define

C 3 xψ7→

∑1≤i<j≤2d

(si(x)∧sj(x))(Eij−Eji) =

0

. . . si(x) ∧ sj(x). . .

0

.





Construction: Proof

Since si ∧ sj ∈∧2 U, by property (iii) the map ψ extends to

Ψ: P2 −→ P(2∧

U)

given by a linear system of plane curves of degree d − 1. Incoordinates it equals to a 2d × 2d skew-symmetric matrixB(x0, x1, x2) with entries from the space of homogeneouspolynomials of degree d − 1. This means that

A =1

F d−2 B.





Canonical Form 1

Proposition

For every pfaffian representation A = x0A0 + x2A2 + x2A2 of Cthere exists a basis of k2d in which A has the canonical form

A = x1

I 0 · · · 00 I · · · 0...

. . ....

0 0 · · · I

− x2

D1 0 · · · 00 D2 · · · 0...

. . ....

0 0 · · · Dd

+ x0A0,

where

I =

[0 1−1 0

]and Di =

[0 pi−pi 0

].





Canonical Form 1: Proof

We can always assume that after a projective change ofcoordinates C intersects the line L : x0 = 0 in distinct pointsP1 = (p1,1,0), . . . ,Pd = (pd ,1,0). By restricting to L, we obtainthe pencil of skew-symmetric matrices

AL = x1A1 + x2A2

with Pf AL = F |L = F (0, x1, x2) =∏d

i=1(x1 − pix2). Note thatE(Pi) is the kernel=cokernel of piA1 + A2. ThusE(Pi), i = 1, . . . ,d are 2-dimensional subspaces in k2d .Condition h0(C, E(−1)) = 0 implies the following: the union ofbases of the vector spaces E(Pi) span the whole space k2d . Inthis basis AL is equivalent to the canonical form above.





Canonical Form 2

Proposition

Another canonical form is

A = x1

[0 Id− Id 0

]− x2

[0 D−D 0

]+ x0A0,

where D is the diagonal matrix {p1, . . . ,pd}.

This canonical form is particularly useful since it naturallyincludes all the decomposable representations. The samecanonical form was obtained by Lancaester and Rodman(2005), where canonical forms for matrix pairs wereclassified purely by the methods of linear algebra.





Canonical Form 2: Proof

The equivalence relation action Q A Qt of

Q =

1 −1 0 0 · · · 00 0 1 −1 0...

. . ....

0 0 · · · 1 −10 1 0 0 · · · 00 0 0 1 0...

. . ....

0 0 · · · 0 1

brings the first two matrices in the Canonical form 1 into theCanonical form 2.




- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations

Notation

The standard notation of vessels will be used (Ball andVinnikov, 1999):

move to affine coordinates (x0, x1, x2) ≡ (1, y1, y2),Pf(y1σ2 − y2σ1 + γ) = c f (y1, y2), where σ1, σ2, γ are2d × 2d skew-symmetric matrices and 0 6= c ∈ k ,E(y1, y2) := Coker(y1σ2− y2σ1 + γ) ∼= ker(y1σ2− y2σ1 + γ).





Admissible Vectors

λ = (λ1, λ2) and µ = (µ1, µ2) distinct regular points on C.

For all vλ ∈ E(λ), uµ ∈ E(µ)

v tλ(λ1σ2 − λ2σ1 + γ)uµ = 0 and v t

λ(µ1σ2 − µ2σ1 + γ)uµ = 0,

implies (λ1 − µ1)v tλσ2uµ = (λ2 − µ2)v t

λσ1uµ. In other words, forany pair of complex parameters t1, t2,

Kvλuµ :=1

t1(λ1 − µ1) + t2(λ2 − µ2)v tλ(t1σ1 + t2σ2)uµ.

is constant whenever the denominator is 0.





Admissible Vectors

DefinitionThe pair of vectors vλ ∈ E(λ), uµ ∈ E(µ) is called admissible ifKvλuµ is not 0.

