Moduli of Abelian varieties, Vinberg θ-groups, and free resolutions
Laurent Gruson

Steven V Sam

Jerzy Weyman

arXiv:1203.2575v2 [math.AG] 21 Jan 2013

January 21, 2013

Dedicated to David Eisenbud on the occasion of his 65th birthday.
Abstract
We present a systematic approach to studying the geometric aspects of Vinberg θ-representations.
The main idea is to use the Borel–Weil construction for representations of reductive groups as
sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci
of these vector bundles. Our main technical tool is to use free resolutions as an “enhanced”
version of degeneracy loci formulas. We illustrate our approach on several examples and show
how they are connected to moduli spaces of Abelian varieties. To make the article accessible
to both algebraists and geometers, we also include background material on free resolutions and
representation theory.

Contents
3
8
17
20
25
28
33
34
36
39

1 Free resolutions.
2 Representation theory.
3 Some geometry.
V
4 C5 ⊗ 2 C5 .
V3 9
C .
5
V4 8
C .
6
V4 8
7
0C .
8 spin(16).
9 C4 ⊗ spin(10).
References

Introduction.
The contents of these notes sit at the crossroads of representation theory, algebraic geometry, and
commutative algebra, so we will explain each of these perspectives on our work before getting into
the details.
From representation theory, we are considering the problem of classifying orbits in Vinberg
θ-representations (G, U ) [Vin1]. From the point of view of (geometric) invariant theory these are
the representations that are the simplest. One naturally gets a representation from a Z-grading of
a Kac–Moody algebra g and so one naturally gets a trichotomy of these representations according
to the structure of g: finite type, affine type, and wild type. The θ-representations come from finite
and affine type. The θ-representations of finite type have finitely many orbits and include many
1

cases of classical interest, such as determinantal varieties. The study of the geometry and algebra
of these orbits is undertaken in work in progress by Kraśkiewicz–Weyman (starting with [KW]). In
the affine case, there are typically infinitely many orbits, but the θ-representation (G, U ) has the
property that the ring of semi-invariants Sym(U ∗ )(G,G) is a polynomial ring and its unstable locus
(nullcone) has finitely many orbits. This class of representations includes the adjoint representations
of semisimple Lie algebras and share many features in common with them.
From the point of view of algebraic geometry, we are giving geometric constructions of (torsors
of) Abelian varieties from the data of a G-orbit in U . The GIT quotient U//G is isomorphic to a
quotient space h/W where W ⊂ GL(h) is a complex reflection group. In most of the cases that
we consider, the quotients h/W were considered previously and contain an open subset isomorphic
to a moduli space of curves with a special kind of marked data. However, the constructions for
Abelian varieties that we give seem to be new. One of the nice features of our work is that it
is possible to make a lot of it explicit V
and turn it into input for the computer algebra system
Macaulay2. For example, in the case of 3 V (V is a vector space of dimension 9), we give in §5.3
a detailed explanation of how to calculate the ideals of the degeneracy loci under study (which
include (3, 3)-polarized Abelian surface torsors) starting with a GL(V )-orbit.
From commutative algebra, this circle of ideas illustrates the power of a systematic use of perfect
resolutions. As we will soon explain, the main tool that we employ is the Eagon–Northcott generic
perfection theorem: using the minimal free resolutions of Kraśkiewicz–Weyman (many of which are
perfect resolutions), we can construct certain global sheafy complexes which specialize to locally
free resolutions of some varieties of interest, such as the Abelian variety torsors mentioned above.
In particular, for θ-representations of affine type, the GIT quotient U//G is a weighted projective
space, and hence rational, and it is of interest to know if the orbit space U/G has a modular
interpretation. The problem of studying the nilpotent orbits requires a different kind of approach
and will not be discussed in these notes. For some of the examples of θ-representations, it is easy to
find such interpretations using constructions from standard linear algebra, such as determinants.
However, many of them do not seem to have any obvious constructions associated with them. We
take up a systematic approach to dealing with these representations which we now outline (see
Construction 3.3 for details). The main idea is to use information of the orbits in representations
of finite type to bootstrap to the affine type case.
1. Using the Borel–Weil construction, one may realize U as the space of sections of a homogeneous
bundle U on a homogeneous space G/P . This can usually be done in many different ways. For
any point x ∈ G/P , the stabilizer of x is a subgroup in G which is conjugate to P , and for
our choices of P , the action of P on the fiber U (x) will have finitely many orbits (and one can
restrict the action to a certain reductive group G′ ⊂ P without affecting the orbit structure).
2. These orbits can be glued together to get “global orbit closures” in the total space of U . Any
vector v ∈ U is then a section of U and each of these global orbit closures gives “degeneracy
loci” in G/P by considering when v(G/P ) intersects a given global orbit closure. Hence the
first step to understanding the geometry of these degeneracy loci should be to understand the
“local” geometry of orbit closures in representations with finitely many orbits.
3. Vinberg’s theory allows one to completely classify the orbits in these finite type representations.
In many cases, one can calculate the minimal free resolutions of the coordinate rings of these
orbit closures and various equivariant modules which are of interest. This program is taken up
in the work of Kraśkiewicz–Weyman.
4. For generic sections, the free resolutions can be turned into locally free resolutions for the
degeneracy loci in question via the Eagon–Northcott generic perfection theorem. The power
in this approach is that it allows one to obtain cohomological information about the varieties.
2

This can sometimes be enough to determine the structure of the variety, such as showing it is
an Abelian variety (which happens in many cases that we consider).
In the case of a representation with finitely many orbits, there is no interesting moduli over
an algebraically closed field, but it is often interesting to consider Z-forms of the group and its
representation and to study the arithmetic orbits. This has been done in a series of papers by
Bhargava [Bha]. Under that perspective, a later goal would be to understand the arithmetic orbits
for the representations that we consider. Some examples of representations which have positivedimensional quotients have been worked out by Ho in her thesis [Ho].
For the contents of the paper: §§1–2 are introductory in nature and provide background on
free resolutions and the requisite representation theory that will be used in the notes. §3 contains
some geometric preliminaries on Abelian varieties and moduli spaces of vector bundles on curves,
as well as a more precise description of the steps outlined above. The rest of the notes are devoted
to studying examples of θ-representations of affine type.

Notation and conventions.
If R is a graded ring, then R(d) is the R-module R with a grading shift: R(d)i = Rd+i . We define
a graded local ring
L to be a positively graded ring R whose degree 0 part is a field. In this case,
we denote m =
i>0 Ri . Given a free K-module E, we think of Sym(E) as a graded ring by
Sym(E)i = S i E. Furthermore, when we talk about modules over graded rings, we will implicitly
assume that they are also graded.
If X is a scheme over a field K and E is a vector space, we let E denote the trivial vector bundle
E ⊗ OX . Also, given a vector bundle E, we let det E denote its top exterior power.
For the sections involving examples, we will work over an algebraically closed field of characteristic 0, which we will just denote by the complex numbers C. Ultimately, the goal is to relax this
assumption to other characteristics, or non-algebraically closed fields, so the introductory sections
are written in this more general context. We will also make some comments throughout about this
point.

Acknowledgements.
We thank Manjul Bhargava, Igor Dolgachev, Wei Ho, Steven Kleiman, Bjorn Poonen, and Jack
Thorne for helpful discussions and Damiano Testa for explaining the proof of Theorem 3.1. We
also thank Igor Dolgachev for pointing out numerous references related to this work. Finally, we
thank an anonymous referee for making some suggestions.
We also thank Federico Galetto and Witold Kraśkiewicz for assistance with some computer
calculations. The software LiE [LiE] and Macaulay2 [M2] were helpful in our work.
Steven Sam was supported by an NDSEG fellowship while this work was done. Jerzy Weyman
was partially supported by NSF grant DMS-0901185.

1
1.1

Free resolutions.
Basic definitions.

A lot of the foundational results in this section can be found in [BV2, Appendix].

