Plane Curves as Pfaffians
Transcript of 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
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
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
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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 .
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
].
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
].
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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 .
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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 .
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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
].
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
Determinantal RepresentationsModuli SpaceExplicit Construction
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- 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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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,
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
The Inverse of Elementary Transformation
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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|.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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|.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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|.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
Ruled Surface ≡ Rank 2 V.B. ≡ Normal Scroll
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
Ruled Surface ≡ Rank 2 V.B. ≡ Normal Scroll
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
Ruled Surface ≡ Rank 2 V.B. ≡ Normal Scroll
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.
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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
.
A. Buckley Plane Curves as Pfaffians
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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).
A. Buckley Plane Curves as Pfaffians
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A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
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.
A. Buckley Plane Curves as Pfaffians
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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.
A. Buckley Plane Curves as Pfaffians
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The idea of the proof 2
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).
A. Buckley Plane Curves as Pfaffians
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The idea of the proof 2
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).
A. Buckley Plane Curves as Pfaffians
Pfaffian RepresentationsElementary Transformations
Plane QuarticGeneralisations to HyperPfaffians
- of Pfaffian Representations- of Vector Bundles- of the Cokernel BundleBridgeing Pfaffian Representations
The idea of the proof 2
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).
A. Buckley Plane Curves as Pfaffians
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Plane QuarticGeneralisations to HyperPfaffians
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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).
A. Buckley Plane Curves as Pfaffians
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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).
A. Buckley Plane Curves as Pfaffians
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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).
A. Buckley Plane Curves as Pfaffians
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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.
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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.
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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 .
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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
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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.
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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.
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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.
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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
.
A. Buckley Plane Curves as Pfaffians
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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)
A. Buckley Plane Curves as Pfaffians
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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)
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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 ).
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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 .
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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.
A. Buckley Plane Curves as Pfaffians
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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.
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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 ).
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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.
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Definition: Pfaffian
Let A = [aij ] be a 2n × 2n skew–symmetric matrix
a1 2 a1 3 · · · a1 2na2 3 a2 2n
. . ....
a2n−1 2n
.
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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).
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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.
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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.
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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.
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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
.
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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.
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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 ± · · ·
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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.
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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 .
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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).
A. Buckley Plane Curves as Pfaffians