For an admissible pair of vectors write:γ = γ − 1

2 Kvλuµσ1uµ ∧ σ2vλ + 1

2 Kvλuµσ2uµ ∧ σ1vλ

γ = γ + ρ σ2vλ ∧ σ1vλ, for arbitrary constant ρ 6= 0which are clearly skew–symmetric matrices,





Elementary Transformations of y1σ2 − y2σ1 + γ

Definition

The Type I elementary transformation y1σ2 − y2σ1 + γ based onthe admissible vectors vλ ∈ E(λ), uµ ∈ E(µ),The Type II elementary transformation y1σ2 − y2σ1 + γ basedon vλ ∈ E(λ) and the constant ρ 6= 0.

Theoremy1σ2 − y2σ1 + γ and y1σ2 − y2σ1 + γ are pfaffianrepresentations of C sincePf(y1σ2−y2σ1 + γ) = Pf(y1σ2−y2σ1 +γ) = Pf(y1σ2−y2σ1 + γ).





The Inverse of Elementary Transformation

The fact that vλ ∈ E(µ), uµ ∈ E(λ) and vλ ∈ E(λ) implies thefollowing

CorollaryThe Type I elementary transformation of y1σ2 − y2σ1 + γ basedon uµ ∈ E(λ), vλ ∈ E(µ) brings us back to y1σ2 − y2σ1 + γ. Thesame way the Type II elementary transformation ofy1σ2 − y2σ1 + γ based on vλ ∈ E(λ) and −ρ brings us back toy1σ2 − y2σ1 + γ.

The Type I and II elementary transformations are special rank 2cases of "the concrete interpolation problem for meromorphicbundle maps" studied by Ball and Vinnikov, 1999.






The fact that vλ ∈ E(µ), uµ ∈ E(λ) and vλ ∈ E(λ) implies thefollowing

CorollaryThe Type I elementary transformation of y1σ2 − y2σ1 + γ basedon uµ ∈ E(λ), vλ ∈ E(µ) brings us back to y1σ2 − y2σ1 + γ. Thesame way the Type II elementary transformation ofy1σ2 − y2σ1 + γ based on vλ ∈ E(λ) and −ρ brings us back toy1σ2 − y2σ1 + γ.

The Type I and II elementary transformations are special rank 2cases of "the concrete interpolation problem for meromorphicbundle maps" studied by Ball and Vinnikov, 1999.





Dfinition: Elem. Transf. of Vector Bundles

Definition (Maruyama, 1973; Abe, 2007)Let E be a rank 2 vector bundle over C. Take an effectivereduced divisor Z on C and consider the canonical surjection

E → k(Z ) → 0,

where k(Z ) is a skyscraper sheaf at Z , i.e. rank 1 OZ –module.Its kernel is a rank 2 vector bundle on C called the elementarytransformation of E at Z . We denote it by E ′ = elemZ (E).






There exists a skyscraper sheaf k(Z )′ that fits into thecommutative diagram

E ⊗ OC(−Z )↓g

E ′ e−→ E → k(Z ) .↓

k(Z )′

Up to tensoring line bundles, i.e. on the level of ruled surfaces,these two elementary transformations are inverse to each other.





Ruled Surface ≡ Rank 2 V.B. ≡ Normal Scroll

On C it is equivalent to consider:(a) Ruled surface π : S → C together with a base-point-free

unisecant complete linear system |H|;(b) Rank 2 vector bundle E over C for which S = PE and

E ∼= π∗OPE(H);(c) Linearly normal scroll R obtained as the image of the

birational map φH : S → R ⊂ PN defined by |H|.
























Analogously we can define elementary transformation at apoint x ∈ C on each of the above:(a) On the ruled surface S we choose a point s ∈ π−1(x).

Denote by B the blow-up of S at s. By Castelnuovotheorem we can contract the starting fibre π−1(x) in B andobtain a new ruled surface π′ : S′ → C;

(b) E ′ = elem{x}(E) at the divisor {x} on C;(c) Pick a point r = φH(s) on the scroll R such that π(s) = x .

Projection from r yields a scroll R′ ⊂ PN−1.