3

Definition 1.1. Let R be a commutative ring and M be a finitely generated R-module. A complex
of R-modules
di
F• : · · · → Fi −→
Fi−1 → · · · → F0
is a projective resolution of M if
• each Fi is a finitely generated projective R-module,
• Hi (F• ) = 0 for i > 0 and H0 (F• ) = M .
The projective dimension of M (denoted pdim M ) is the minimum length of any projective
resolution of M . Free resolutions are projective resolutions where the Fi are free modules. If R
is a (graded) local ring with maximal ideal m, then F• is minimal if
• di (Fi ) ⊆ mFi−1 for all i > 0.
In the graded case, we will shift the gradings of the Fi to assume that the differentials are homogeneous of degree 0.
Definition 1.2. Let R be a Noetherian ring and let M be a finitely generated R-module. A
sequence (r1 , . . . , rn ) of elements in R is a regular sequence on M if
• r1 is not a zerodivisor or unit on M , and
• ri is not a zerodivisor or unit on M/(r1 , . . . , ri−1 )M for all i > 1.
For an ideal I ⊂ R, the depth of M (with respect to I) is the length of the longest regular sequence
for M which is contained in I. It is denoted by depthI M . If R is local with maximal ideal m,
we denote depth M = depthm M . For an ideal I ⊂ R, the grade of I is the length of the longest
regular sequence in I for R. M is perfect of grade g if g = pdim M = grade Ann M . (In general,
one has pdim M ≥ grade Ann M .)
Over a local Noetherian ring R, a finitely generated module M is Cohen–Macaulay if it is 0 or
depth M = dim M := dim(R/ Ann(M )). For a general Noetherian ring R, M is Cohen–Macaulay
if the localization Mp is Cohen–Macaulay over (Rp , p) for all prime ideals p of R. A Noetherian
ring is Cohen–Macaulay if it is a Cohen–Macaulay module over itself.
Theorem 1.3. Let R be a Noetherian Cohen–Macaulay ring.
1. For every ideal I ⊂ R, we have grade I = codim I = dim R − dim(R/I).
2. The polynomial ring R[x] is Cohen–Macaulay.
3. If an R-module M is perfect, then it is Cohen–Macaulay.
(The distinction between perfect and Cohen–Macaulay for a module over a Cohen–Macaulay
ring is the property of having finite projective dimension.)
Theorem 1.4 (Auslander–Buchsbaum formula). Suppose R is a Noetherian (graded) local ring
and that M is a finitely generated R-module with pdim M < ∞. Then
depth M + pdim M = depth R.
Theorem 1.5. Let R be a Noetherian (graded) local ring and M be a perfect R-module of grade
g with minimal free resolution F• . Then Hom(F• , R) is a minimal free resolution of the perfect
module M ∨ = ExtgR (M, R), and (M ∨ )∨ ∼
= M.
Definition 1.6. If M = R/I, for an ideal I, is perfect, then we write ωR/I = M ∨ and call it the
canonical module of R/I. If R/I is perfect and ωR/I ∼
= R/I (ignoring grading if it is present),
then we say that I is a Gorenstein ideal. This is equivalent to the last term in the minimal free
resolution of R/I having rank 1.

4

Theorem 1.7 (Eagon–Northcott generic perfection). Let R be a Noetherian ring and M a perfect
R-module of grade g, and let F• be an R-linear free resolution of M of length g. Let S be a
Noetherian R-algebra. If M ⊗R S 6= 0 and grade(M ⊗R S) ≥ g, then M ⊗R S is perfect of grade g
and F• ⊗R S is an S-linear free resolution of M ⊗R S. If M ⊗R S = 0, then F• ⊗R S is exact.
See [BV2, Theorem 3.5]. It is natural to ask what happens if the grade of M ⊗R S is some
value less than g, and this can be answered by the Buchsbaum–Eisenbud acyclicity criterion [Eis,
Theorem 20.9].
Remark 1.8. In particular, if R is Cohen–Macaulay (for example, R = K[x1 , . . . , xn ]) then we
can replace grade in the above theorem with codimension. It is often much easier to calculate
codimension. We will use it as follows. We first construct graded minimal free resolutions of
perfect modules M over A = K[x1 , . . . , xn ]. Then we specialize the variables xi to elements of a
Cohen–Macaulay K-algebra S in such a way that the codimension of M is preserved. Then the
resulting specialized complex is still a resolution.

1.2

Examples.

For this section, let K be a commutative ring and let E be a free module of rank N . In the following
examples, we will construct some complexes that are functorial in E and compatible with change
of rings. Two consequences of these properties is that the complex carries an action of the general
linear group GL(E) and that the constructions make sense for vector bundles over an arbitrary
scheme.
V
Example 1.9. Write A = Sym(E). We define a complex F• by setting Fi = i E ⊗ A(−i). The
differential is defined by
i
^

E ⊗ A(−i) →

e1 ∧ · · · ∧ ei ⊗ f 7→

i−1
^

E ⊗ A(−i + 1)

j=1

(−1)j e1 ∧ · · · ebj · · · ∧ ei ⊗ ej f.

i
X

This is the Koszul complex. It is a resolution of K = A/m, and is functorial with respect to E.
See [Eis, Chapter 17] for basic properties.
V
Example 1.10 (Buchsbaum–Eisenbud). We assume N = 2n + 1 is odd. Set A = Sym( 2 E),
which we can interpret as the coordinate ring of the space of all skew-symmetric matrices of size
2n + 1 with entries in K if we fix a basis e1 , . . . , e2n+1 of E. Let Φ be the generic skew-symmetric
matrix of size 2n + 1 whose (i, j) entry is xij = ei ∧ ej ∈ A1 . We construct a complex
F• : 0 → (det E)⊗2 ⊗ A(−2n − 1) → (det E) ⊗ E ⊗ A(−n − 1) →

2n
^

E ⊗ A(−n) → A.

For j = 1, . . . , 2n + 1, let e′j = e1 ∧ · · · eˆj · · · ∧ e2n+1 . We also define Pf(̂) to be the Pfaffian of the
submatrix of Φ obtained by deleting row and column j. Then we have
2n
^

d

1
E ⊗ A(−n) −→
A

e′j ⊗ f 7→ Pf(̂)f,
5

d

2
(det E) ⊗ E ⊗ A(−n − 1) −→

(e1 ∧ · · · ∧ e2n+1 ) ⊗ ej ⊗ f 7→

2n
^

E ⊗ A(−n)

2n+1
X
i=1

(−1)i e′i ⊗ xij f,

d3

(det E)⊗2 ⊗ A(−2n − 1) −→ (det E) ⊗ E ⊗ A(−n − 1)
2

(e1 ∧ · · · ∧ e2n+1 ) ⊗ f 7→ (e1 ∧ · · · ∧ e2n+1 ) ⊗

2n+1
X

(−1)j Pf(̂)ej f.

j=1

This is the Buchsbaum–Eisenbud complex. It is a resolution of A/I where I is the ideal
generated by the 2n × 2n Pfaffians of Φ, and it is functorial with respect to E. Furthermore, A/I
is a free K-module, and I is a Gorenstein ideal of codimension 3. We can identify d2 with the map
Φ. Buchsbaum and Eisenbud showed that given a codimension 3 Gorenstein ideal I, there is an
n such that its free resolution is a specialization of the above complex. See [BE, Theorem 2.1] for
more details.
V
Example 1.11 (Józefiak–Pragacz). We assume N = 2n is even. Again we set A = Sym( 2 E)
and let Φ be the generic skew-symmetric matrix. We will give the resolution for the ideal generated
by the Pfaffians of size 2n − 2. We just give the functorial terms in the complex when K contains
the field of rational numbers (the definition of the Schur functors S are given in §2.1):
F0 = A
F1 =

2n−2
^

E ⊗ A(−n + 1)

F2 = S2,12n−2 E ⊗ A(−n)

F3 = (det E) ⊗ S 2 E ⊗ A(−n − 1) ⊕ (det E)2 ⊗ (S 2 E)∗ ⊗ A(−2n + 1)

F4 = (det E) ⊗ S2,12n−2 E ⊗ A(−2n)
F5 = (det E)2 ⊗

2
^

E ⊗ A(−2n − 1)

F6 = (det E)3 ⊗ A(−3n).

When K is an arbitrary commutative ring, the functors must be defined differently, but the ranks
of the modules remain the same. We refer the reader to [Pra] for the details.
Example 1.12 (Goto–Józefiak–Tachibana). We set A = Sym(S 2 E), which we can interpret as the
coordinate ring of the space of symmetric matrices of size N . We let Φ be the generic symmetric
matrix. We give the terms of the resolution of the ideal generated by the minors of size N − 1:
F0 = A
F1 = (det E)2 ⊗ (S 2 E)∗ ⊗ A(−N + 1)
eval

F2 = (det E)2 ⊗ ker(E ⊗ E ∗ −−→ K) ⊗ A(−N )
2

F3 = (det E) ⊗

2
^

E ⊗ A(−N − 1).
eval

See [Joz, §3] for details. Over a field of characteristic 0, the term det E ⊗ ker(E ⊗ E ∗ −−→ K) can
be replaced by the Schur functor S2,1n−2 E (see §2.1 for the definition).
6

1.3

The geometric technique.

The material in this section is not logically necessary for the rest of the paper, but it is the main
tool behind the work of Kraśkiewicz–Weyman, so we include it for completeness. For a reference,
see [Wey, Chapter 5]. Note that we have changed notation.
Let K be a field, let X be a projective K-variety, and let V be a vector space. Suppose we are
given a short exact sequence of locally free sheaves
0 → S → V → T → 0.
We let p1 : V → V and p2 : V → X be the projection maps. Set Y = p1 (S) ⊂ V and A = OV =
Sym(V ∗ ). Note that Y is the affine cone over some projective variety in P(V ). Also, let E be any
vector bundle on X.
Theorem 1.13. There is a minimal A-linear complex F• whose terms are
Fi =

M
j≥0

j

H (X;

i+j
^

(T ∗ ) ⊗ E) ⊗ A(−i − j).