Pfaffian Representation ↔ Cokernel Bundle

Theorem

Let C be defined by F = Pf(x1σ2 − x2σ1 + x0γ) and letx1σ2 − x2σ1 + x0γ, x1σ2 − x2σ1 + x0γ be elementarytransformations of Type I and II respectively. Denote byE(x), E(x), E(x) the corresponding cokernels. The relatingmorphisms P,S,T ,R,Q can be expressed by elementarytransformations of vector bundles,

ES ↑ Rt ↓

E ′P ↑ T t ↓

E

and Q

xE↓E ′′↑E

.





Matrices with rational elements:

T (x) = Id +x0

Kvλuµ (t1(x1 − λ1x0) + t2(x2 − λ2x0))(t1σ1 + t2σ2)uµv t

λ,

S(x) = Id +x0

Kvλuµ (t1(x1 − λ1x0) + t2(x2 − λ2x0))uµv t

λ(t1σ1 + t2σ2),

P(x) = Id +x0

Kvλuµ (t1(x1 − µ1x0) + t2(x2 − µ2x0))vλut

µ(t1σ1 + t2σ2),

R(x) = Id +x0

Kvλuµ (t1(x1 − µ1x0) + t2(x2 − µ2x0))(t1σ1 + t2σ2)vλut

µ

and Q(x) = Id +2ρx0

t1(x1 − λ1x0) + t2(x2 − λ2x0)vλv t

λ(t1σ1 + t2σ2).

S(x)P(x) E(x) = E(x) and Q(x) E(x) = E(x).





Show Picture





Bridgeing Pfaffian Representations

Theorem

From any given pfaffian representation of C we can build all thenonequivalent pfaffian representations of C by finite sequencesof Type I and Type II elementary transformations.





The idea of the proof 1

Step 1: bridge the cokernel with a decomposable vectorbundle by applying a finite number of Type II elementarytransformations.A finite sequence of m Type II elementary transformations byrecursion yields a new representation x1σ2 − x2σ1 + x0γm,where γm = γ +

∑mj=1 ρj σ2vλj ∧ σ1vλj . The above constants

ρj ∈ k and points λj ∈ C are arbitrary andvλj ∈ Ej−1(λ

j) := Coker(λj1σ2 − λj

2σ1 + λj0γj−1) with γ0 = γ.

Since every union {Ej(x)}x∈C spans the whole k2d , we can (bysuitable choices of vλj ) generate enough independent rank 2matrices σ2vλj ∧ σ1vλj whose linear combination will yield adecomposable matrix γm.






Step 2: bridge any two decomposable cokernels by applying afinite number of Type I elementary transformations.

decomposable cokernel bundles ≡decomposable pfaffian representations ≡determinantal representations

Vinnikov, 1990: any two determinantal representations can bebridged by a a finite sequence of elementary transformations.Induction: Type I elementary transformation based on[

vn0

]∈ dEn−1(λ

n),

[0un

]∈ dEn−1(µ

n).









vn0

]∈ dEn−1(λ

n),

[0un

]∈ dEn−1(µ

n).









vn0

]∈ dEn−1(λ

n),

[0un

]∈ dEn−1(µ

n).




Theta CharacteristicAronhold bundles

Introduction

DefinitionA nonsingular plane quartic C is a non-hyperelliptic genus 3curve embedded by its canonical linear system |KC |.

The moduli space MC(2,OC(1)) ∼= MC(2,OC) can be embededas a Coble quartic hypersurface in P7 with singularities alongthe 3–dimensional Kummer variety KC .

Using the canonical pfaffian representations of C, we canexplicitly parametrise MC(2,OC(1)) \ Θ2,OC(1).





Introduction








Introduction








Even Theta Characteristic

DefinitionAn even theta characteristic of C is a line bundle Lϑ with theproperty

L⊗2ϑ∼= ωC

∼= OC(1) and dim H0(C,Lϑ)is even.

Dolgachev: There are exactly 36 even theta characteristicson a smooth plane quartic, all with dim H0(C,Lϑ) = 0.





Even Theta Characteristic

DefinitionAn even theta characteristic of C is a line bundle Lϑ with theproperty

L⊗2ϑ∼= ωC

∼= OC(1) and dim H0(C,Lϑ)is even.

Dolgachev: There are exactly 36 even theta characteristicson a smooth plane quartic, all with dim H0(C,Lϑ) = 0.





Even Theta Characteristic ≡ Symmetric DeterminantalRepresentation

For a line bundle L on a nonsingular plane quartic C withH0(C,L) = 0 the following are equivalent:

L is an even theta characteristics on C,L ∼= L−1 ⊗OC(1),L = Coker M ⊗OC(−1) where M is a symmetricdeterminantal representation of C with the propertyCoker M ∼= Coker M t .





Example

27x30 x1 − 432x0x3

1 − x41 − 72 1071/3x2

0 x1x2−9 1071/3x0x2

1 x2 + 81 107−1/3x20 x2

2 − 108x0x32 − 27x1x3

2

Mϑ = x1 Id4−x2 Diag [0,−3,3(−1)1/3,−3(−1)2/3]+

x0

4 −24.296‘ 23.685‘+0.336‘i −23.685‘+0.336‘i

4283 −1071/3 −141.449‘+2.004‘i 141.449‘+2.004‘i

4283 −1071/3(−1)2/3 −145.099‘

4283 +1071/3(−1)1/3





Definition: the Scorza Map

DefinitionThe Scorza map between plane quartics

F 7→ the Clebsch covariant quartic S(F )

is

F 7→ polar cubic Px(F ) at x ∈ P2 7→ Aronhold invariant(Px(F )) .

Note that in this notation the coefficients wijk of the cubicP(x0,x1,x2)(F ) are linear in x0, x1, x2.





Definition: the Scorza Map

DefinitionThe Scorza map between plane quartics

F 7→ the Clebsch covariant quartic S(F )

is

F 7→ polar cubic Px(F ) at x ∈ P2 7→ Aronhold invariant(Px(F )) .

Note that in this notation the coefficients wijk of the cubicP(x0,x1,x2)(F ) are linear in x0, x1, x2.





The Scorza Map

Theorem (Dolgachev and Kanev, 1993)

The curve S(F ) carries a unique even theta characteristic ϑ,more precisely, the Scorza map

F 7→ (S(F ), ϑ)

is an injective birational isomorphism and the natural projectionto the first component is an unramified covering of degree 36.





Aronhold Invariant

Ottaviani, 2009: The Aronhild invariant evaluated in

w000x3 + w111y3 + w222z3 + 6w012xyz +

3w001x2y + 3w002x2z + 3w011xy2 + 3w022xz2 + 3w112y2z + 3w122yz2

equals Pf Ar of the Aronhold pfaffian representation

Ar =

0 w222 −w122 0 w112 0 w022 −w0120 w022 w122 −w012 −w022 0 w002

0 −w112 0 w012 −w002 00 −w111 0 −w012 w011

0 −w011 w001 00 w002 −w001

0 w0000

.





Aronhold Bundles

Pauly, 2002:The Aronhold bundles (cokernels of Aronholdrepresentations) are in 1-to-1 correspondence with the 288unordered Aronhold sets of bitangents on C.

They are stable (thus indecomposable) rank 2 bundle withcanonical determinant OC(1)





Aronhold Bundles

Pauly, 2002:The Aronhold bundles (cokernels of Aronholdrepresentations) are in 1-to-1 correspondence with the 288unordered Aronhold sets of bitangents on C.

They are stable (thus indecomposable) rank 2 bundle withcanonical determinant OC(1)





Aronhold Bundle ↔ Theta Characteristic

Proposition

From the Aronhold pfaffian representation of S(F ) it is possibleto explicitly recover the unique theta characteristic on S(F ).





Proof

The Scorza correspondence is

Rϑ := {(λ, µ) ∈ S(F )× S(F ) : h0(ϑ+ λ− µ) > 0}.