Furthermore, Hi (F• ) = 0 for i > 0 and for i ≤ 0, we have
Hi (F• ) = R−i (OS ⊗OV p∗2 E) = H−i (X; Sym(S ∗ ) ⊗OX E).
In particular, if the higher direct images of OS ⊗ p∗2 E vanish, then F• is a free resolution of the
pushforward. In the case that E = OX , this pushforward is an A-algebra. The vanishing of the
higher direct images is an intrinsic property of the variety Y .
The idea behind this theorem is to start with an affine cone variety Y and to find X and S that
fit into the above framework. Then the theorem above gives a tool for calculating the minimal free
resolution of Y .
Example 1.14 (Eagon–Northcott complex). Let E and F be vector spaces of dimensions m and
n and assume that m ≥ n. We set V = Hom(E, F ) and let Y ⊂ V be the subvariety of linear maps
of rank at most n − 1.
This fits into the previous setup by taking X = Gr(n − 1, F ) ∼
= P(F ∗ ). This has a tautological
exact sequence of vector bundles
0 → R → F → O(1) → 0
where R = {(x, W ) ∈ F × X | x ∈ W }. Then we can take S = Hom(E, R) = E ∗ ⊗ R. We will see
in §2.3 that the higher direct images of Sym(S ∗ ) vanish and so F• gives a minimal free resolution
for OY .
V
V
Since T ∗ = E ⊗ O(−1), we have d T ∗ = d E ⊗ O(−d). So we can calculate the terms of F•
explicitly. For i > 0, we have
Fi =

n+i−1
^

E ⊗ det F ∗ ⊗ (S i−1 F )∗ ⊗ A(−n − i + 1).

The differentials can be calculated by noting that they will preserve the natural GL(E) × GL(F )
action on V and Y and that equivariant maps of this form are unique up to a choice of scalar. In
fact, this construction works with K = Z, in which case we get uniqueness of scalars up to a choice
of sign. A multilinear generalization of this complex, constructed using similar ideas, can be found
in [BEKS].
7

2
2.1

Representation theory.
Schur functors.

For the material in this section, see [Wey, Chapter 2]. What we call Sλ is denoted by Lλ′ there.
Definition 2.1. A partition λ is a decreasing sequence of positive integers λ1 ≥ λ2 ≥ · · · ≥ λn .
We represent this as a Young diagram by drawing λi boxes left-justified in the ith row, starting
from top to bottom. The dual partition λ′ is obtained by letting λ′i be the number of boxes in the
ith column of λ. Given a box b = (i, j) ∈ λ, its content is c(b) = j − i and its hook length is
h(b) = λi − i + λ′j − j + 1. If we have a sequence (i, i, . . . , i) repeated j times, we abbreviate by the
notation (ij ).
Example 2.2. Let λ = (4, 3, 1). Then λ′ = (3, 2, 2, 1). The contents and hook lengths are given
as follows:
0 1 2 3
6 4 3 1
c : −1 0 1
h: 4 2 1
−2
1
Definition 2.3. Let R be a commutative ring and E a free R-module. Let λ be a partition with
n parts and write m = λ1 . We use S n E to denote the nth symmetric power of E. The Schur
functor Sλ (E) is the image of the map
′

λ1
^

′

E ⊗ ··· ⊗

λm
^

∆

′

µ

′

E−
→ E ⊗λ1 ⊗ · · · ⊗ E ⊗λm = E ⊗λ1 ⊗ · · · ⊗ E ⊗λn −
→ S λ1 E ⊗ · · · ⊗ S λn E,

where
the maps are defined as follows.
First, ∆ is the product of the comultiplication maps
P
Vi
E → E ⊗i given by e1 ∧ · · · ∧ ei 7→ w∈Σi sgn(w)ew(1) ⊗ · · · ⊗ ew(i) . The equals sign is interpreted
′
′
as follows: pure tensors in E ⊗λ1 ⊗ · · · ⊗ E ⊗λm can be interpreted as filling the Young diagram of λ
with vectors along the columns, which can be thought of as pure tensors in E ⊗λ1 ⊗ · · · ⊗ E ⊗λn by
reading via rows. Finally, µ is the multiplication map E ⊗i → S i E given by e1 ⊗ · · · ⊗ ei 7→ e1 · · · ei .
In particular, note that Sλ E = 0 if the number of parts of λ exceeds rank E.
Example 2.4. Take λ = (3, 2). Then the map is given by
e2 e3 e5
e1 e4 e5
e2 e4 e5
e1 e3 e5
−
−
+
e2 e4
e1 e4
e2 e3
e1 e3
7→ (e1 e3 e5 ⊗ e2 e4 ) − (e2 e3 e5 ⊗ e1 e4 ) − (e1 e4 e5 ⊗ e2 e3 ) + (e2 e4 e5 ⊗ e1 e3 ).

(e1 ∧ e2 ) ⊗ (e3 ∧ e4 ) ⊗ e5 7→

Theorem 2.5. The Schur functor Sλ E is a free R-module. If rank E = n, then
rank Sλ E =

Y n + c(b)

b∈λ

h(b)

.

The construction of Sλ E is functorial with respect to E. This has two consequences: Sλ E is
naturally a representation of GL(E), and we can also construct Sλ E when E is a vector bundle.
If rank E = n, then we have Sλ E ⊗ det E = S(λ1 +1,...,λn +1) E. Using this, it makes sense to
define Sλ E when λ is any weakly decreasing sequence of integers. Furthermore, we have Sλ (E ∗ ) =
S−λn ,...,−λ1 E [Wey, Exercise 2.18] and over a field of characteristic 0, the isomorphism Sλ (E ∗ ) =
(Sλ E)∗ [Wey, Proposition 2.1.18].
8

2.2

Descriptions of some homogeneous spaces.

Let E be a vector space of rank N . We let Fl(E) be the flag variety of E. Its K-valued points are
complete flags of subspaces E• : E1 ⊂ E2 ⊂ · · · EN = E such that rank Ei = i. The trivial bundle
E contains a tautological flag of subbundles R1 ⊂ R2 ⊂ · · · ⊂ RN = E where
Ri = {(x, E• ) ∈ E × Fl(E) | x ∈ Ei }.
Given a subset S ⊂ {1, . . . , N − 1}, we can also consider the partial flag varieties Fl(S; E) whose
K-valued points only are partial flags whose ranks are the elements in S. Then E has a tautological
partial flag of subbundles Ri (i ∈ S).
Now assume that E is equipped with a symplectic form ω and set n = N/2. We say that a
subspace U ⊂ E is isotropic if ω(u, u′ ) = 0 for all u, u′ ∈ U . Also, for any subspace U , we set
U ⊥ = {x ∈ E | ω(x, u) = 0 for all u ∈ U }. Then rank U + rank U ⊥ = rank E and U is isotropic if
and only if U ⊆ U ⊥ .
We define the symplectic flag variety to be the subvariety Flω (E) of the partial flag variety
Fl(1, . . . , n; E) consisting of flags E1 ⊂ E2 ⊂ · · · ⊂ En such that each Ei is isotropic. We also let
R1 ⊂ R2 ⊂ · · · ⊂ Rn denote the restriction of the tautological bundles to Flω (E). We can also
⊥
define the subbundles R⊥
i = {(x, E• ) ∈ E × Flω (E) | x ∈ Ei }. Note that we have
⊥
⊥
0 ⊂ R1 ⊂ R2 ⊂ · · · ⊂ Rn = R⊥
n ⊂ Rn−1 ⊂ · · · ⊂ R1 ⊂ E.

Given a subset S ⊂ {1, . . . , n}, we can also define partial ω-isotropic flag varieties Flω (S; E).
When S = {i} is a singleton, we also write Grω (i, E) = Flω ({i}; E) and call it the ω-isotropic
Grassmannian.

2.3

Borel–Weil–Bott theorem.

For the material in this section, see [Wey, Chapter 4].
Let S = {i1 < i2 < · · · < ik } be a subset of {1, . . . , N } and consider the partial flag variety Fl(S; E). For each j = 1, . . . , k + 1, let λ(j) be a weakly decreasing sequence of integers of
length ij − ij−1 (set i0 = 0 and ik+1 = N ). Let λ be the sequence obtained by concatenating
λ(1) , λ(2) , . . . , λ(k+1) . Set
k+1
O
Sλ(j) ((Rij /Rij−1 )∗ ).
R(λ) =
j=1

Theorem 2.6 (Borel–Weil). If λ is a weakly decreasing sequence of integers, then we have a
GL(E)-equivariant isomorphism
H0 (Fl(S; E); R(λ)) = Sλ (E ∗ ),
and all higher cohomology of R(λ) vanishes.
Now we consider the analogue for the symplectic group. Let λ be a partition with at most n
parts. We restrict the line bundle R(λ) on the flag variety Fl(E) to the symplectic flag variety
Flω (E) and denote its space of sections by
S[λ] (E) = H0 (Flω (E); R(λ)).