λRϑ µ iff v t Mϑ(x) u ≡ 0 for allv ∈ Coker Mϑ(λ), u ∈ Coker Mϑ(µ), x ∈ P2.λ, µ ∈ S(F ) are related if the second polar Pλ,µ(F ) = g2

i forsome i = 1,2,3 such that Pλ(F ) = g3

1 + g32 + g3

3 .





Example

The Scorza map sends F = x4 + x3y − y4 − yz3 + 1071/3xy2zto S(F ) = Pf[Ar ] given in our first Example. We will computethe unique theta characteristic on S(F ). We calculate inWolfram Mathematica to precision 10−10.





Example

For λ = (1,0, 34107−1/3) ∈ S(F ) we get Pλ(F ) = g3

1 + g32 + g3

3for

g1 = (4x + y)(−0.198‘− 0.344‘i),g2 = (0.002‘− 2.089‘i)y + (−0.181‘ + 0.104‘i)z,g3 = (0.002‘ + 2.089‘i)y + (−0.181‘− 0.104‘i)z,

which is explicitly obtained from the equalitydet Hess (Pλ(F )) = g1g2g3.





Example

The intersections

µ1 = g2 ∩ g3 = (1,0,0),µ2 = g1 ∩ g3 = (1,−4,−20.034‘ + 34.609‘i),µ3 = g1 ∩ g2 = (1,−4,−20.034‘− 34.609‘i)

determine the polar triangle R(λ) of λ. This proves that λ is inrelation with µ1, µ2 and µ3 on S(F ).





Example

On the other hand it is easy to compute all the 36 symmetricdeterminantal representations of S(F ). For Mϑ in Example 1we have

vλ = [−0.006−0.009i,−0.335−0.482i,−0.571+0.04i,−0.236+0.521i]t ∈ Coker Mϑ(λ),vµ1 = [0,−0.543−0.164i,−0.419+0.404i,0.124+0.569i]t ∈ Coker Mϑ(µ

1),

vµ2 = [0.602−0.73i,−0.186−0.025i,−0.124+0.147i,0.062+0.172i]t ∈ Coker Mϑ(µ2),

vµ3 = [0.613+0.72i,−0.185+0.028i,−0.059+0.173i,0.127+0.145i]t ∈ Coker Mϑ(µ3).

We check that

v tλ Mϑ(x0, x1, x2) vµi = 0 for any (x0, x1, x2) ∈ P2, i = 1,2,3.

This proves that the corresponding Lϑ is the unique thetacharacteristic on S(F ) that we were looking for.




PfaffiansHyperPfaffians

Definition: Pfaffian

Let A = [aij ] be a 2n × 2n skew–symmetric matrix

a1 2 a1 3 · · · a1 2na2 3 a2 2n

. . ....

a2n−1 2n

.





Definition: Pfaffian

Consider permutations

P2n := {σ ∈ S2n : σ(2i − 1) < σ(2i) and σ(2i − 1) < σ(2i + 1)} .

DefinitionThe Pfaffian of A is

Pf(A) =∑σ∈P2n

sgn(σ) aσ(1)σ(2) · . . . · aσ(2n−1)σ(2n)

=1

2nn!

∑σ∈S2n

sgn(σ) aσ(1)σ(2) · . . . · aσ(2n−1)σ(2n).





Alternative Definition: Pfaffian

To A one can associate a bivector

ω =∑i<j

aij ei ∧ ej ,

where e1,e2, . . . ,e2n is the standard basis of k2n.

DefinitionThe Pfaffian of A is given by

1n!ω ∧ · · · ∧ ω︸︷︷︸

n

= Pf(A) e1 ∧ e2 ∧ · · · ∧ e2n.





Hyper Matrix

An r -dimensional rn × · · · × rn matrix A = [ai1...ir ]1≤ij≤rn isskew–symmetric, if for any permutation σr of r elementsi1, . . . , ir holds

ai1...ir = sgn(σr ) aσr (i1)...σr (ir ).

A can be presented by the upper r -tetraheder, obtained asthe intersection of r − 1 diagonal hyperplanes.