(2.7)

By considering a relative situation, this definition makes sense for any vector bundle
over a scheme
V2 E
∗
E → OX . (More
X which is equipped with a nondegenerate skew-symmetric OX -bilinear form
9

generally, the form could take values in a line bundle, but we will not use this generalization.) For
all tautological subbundles R on any symplectic flag variety, the restriction of the symplectic form
to the quotient bundle R⊥ /R is nondegenerate.
Now let S = {i1 < · · · < ik } be a subset of {1, . . . , n} and let λ(j) be as before, except now we
assume that λ(k+1) only has nonnegative integers. We set
R[λ] = S[λ(k+1) ] (R⊥
ik /Rik ) ⊗

k
O

Sλ(j) ((Rij /Rij−1 )∗ ).

j=1

Theorem 2.8 (Borel–Weil). If λ is a weakly decreasing sequence of nonnegative integers, then we
have a Sp(E)-equivariant isomorphism
H0 (Flω (S; E); R[λ]) = S[λ] E
and all higher cohomology of R[λ] vanishes.
Now we discuss what happens when λ is not a weakly decreasing sequence of integers. For this,
we now need to assume that the characteristic of the ground field is 0. We first handle the partial
flag varieties.
Define the vector ρ = (N − 1, N − 2, . . . , 1, 0). Given a permutation w ∈ ΣN , we define the
dotted action of w on a sequence of integers α of length N by
w• (α) = w(α + ρ) − ρ.
We define the length of a permutation w to be ℓ(w) = #{(i, j) | i < j, w(i) > w(j)}. Alternatively,
let si be the simple transposition that swaps i and i + 1. Then s1 , . . . , sN −1 generate ΣN , and we
could also define ℓ(w) to be the minimal number k so that we can write w = si1 · · · sik .
Theorem 2.9 (Bott). Assume that the field K has characteristic 0 and let λ be as before. Then
exactly one of the following two cases occurs:
• There exists a non-identity w ∈ ΣN such that w• (λ) = λ. In this case, all cohomology of
R(λ) vanishes.
• There is a unique w ∈ ΣN such that w• (λ) = µ is a decreasing sequence. In this case, we
have a GL(E)-equivariant isomorphism
Hℓ(w) (Fl(S; E); R(λ)) = Sµ (E ∗ )
and all other cohomology of R(λ) vanishes.
An example that uses the previous theorem is given in the proof of Theorem 5.5.
Now we consider Bott’s theorem for the symplectic flag varieties Flω (S; E). Now we set ρ =
(N, N − 1, . . . , 2, 1). We replace the symmetric group ΣN with the group of signed permutations
W = ΣN ⋉ (Z/2)N , which we think of as N × N signed permutation matrices.
We consider the generators s1 , . . . , sN for W . The meaning of s1 , . . . , sN −1 is the same as for
the symmetric group ΣN , and sN is the diagonal matrix diag(1, . . . , 1, −1). Now given w ∈ W , we
define ℓ(w) to be the minimal number k so that we can write w = si1 · · · sik .
Then the definition of the dotted action of W remains the same, and the analogue of Bott’s
theorem holds for the bundles R[λ] on Flω (S; E).
Theorem 2.10 (Bott). Assume that the field K has characteristic 0 and let λ be as before. Then
exactly one of the following two cases occurs:
10

• There exists a non-identity w ∈ W such that w• (λ) = λ. In this case, all cohomology of R[λ]
vanishes.
• There is a unique w ∈ ΣN such that w• (λ) = µ is a weakly decreasing sequence of nonnegative
integers. In this case, we have an Sp(E)-equivariant isomorphism
Hℓ(w) (Flω (S; E); R[λ]) = S[µ] (E ∗ )
and all other cohomology of R[λ] vanishes.
The symplectic form gives Sp(E)-equivariant isomorphisms S[µ] (E ∗ ) = S[µ] E for all µ.

2.4

Vinberg θ-representations.

For this section, we refer to [Kac1, Kac2, Vin1, Vin2] for reference.
Let Xn be a Dynkin diagram and let g be the corresponding simple Lie algebra. Let us distinguish a node x ∈ Xn . Let αk be a corresponding simple root in the root system Φ corresponding
to Xn . The choice of αk determines a Z-grading on Φ by letting the degree of a root β be equal to
the coefficient of αk when we write β as a linear combination of simple roots. On the level of Lie
algebras this corresponds to a Z-grading
M
gi .
g=
i∈Z

We define the group G0 := (G, G) × C∗ where (G, G) is a connected semisimple group with
the Dynkin diagram Xn \ x. A representation of type I is the representation of G0 on g1 , and a
representation of type II is what we get when we replace Xn with an affine Dynkin diagram. This
notation follows [Kac2, Proposition 3.1].
Remark 2.11. In the case of a type II representation (i.e., when Xn is an affine Dynkin diagram),
each θ-representation (G0 , g1 ) has a Chevalley isomorphism: there is a subspace h ⊂ g1 and a
complex reflection group W ⊂ GL(h) (defined as the normalizer of h modulo the centralizer of h),
which we call the graded Weyl group, such that the restriction map is an isomorphism
∼
=

Sym(g∗1 )(G,G) −
→ Sym(h∗ )W .
In particular, Sym(g∗1 )(G,G) is always a polynomial ring. The complex reflection groups were classified by Shephard–Todd in [ST] and we will refer to their numbering system when using names for
these groups.
In [Vin1], Vinberg gave a description of the G0 -orbits in the representations of type I in terms
of conjugacy classes of nilpotent elements in g. Let e ∈ g1 be a nilpotent element in g. Consider
the irreducible components of the intersection of the conjugacy class of e in g
C(e) ∩ g1 = C1 (e) ∪ · · · ∪ Cn(e) (e).
The sets Ci (e) are clearly G0 -stable. Vinberg’s result shows that these are precisely the G0 -orbits
in g1 .
Theorem 2.12 (Vinberg). The G0 -orbits of the action of G0 on g1 are the components Ci (e), for
all choices of the conjugacy classes C(e) and all i, 1 ≤ i ≤ n(e).

11

Theorem 2.12 makes a connection between the orbits in g1 and the nilpotent orbits in g. The
classification of nilpotent orbits in simple Lie algebras was obtained by Bala and Carter in the
papers [BC]. A good account of this theory is the book [CM]. Here we recall that the nilpotent
orbit of an element e in a simple Lie algebra g is characterized by the smallest Levi subalgebra l
containing e. One must be careful because sometimes l is equal to g. If the element e is a principal
element in l, then this orbit is denoted by the Dynkin diagram of l (but there might be different
ways in which the root system R(l) sits as a subroot system of R(g)).
There are, however, the non-principal nilpotent orbits that are not contained in a smaller
reductive Lie algebra l. These are called the distinguished nilpotent orbits and are described in
[CM, §§8.2–8.4]. They are characterized by their associated parabolic subgroups (as their Dynkin
characteristics are even, [CM, §8]). Let us remark that for Lie algebras of classical types, for type
An the only distinguished nilpotent orbits are the principal ones, and for types Bn , Cn , Dn these
are orbits corresponding to the partitions with different parts. For exceptional Lie algebras the
distinguished orbits can be read off the tables in [CM, §8.4].
Theorem 2.12 is not easy to use because it is not very explicit. In the next section we describe
a more precise method from another of Vinberg’s papers [Vin2].

2.5

The Vinberg method for classifying orbits.