Hyper Matrix

An r -dimensional rn × · · · × rn matrix A = [ai1...ir ]1≤ij≤rn isskew–symmetric, if for any permutation σr of r elementsi1, . . . , ir holds

ai1...ir = sgn(σr ) aσr (i1)...σr (ir ).

A can be presented by the upper r -tetraheder, obtained asthe intersection of r − 1 diagonal hyperplanes.





Definition: HyperPfaffian

Consider the permutations

Prn :=

{σ ∈ Srn :

σ(ri − r + 1) < · · · < σ(ri) for 1 ≤ i ≤ n, andσ(ri − r + 1) < σ(ri + 1) for 1 ≤ i ≤ n − 1

}.

DefinitionFor a skew–symmetric A, define the HyperPfaffian by

HyPf A =1

(r !)n n!

∑σ∈Srn

sgn(σ) aσ(1)...σ(r) · . . . · aσ(rn−r+1)...σ(rn)

=

{ ∑σ∈Prn

sgn(σ) aσ(1)...σ(r) · . . . · aσ(rn−r+1)...σ(rn), if r even0, if r odd ,n > 1

.





Alternative Definition: HyperPfaffian

The wedge product in∧

V is anticommutative:v1 ∧ v2 = (−1)l1l2 v2 ∧ v1, for v1 ∈

∧l1 V , v2 ∈∧l2 V .

This gives another description of A and HyPf A in the standardbasis e1,e2, . . . ,ern of k rn:

A corresponds to an r–vectorω =

∑i1<···<ir ai1...ir ei1 ∧ · · · ∧ eir ∈

∧r k rn

the HyperPfaffian is given by the equation

1n!ω ∧ · · · ∧ ω︸︷︷︸

n

= HyPf(A) e1 ∧ e2 ∧ · · · ∧ ern.





Examples

A 3n × 3n × 3n matrix A = [aijk ]1≤i,j,k≤3n withaijk = ajki = akij = −ajik = −akji = −aikj can be presentedby the upper tetraheder, obtained as intersection of twodiagonal planes.In a 9× 9× 9 matrix A(3·3)3 the HyperPfaffian contains 280cubic monomials that sum into 0:

HyPf A(3·3)3 =a123a456a789 − a123a457a689 + · · ·+ a148a236a579 ± · · ·





Examples

In a 8× 8× 8× 8 matrix HyPf A(4·2)4 =

a1234a5678 − a1235a4678 + a1236a4578 − a1237a4568 + a1238a4567−a1245a3678 + a1246a3578 − a1247a3568 + a1248a3567−a1256a3478 + a1257a3468 − a1258a3467 + a1267a3458 − a1268a3457 + a1278a3456−a1345a2678 + a1346a2578 − a1347a2568 + a1348a2567−a1356a2478 + a1357a2468 − a1358a2467 + a1367a2458 − a1368a2457 + a1378a2456−a1456a2378 + a1457a2368 − a1458a2367 + a1467a2358 − a1468a2357 + a1478a2356−a1567a2348 + a1568a2347 − a1578a2346 + a1678a2345.





Sub–HyperPfaffian HyPfi1...ir

DefinitionGiven an r–dimensional skew–symmetric matrix A of sizern × · · · × rn, let Ai1...ir denote the r–dimensionalskew–symmetric matrix of size r(n − 1)× · · · × r(n − 1)obtained from A by removing all aj1...jr for which at least one ofjl ∈ {i1, . . . , ir}. Denote the HyperPfaffian of Ai1...ir by HyPfi1...ir .





Interesting Questions

For a fixed integer j , 1 ≤ j ≤ rn, prove that

HyPf A =∑

j∈{i1<···<ir}

ai1...ir HyPfi1...ir .

Define the adjoint of A via HyPfi1...ir .Show that A ∗ adj A = HyPf A · Idrn × · · · × rn︸︷︷︸

2(r−1)

.

What does it mean for adj A to have rank r?Use other immanants instead of sgn (σ).


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Plane Curves as Pfaffians

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