In this section we describe the second paper of Vinberg [Vin2] in which he describes orbits of
nilpotent elements in g1 . Similar to the Bala–Carter classification, the nilpotent elements in g1 are
described by means of some graded subalgebras of g. We need some preliminary notions.
All Lie algebras g we will consider will be Lie algebras of some algebraic group G.
Let (Xn , αk ) be one of the representations from the previous section. As before, it defines the
grading
M
g=
gi
i∈Z

where gi is the span of the roots which, written as a combination of simple roots, have αk with
coefficient i. The component g0 contains a Cartan subalgebra. G0 denotes the connected closed
subgroup of G corresponding to g0 .
In the sequel, Z(x) denotes the centralizer of an element x ∈ G and Z0 (x) = Z(x) ∩ G0 , and
z, z0 denote the corresponding Lie algebras. Similarly, N (x) denotes the normalizer of an element
x ∈ G and N0 (x) = N (x) ∩ G0 .
We let R(g) denote the set of roots of g, and Π(g) denotes a set of simple roots.
Definition 2.13. A graded Lie subalgebra s ⊂ g is regular if it is normalized by a maximal torus
in g0 . A reductive graded Lie algebra s ⊂ g is complete if it is not a proper graded Lie subalgebra
of any regular reductive Z-graded Lie algebra of the same rank.
Definition 2.14. A Z-graded Lie algebra g is locally flat if any of the following equivalent
conditions is satisfied, for e a point in general position in g1 :
(1) the subgroup Z0 (e) is finite,
(2) z0 (e) = 0,
(3) dim g0 = dim g1 .
Fix a nonzero nilpotent element e ∈ ga and choose some maximal torus H in N0 (e). Its Lie
algebra h is the accompanying torus of the element e. We denote by ϕ the character of the
torus H defined by the condition
[u, e] = ϕ(u)e
12

for u ∈ h. Consider the graded Lie subalgebra g(h, ϕ) of g defined by
M
g(h, ϕ) =
g(h, ϕ)i
i∈Z

where
g(h, ϕ)i = {x ∈ gia | [u, x] = iϕ(u)x for all u ∈ H}.
Definition 2.15. The support s of the nilpotent element e ∈ ga is the commutator subalgebra of
g(h, ϕ) considered as a Z-graded Lie algebra.
Clearly e ∈ s1 . We are ready to state the main theorem of [Vin2].

Theorem 2.16 (Vinberg). The supports of nilpotent elements of the space gi are exactly the complete regular locally flat semisimple Z-graded subalgebras of the algebra g. The nilpotent element e
can be recovered from the support subalgebra s as the generic element in s1 .
It follows from the theorem that the nilpotent element e is defined uniquely (up to conjugation
by an element of G0 ) by its support. This means it is enough to classify the regular semisimple
Z-graded subalgebras s of g.
Let us choose a maximal torus t ⊂ g0 . The Z-graded subalgebra s is standard if it is normalized
by t, i.e., if for all i ∈ Z we have
[t, si ] ⊂ si .

Vinberg also proves that every Z-graded subalgebra s is conjugated to a standard subalgebra
by an element of G0 . Moreover, he shows that if two standard Z-graded subalgebras are conjugated
by an element of G0 , then they are conjugated by an element of N0 (t). This gives combinatorial
method for classifying regular semisimple Z-graded subalgebras of g.
Let s be a standard semisimple Z-graded subalgebra of g. The grading on the subalgebra s
defines a degree map deg : R(s) → Z. For a standard Z-graded subalgebra s we also get the map
f : R(s) → R(g).
The map f has to be additive, i.e., it satisfies
f (α + β) = f (α) + f (β) ∀α, β ∈ R(s),
f (−α) = −f (α) ∀α ∈ R(s).

Moreover we have
Proposition 2.17. The map f satisfies the following properties:
(a)
(α, β)
(f (α), f (β))
=
∀α, β ∈ R(s).
(f (α), f (α))
(α, α)
(b) f (α) − f (β) ∈
/ R(g) ∀α, β ∈ Π(s),
(c) deg f (α) = deg α, ∀α ∈ Π(s).
Conversely, every map satisfying these conditions defines a standard regular Z-graded subalgebra s
of g.
Remark 2.18. The subalgebra s corresponding to the map f is complete if and only if there exists
an element w in the Weyl group W of g such that wf (Π(s)) ⊂ Π(g) (see [Vin2, p.25]).

Theorem 2.16 means that in order to classify the nilpotent elements e ∈ g1 we need to classify
the possible maps f corresponding to its support, i.e., the corresponding complete regular Z-graded
subalgebra s. Since we are interested in the nilpotent elements e ∈ g1 , we need to classify the maps
f for which deg(f (α)) ∈ {0, 1} for every α ∈ Π(s).
13

2.6

Example:

V3

C7 .

Let Xn = E7 , and αk = α2 in Bourbaki numbering.
The graded Lie algebra corresponding to the resulting grading is
g(E7 ) = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2
V
V
with G0 = GL7 (C), g0 = gl7 (C), g1 = 3 C7 , g2 = 6 C7 .
V
We choose a basis {e1 , . . . , e7 } of C7 . The weight vectors in g1 = 3 C7 are the vectors ei ∧ej ∧ek
of weight εi + εj + εk for 1 ≤ i < j < k ≤ 7.
There is a natural bijection Σ of these weight vectors with the positive roots with label 1 on the
node α2 . In order to describe it we identify the subroot system of g0 with the root system of type A6
with the simple roots β1 , . . . , β6 (corresponding to permutations (1, 2), (2, 3), . . . , (6, 7) respectively)
by identifying β1 with α1 and βs with αs+1 for 2 ≤ s ≤ 6. We denote this identification map by Λ.
We set Σ(e5 ∧ e6 ∧ e7 ) = α2 and extend it to a unique bijection satisfying Σ(ei ∧ ej ∧ ek + βs ) =
Σ(ei ∧ ej ∧ ek ) + Λ(βs ) for every simple root βs , for 1 ≤ s ≤ 6.
The invariant scalar product on h restricted to the roots from g1 and transferred by Σ is
(ei ∧ ej ∧ ek , ep ∧ eq ∧ er ) = #({i, j, k} ∩ {p, q, r}) − 1.
This can be checked by observing that both the invariant scalar product and the one defined by
the formula above are invariant with respect to the Weyl group of g0 , i.e., the permutation group
S7 , and checking the equality of one scalar product directly.
V
Next using the table from [Spa, p.248] we see that the general element of 3 C7 has the Bala–
Carter label A2 + 3A1 . Notice that the labeling of this orbit given on [CM, p.130] has weighted
Dynkin diagram with labels 0 on all nodes except for αk and label 2 on the node αk (this also
happens in the other cases except the case (E6 , α3 ) where the weighted Dynkin diagram of the
nilpotent orbit intersecting g1 in the open orbit has labelings 1 at α3 and α5 and zero elsewhere).
The support subalgebras of the smaller orbits in g1 have Vinberg labels exactly matching the Bala–
Carter labels that can be read off the Spaltenstein tables. In our analysis we checked also that the
other types of support algebras given in [Vin2] do not exist.
The combinatorial analysis is not difficult. We just give a few examples. Let us identify the
support subalgebra of type A2 + 3A1 . According
V to Proposition 2.17 we need to find all choices (up
to the action of S7 ) of five weight vectors in 3 C7 whose pairwise scalar products match those of
the five simple roots of the root system A2 + 3A1 . It is clear that for simple roots of the system A2
we can choose e1 ∧ e2 ∧ e3 and e4 ∧ e5 ∧ e6 . Then we need a choice of three subsets [i, j, k] which have
one element intersections with both [1, 2, 3], [4, 5, 6] and with each other. Up to the permutation
group S7 it is clear that there is only one choice [1, 4, 7], [2, 5, 7], [3, 6, 7]. One can check that the
resulting orbit is open by calculating its dimension. This is an easy linear algebra exercise since
the tangent space is obtained by hitting this element with all vectors in gl7 (C).
Similarly, looking at the subsystems of type 3A1 we need choices of three subsets [i, j, k] such
that every pair of subsets intersects in one element. This element can be common to all three
subsets or not, which gives two possibilities {[1, 2, 3], [1, 4, 5], [1, 6, 7]} and {[1, 2, 3], [1, 4, 5], [2, 4, 6]}
up to the S7 -action.
Finally, looking at the possibilities for 4A1 we notice that the triple [1, 2, 3], [1, 4, 5], [1, 6, 7] can
be only complemented in eight ways equivalent to [2, 4, 6]. The triple [1, 2, 3], [1, 4, 5], [2, 4, 6] can
be complemented in three ways equivalent to [1, 6, 7]. But theseVtwo choices of roots of 4A1 are the
same. We conclude that there is only one orbit of type 4A1 in 3 C7 .
V
A subtle point is checking that the obtained support algebra is complete. In the case of 3 C7
it also follows from the fact that all the representatives give different orbits. We omit this.
14

The result of the analysis is presented in the following table (writing [i, j, k] for ei ∧ ej ∧ ek ).
Label
s
dim
Representative
0
0
0
0
1
A1
13
[123]
2
2A1
20
[123] + [145]
3
3A1
21
[123] + [145] + [167]
4
3A1
25
[123] + [145] + [246]
5
A2
26
[123] + [456]
6
4A1
28
[123] + [145] + [167] + [357]
7
A2 + A1
31
[123] + [456] + [147]
8
A2 + 2A1 34
[123] + [456] + [147] + [257]
9
A2 + 3A1 35 [123] + [456] + [147] + [257] + [367]
The containment diagram is
35

O9

34

O8

31

O7✲
✄✄✄ ✲✲
✲✲
O6 ❀
✲✲
❀❀
❀❀
❀❀ O5
❀❀

28
26

O4

25
21
20

✑✑
O3 ❀ ✑✑✑
❀❀ ✑
✑

O2

13

O1

0

O0

and it happens to be the same as the corresponding containment of the nilpotent orbits inVg1 .
∗
We can interpret the orbits in terms of skew-symmetric tensors as follows. Let A = Sym( 3 C7 )
be the coordinate ring of our representation. The orbit O1 is the orbit of the highest weight vector,
and its closure is the set of decomposable tensors, i.e., the tensors t = v1 ∧ v2 ∧ v3 (v1 , v2 , v3 ∈ C7 ).
There is an SL7 (C)-invariant ∆ of degree 7, the hyperdiscriminant. This is the projective dual
∗
variety of the decomposable tensors in C7 . In fact, we can canonically identify the orbits in C7
∨
with those in its dual, so it makes sense to consider
the projective dual O of an orbit closure O as a
V
subset of C7 itself. By the rank of a tensor t ∈ 3 C7 we mean the subspace rank,
V minimal
V i.e., the
number s such that there exists a subpace V of dimension s in C7 such that t ∈ 3 V ⊂ 3 C7 . An
orbit
GL6 (C)-orbits
V is degenerate if it consists of tensors of rank ≤ 6. Such orbits correspond to V
in 3 C6 . By a 1-decomposable tensor we mean a tensor t = v ∧ q with v ∈ C7 , q ∈ 2 C7 . Finally
in the projective picture we interpret orbit closures as secant and tangential varieties of the orbit
O1 . We denote by σk (O1 ) the kth secant variety of O1 and by τ (O1 ) the tangential variety of O1 .
With this terminology, we have the following table describing the orbits.

15

Number
0
1
2
3
4
5
6
7
8
9

Projective picture
0
cone(Gr(3, 7))
σ2 (O1 )∨
τ (O1 )
σ2 (O1 )
J(O1 , τ (O 1 ))
σ3 (O1 )
∨
O1

Tensor picture
0
decomposable tensors
tensors of rank ≤ 5
1-decomposable tensors
V
degenerate and zero hyperdiscriminant for 3 C6
tensors of rank ≤ 6
V
polarizations of hyperdiscriminant for 3 C6
singular locus of O1
hyperdiscriminant is zero
generic

Remark V
2.19. The geometric
description of these orbits is V
also considered in [Hol, §5] (he also
V
considers 3 C6 and 3 C8 ). The classification of orbits of 3 K 7 is studied for many kinds of
fields K (including algebraically closed fields of positive characteristic and finite fields) in [CH].
In particular, the classification of orbits is independent of characteristic if the field is algebraically
closed.
V
We will describe in detail the non-degenerate orbit closures in 3 C7 . These are the orbits O9 ,
O8 . O7 , O6 and O3 . The first of these is generic so there is not much to say. We also describe the
generic degenerate orbit of tensors of rank ≤ 6. V
∗
∗
We use the usual notation. Let A = Sym( 3 C7 ) and λ is notation for Sλ (C7 ). Also, let
∗
x1 , . . . , x7 be a basis of C7 dual to the basis e1 , . . . , e7 of C7 . We will also describe vector bundle
desingularizations for these orbit closures. The bundles η and ξ correspond to S ∗ and T ∗ in the
notation of §1.3.
• The hyperdiscriminant orbit O8 .

This is the hypersurface given by the tensors with vanishing hyperdiscriminant ∆. The orbit closure O8 is characterized (set-theoretically) by the condition that the determinant of the
multiplication map
2
5
^
^
∗
7∗
C7 ⊗ A
C ⊗ A(−1) →
given by multiplication is zero. The determinant of this matrix is equal to ∆3 .

• The codimension 4 orbit O7 .

This orbit closure is the singular locus of the hyperdiscriminant orbit O8 .

We can find a desingularization by a vector bundle
over G/P = Fl(2, 6; C7 ). The bundle ξ ⊂
V
V
∗
∗
3 7
C is induced from the largest P -submodule of 3 C7 which does not contain x1 ∧x2 ∧x7 and
x2 ∧ x5 ∧ x7 . The bundle η has rank 17, so the dimension of the desingularization is 17 + 14 = 31
as needed. One gets a very nice complex describing the resolution of C[O7 ]. The terms of the
complex F(7)• are as follows
0 → (6, 56 ) → (52 , 45 ) → (4, 35 , 2) → (34 , 23 ) → (07 ).

The orbit closure is normal and has rational singularities and the complex F(7)• is pure.
• The codimension 7 orbit O6 .

7
We can find a desingularization by a vector bundle
V3 7 ∗over G/P = Fl(1, 4; C ). The bundle ξ ⊂
V
3 7∗
C which does not contain x1 ∧ x4 ∧ x7 and
C is induced from the P -submodule of

16

x2 ∧ x3 ∧ x4 . The bundle η has rank 13, so the dimension of the desingularization is 13 + 15 = 28
as needed. The terms in the resulting complex F(6)• are
0 → (76 , 6) → (7, 65 , 5) → (62 , 54 , 4) → (53 , 43 , 3) → (44 , 32 , 2) → (35 , 2, 1) → (26 , 0) → (07 ).
The orbit closure is normal, with rational singularities and the complex F(6)• is pure.
• The generic degenerate orbit closure O5 of tensors of rank ≤ 6 (codimension 9).

∗

7
This orbit closure has a desingularization Z(5) that lives on the Grassmannian Gr(1,
V2 C ).
Q. It
Denoting the tautological bundles R, Q (rank R = 1, rank Q = 6), we have ξ = R ⊗
is normal and has rational singularities. Calculating the resolution is straightforward, as ξ is
irreducible.

• The orbit O3 of 1-decomposable tensors (codimension 14).
V
ThisVorbit closure is the set of tensors t ∈ 3 C7 that can be expressed as t = ℓ ∧ t where ℓ ∈ C7 ,
7∗
t ∈ 2 C7 . The desingularization Z(3) lives on the Grassmannian Gr(6,
V3 C ). Denoting the
R. It is normal and
tautological bundles as R, Q (rank R = 6, rank Q = 1), we have ξ =
has rational singularities. Calculating the resolution is straightforward, as ξ is irreducible. The
defining ideal is generated by the representation (23 , 13 , 0) in degree 3.
V3 7
C , how many lines ℓ ⊂ C7
Remark 2.20. Since we will need it later
on,
we
ask,
for
a
fixed
v
∈
V3 7
(C /ℓ) is a pure tensor. This only depends on the orbit,
there are such that the image of v in
so we can study specific representatives.
No such line exists for vectors in the generic orbit or codimension 1 orbit. A representative for
the codimension 4 orbit is

This is a pure tensor in
codimension 7 orbit is

V3

e1 ∧ e2 ∧ e3 + e4 ∧ e5 ∧ e6 + e1 ∧ e4 ∧ e7 .
(C7 /ℓ) exactly for ℓ = he1 i and ℓ = he4 i. A representative for the

e1 ∧ e2 ∧ e3 + e1 ∧ e4 ∧ e5 + e1 ∧ e6 ∧ e7 + e3 ∧ e5 ∧ e7 .
V
This is a pure tensor in 3 (C7 /ℓ) exactly for ℓ = he1 i.

3
3.1

Some geometry.
Abelian varieties.

The following result is most likely well-known, but we could not find it in the literature. The main
points of the proof were communicated to us by Damiano Testa.
Theorem 3.1. Let X be a g-dimensional geometrically connected projective nonsingular variety
over a field K of characteristic 0. If ωX ∼
= OX and dim H1 (X; OX ) = g, then X is a torsor over
an Abelian variety (namely, its Albanese variety).
There are two reasons that we emphasize the fact that X is only a torsor: in our applications of
this theorem in the later sections, there may be no natural choice of base point (which is relevant
when working over non-algebraically closed fields), and in later work we will be interested in working
over families (in which case the existence of a section may be more subtle).
17

Proof. First, extend scalars to the algebraic closure K of K. By [Kaw, Corollary 2], XK is birationally equivalent to its Albanese variety X0,K , let f : XK → X0,K be a birational morphism. In
fact, f can be (uniquely) extended to a morphism on all of XK [BL, Theorem 4.9.4]. One has an
induced map df on cotangent bundles. The determinant of df is a map between the canonical
bundles, which are trivial by assumption. The endomorphisms of the trivial bundle must be scalars
since we assumed that XK is projective, so det df is a scalar. This scalar is nonzero since f is
generically an étale morphism. So in fact f is étale (and hence open). Also f is proper (and hence
closed), so f is an étale covering, which implies that XK is an Abelian variety. Furthermore, f
must be an isomorphism since it is birational.
Hence, choosing any point P ∈ X(K), we have a K-isomorphism X0,K → XK via x 7→ P + x.
This map descends to a K-rational map Y → X where Y is a K-rational X0 -torsor given by the
cocycle γ 7→ γ P − P , which gives the claim.
Remark 3.2. If we drop the assumption that K be of characteristic 0, then this theorem already
fails for g = 2. In particular, it is valid if the characteristic is different from 2 or 3, but in these
small characteristics, there are new exotic examples, known as quasi-hyperelliptic surfaces which
come from the Bombieri–Mumford classification of surfaces (the quasi-hyperelliptic surfaces have
the property that their Picard varieties are non-reduced), see [BM, p.25, Table].
Given a smooth curve C of genus g, we let Jac(C) denote its Jacobian, which is an Abelian
variety of dimension g (see [BL, Chapter 11] for an analytic construction of Jac(C)).

3.2

Moduli space of vector bundles.

The material in this section is provided for convenience and informative purposes, since later we will
see some examples of the moduli spaces discussed in this section (see Example 3.6, Remark 6.10,
and §8). We refer the reader to [Be3] for a more in-depth survey of the following.
Let C be a smooth curve of genus g ≥ 2. There is a moduli space SUC (n, d) which parametrizes
rank n semistable vector bundles of degree d on C. It has dimension n2 (g − 1) + 1. Let SUC (n, L)
denote the moduli space of rank n semistable vector bundles on C with determinant equal to L.
We write SUC (n) for SUC (n, OC ). Then SUC (n) has dimension (g − 1)(n2 − 1) and is Gorenstein
and has rational singularities. The Picard group of SUC (n) is infinite cyclic. Let L be its ample
generator, which we call the theta divisor. The canonical bundle is L−2n .
Now focus on n = 2. If g = 2, then SUC (2) ∼
= P3 , and for g > 2, the singular locus of SUC (2) is
−1
bundles of the form L⊕L and is naturally identified with the Kummer quotient of the Jacobian of
degree g − 1 line bundles on C. Also, h0 (SUC (n); L) = 2g and the map given by L is an embedding
if C is not hyperelliptic. Furthermore, the restriction of L is a (2, . . . , 2)-polarization. See [BV1]
and [vGI] for more details.
Finally, we state the Verlinde formula, which gives the dimension of the space of sections of
powers of L. For n = 2, we have
0

k

h (SUC (2); L ) =



k+2
2

g−1 X
k+1

(sin

j=1

see [Be1, §5] for the case of general n.

3.3

Degeneracy loci.

The main idea of the paper is in the following construction.
18

πj −2g+2
)
,
k+2

Construction 3.3. Start with a Vinberg representation (G, U ) of affine type (see §2.4). Choose a
parabolic subgroup P of G. We can realize U as the sections of a homogeneous bundle U over the
homogeneous space G/P using §2.3. In each case that we consider, the fibers of U can naturally be
interpreted as another Vinberg representation (G′ , U ′ ) of finite type (more specifically, G′ will be
the Levi subgroup of the stabilizer of the point where the fiber lives). We use information about
the orbit closures in (G′ , U ′ ) and patch them together to get subvarieties Y of the total space of U .
Given a section v ∈ H0 (X; U ), we consider the subvarieties v(G/P ) ∩ Y, and in particular, when
the grade of the ideal sheaf does not change. In all of the orbit closures in U ′ of relevance, we
give free resolutions for their coordinate rings and some related modules. Then this gives a locally
free resolution of v(X) ∩ Y ⊂ v(G/P ) ∼
= G/P via Theorem 1.7, and this will allow us to read off
properties of this variety. In particular, we can try to use this resolution to calculate the canonical
sheaf of v(G/P ) ∩ Y and the cohomology of its structure sheaf.
In all cases, we will find a choice of P so that one of the degeneracy loci (or a variety closely
related to it) is a torsor over an Abelian variety. We will also explore what happens when we vary
the choice of P . In some cases, we are able to establish a direct link between these other degeneracy
loci and the torsor via some classical geometric constructions (such as projective duality).
Lemma 3.4. Let Y ◦ be a G′ -orbit in U ′ and let Y ◦ be the union of these G′ -orbits over all fibers.
There is a nonempty open subset UYgen ⊂ U such that either codim(v(G/P ) ∩ Y ◦ , v(G/P )) =
codim(Y ◦ , U ) for all v ∈ UYgen , or v(G/P ) ∩ Y ◦ = ∅ for all v ∈ UYgen . In particular, this intersection
either has expected codimension or is empty. Furthermore, if the base field has characteristic 0,
there is a nonempty open subset UYsm such that v(G/P ) ∩ Y ◦ is smooth.
Proof. Define Z = {(v, x) ∈ U × G/P | v(x) ∈ Y ◦ }. Let π1 : Z → U and π2 : Z → G/P be the
projection maps. We claim that π2 is a fiber bundle with smooth fibers. Fix x ∈ G/P and let U (x)
be the fiber of U at x. Then the restriction map ρx : U → U (x) is G′ -linear and it is surjective since
◦
◦
◦
U (x) is irreducible as a G′ -module. Hence the map ρx : ρ−1
x (Y ) → Y is an affine bundle. But Y
−1
−1
◦
is smooth, and ρx (Y ) = π2 (x), so the claim follows. In particular, Z is smooth and
dim Z = dim G/P + dim U − codim(Y ◦ , U ′ ).
Now consider the map π1 . If it is dominant, then the fibers over a nonempty open subset
UYgen have dimension dim G/P − codim(Y ◦ , U ′ ) and hence have expected codimension. Otherwise,
the fibers are empty over a nonempty open subset UYgen . Given v ∈ U , we have an identification
π1−1 (v) = v(G/P ) ∩ U , so this proves the first claim.
The last statement follows from generic smoothness applied to π1 .
We will define U gen ⊂ U to be the intersection of UYgen over all orbits Y in U ′ , and similarly we
define U sm .
Remark 3.5. The idea of studying degeneracy loci using perfect complexes rather than cohomology class formulas has been considered by the second author in [Sam] for the class of Schubert
determinantal loci.
Example 3.6. Let V be a vector space of dimension 2n and let q ∈ S 2 V be a nondegenerate
quadratic form. Denote the quotient S 2 V /hqi by S02 V .
Consider the action of SO(V ) × C∗ on S02 V . Given a nondegenerate quadric q ′ ∈ S02 C2n , we
can form the pencil xq + yq ′ . The determinant of the associated symmetric matrix gives us 2n
points in P1 and hence a hyperelliptic curve C of genus n − 1, and this process is reversible. This
situation was considered by Weil.
19

Now consider the intersection of the quadrics defined by q and q ′ in P(V ∗ ). Then the variety
of Pn−2 ’s in q ∩ q ′ is isomorphic to the Jacobian of C (after fixing a base point), and the variety of
Pn−3 ’s in q ∩ q ′ is isomorphic to the moduli space SUC (2, L) (see §3.2) where L is any line bundle
of odd degree. See [DR, Theorems 1, 2] for details.
These constructions can be interpreted as degeneracy loci as follows. Consider the orthogonal
Grassmannian OGr(n − 1, V ∗ ), which is the subvariety of Gr(n − 1, V ∗ ) whose points are totally
isotropic subspaces for q. The trivial bundle Gr(n − 1, V ∗ ) × V ∗ has a tautological rank n − 1
subbundle R = {(x, W ) | x ∈ W }. Then we have S02 V = H0 (OGr(n − 1, V ∗ ); S 2 R∗ ) and q ′ gives a
generic section whose zero locus is the variety of Pn−2 ’s in q ∩ q ′ . Similar comments apply to the
variety of Pn−3 ’s using OGr(n − 2, V ∗ ). Modular interpretations for the degeneracy loci for the
other Grassmannians OGr(k, V ∗ ) are given by Ramanan [Ram, §6, Theorem 3].

C5 ⊗

4

V2

C5 .

In the rest of the article, we will work over the complex numbers C. In fact, many results will
hold over more general fields, but as we have not done a systematic investigation of the correct
hypotheses, we will not make any attempt toVbe more general.
The analysis of the representation C5 ⊗ 2 C5 is in fact easy to handle by more direct means,
but we want to illustrate our approach. We also mention that Fisher has examined this case as
well, see [Fis1] and [Fis2].
Let A and B
V2be vector spaces of dimension 5. The relevant data:
B
• U =A⊗
• G = (GL(A) × GL(B))/{(x, x−1 ) | x ∈ C∗ }
∗ ) = Gr(1, A∗ )
• G/P = P(A
V
V
∗ ⊗ 2 B ∼ O(−1) ⊗ 2 B
• U =R
=
V
• U ′ = 2 C5
• G′ = GL5 (C)
The ring of invariants Sym(U ∗ )(G,G) is a polynomial ring with generators of degrees 20, 30, and
the graded Weyl group is Shephard–Todd group 16 [Vin1, §9].

4.1

Modules over OU ′ .

We are only interested in the ideal of 4 × 4 Pfaffians of U ′ . This situation was explained in
Example 1.10.

4.2

Geometric data from a section.

The constructions in this section work over an arbitrary field. V
We get the following locally free resolution over OU = Sym( 2 B ∗ ⊗ OU (1)):
∗ ⊗2

0 → (det B )

∗

∗

⊗ OU (−5) → (det B ) ⊗ B ⊗ OU (−3) →

4
^

B ∗ ⊗ OU (−2) → OU → OC → 0

where C has codimension 3 in the total space of U . Its singular locus is the zero section of U , and
has codimension 10 in U .
For v ∈ U gen , Cv = C ∩ v(P(A∗ )) will have codimension 3 in v(P(A∗ )) ∼
= P(A∗ ). By generic

20

perfection, we get a locally free resolution for OC :
0 → (det B ∗ )⊗2 ⊗ OP(A∗ ) (−5) → (det B ∗ ) ⊗ B ∗ ⊗ OP(A∗ ) (−3)
→

4
^

B ∗ ⊗ OP(A∗ ) (−2) → OP(A∗ ) → OC → 0.

This gives enough information to see that ωC = OC , dim H0 (C; OC ) = 1, and that dim H1 (C; OC ) =
1. In particular, C is a curve of genus 1. We can also deduce that C is projectively normal and
embedded by a complete linear series.
Conversely, given a smooth curve C of genus 1 embedded in P(A∗ ) by a complete linear series,
its homogeneous ideal I is generated by 5 quadrics and is a codimension 3 Gorenstein ideal. The
Buchsbaum–Eisenbud classification of such ideals says that we can recover a section v ∈ U which
gives rise to C.
Theorem 4.1. We have a bijection between G-orbits in U sm and the set of pairs (C, L) where C
is a genus 1 curve and L is a degree 5 line bundle on C.

4.3

Examples of singular quintic curves.

In this section, we give a few examples of degenerations of the smooth elliptic quintic C. Describing
degenerations of the Abelian varieties in the later examples will require more effort and will appear
in future work.
We pick homogeneous coordinates z1 , . . . , z5 on P4 .
Example 4.2. Here is one example of a section that

0
z5
z1
−z5
0
z2

−z1 −z2
0

−z2 −z3 −z4
−z3 −z4 −z 5

gives a rational nodal curve:

z2 z3
z3 z4 

z4 z5 

0 0
0 0

It is given by the parametrization [a : b] 7→ [a5 + b5 : ab4 : a2 b3 : a3 b2 : a4 b] and its node is the
point [1 : 0 : 0 : 0 : 0]. The stabilizer subgroup in G of this curve is the dihedral group of size 10
generated by the transformations [a : b] 7→ [b : a] and [a : b] 7→ [a : ζb] where ζ is a primitive 5th
root of unity [Fis2, proof of Lemma 2.3]. This is not an unstable orbit.
Furthermore, its secant variety is an irreducible quintic hypersurface.
Example 4.3. We can get a triangle consisting
is one example:

0
0
 0
0

−z4
0

−z3 −z2
−z2 −z1

of two smooth rational quadrics and a line. Here

z4 z3 z2
0 z2 z1 

0 0 −z5 

0 0 −z4 
z5 z4
0

The quadrics are [a : b] 7→ [a2 : ab : b2 : 0 : 0] and [a : b] 7→ [0 : 0 : a2 : ab : b2 ] and the line
is [a : b] 7→ [a : 0 : 0 : 0 : b]. The secant variety is the union of z4 = 0, z2 = 0, and the cubic
z1 z42 + z22 z5 − z1 z3 z5 = 0.

21

Example 4.4. We can also get a union of 5 P1 ’s which are labeled with i ∈ Z/5 such that P1i
intersects P1j if and only if j = i ± 1, and the intersection points are distinct. This is a Néron
pentagon. All Néron pentagons form a single orbit since they are determined by their points of
intersection. Here is one example of a section that gives a Néron pentagon:


0
z1
z2
0
0
−z1
0
0
z3
0


−z2
0
0
0
z4 

.
 0 −z3
0
0
z5 
0
0
−z4 −z5 0
This is the set of points in P(A∗ ) with at least 3 coordinates equal to 0. Its stabilizer subgroup
contains the normalizer of the diagonal subgroup in SL(A). So the orbit of Néron pentagons has
codimension at least 5.
Its secant variety is the hypersurface z1 z2 z3 z4 z5 = 0.
Example 4.5. Here is a non-reduced example of a
reduced line which intersect with multiplicity 2:

0
0
z5
 0
0
0

−z5
0
0

−z3 −z2 −z4
−z2 −z1 −z3

union of a rational normal cubic and a non
z3 z2
z2 z1 

z4 z3 
.
0 0
0 0

The cubic is [a : b] 7→ [a3 : a2 b : ab2 : b3 : 0] and the line is given by the ideal (z1 , z2 , z32 ).
Its secant variety is the union of the hyperplane z5 = 0 and the non-reduced quartic (z22 −
z1 z3 )2 = 0.
Example 4.6. Here is a rational cuspidal cubic:

0
z1
z4
0
−z1
0
0
z
5

−z4
0
0
z
2

 0 −z5 −z2
0
−z5 −z2 −z3 −z4


z5
z2 

z3 
.
z4 
0

Its cusp point is [0 : 0 : 1 : 0 : 0] and it has the parametrization
√
√
[s : t] 7→ [ −1t5 : s3 t2 : s5 : −1s2 t3 : st4 ].
This vector lies in the unstable locus of the representation. The group of automorphisms of this
curve V
that extend to automorphisms of P4 is generated by scaling t, so the orbit of this curve in
5
C ⊗ 2 C5 has codimension 2. In particular, it gives a generic point of the unstable locus.
Its secant variety is an irreducible quintic hypersurface.

4.4

Secant and tangential varieties.

Here is a different approach:
• G/P = Gr(2,
A∗ )
V
2
B
• U = R∗ ⊗
22

V
• U ′ = C2 ⊗ 2 C5
• G′ = (GL2 (C) × GL5 (C))/{(x, x−1 ) | x ∈ C∗ }
The relevant G′ -orbit closures in U ′ are of codimensions 2, 4, and 5. The singular locus and
the non-normal locus of the codimension 2 orbit closure are both the codimension 4 orbit closure
S ′ . Also, S ′ is smooth along the codimension 5 orbit closure T ′ . Furthermore, S ′ and T ′ can be
identified as the secant and tangential varieties of the affine cone over the Segre variety P1 ×Gr(2, 5).
Let S and T be the global versions of these varieties, and given a section v ∈ U , let S and T be
the corresponding degeneracy loci.
Proposition 4.7. S is the locus of planes W such that deg(P(W ) ∩ C) ≥ 2 and T is the locus of
planes W such that P(W ) is tangent to some point of C.
V
Proof. Pick W ∈ Gr(2, A∗ ). Then W ∈ S \ T if and only if v(S) ∈ (R∗ ⊗ 2 B)(W ) is a sum of
the form a1 ⊗ (b1 ∧ c1 ) + a2 ⊗ (b2 ∧ c2 ) where aV1 and a2 are linearly independent. But we can also
identify this fiber with H0 (P(W ); OP(A∗ ) (1) ⊗ 2 B) when we identify P(W ) with a line in P(A∗ ).
This means that P(W ) ∩ C consists of two points corresponding to the vanishing of a1 and a2 .
Since T is in the closure of S \ T , we finish via a limiting argument.
In particular, T is a smooth curve. To calculate its genus, we can use a free resolution for T ′
(here we use (λ; µ)(−i) as shorthand for Sλ (C2 ) ⊗ Sµ (C5 ) ⊗ Sym(U ′∗ )(−i)):
F1 = (2, 1; 2, 1, 1, 1, 1)(−3) + (2, 2; 2, 2, 2, 2, 0)(−4)
F2 = (2, 2; 2, 2, 2, 1, 1)(−4) + (4, 1; 2, 2, 2, 2, 2)(−5) + (3, 2; 3, 2, 2, 2, 1)(−5)
F3 = (4, 2; 3, 3, 2, 2, 2)(−6) + (3, 3; 4, 2, 2, 2, 2)(−6) + (4, 3; 3, 3, 3, 3, 2)(−7)
F4 = (4, 2; 4, 3, 3, 3, 3)(−8) + (4, 4; 4, 3, 3, 3, 3)(−8)
F5 = (6, 4; 4, 4, 4, 4, 4)(−10).
So a locally free resolution for T is
F1 = R(−1)5 + O(−2)15

F2 = O(−2)5 + S 3 (R)(−1) + R(−2)24

F3 = S 2 (R)(−2)10 + O(−3)15 + R(−3)5
F4 = S 2 (R)(−2)5 + O(−4)5
F5 = S 2 (R)(−4),

and we see that T has genus 1.
Proposition 4.8. C ∼
=T
Proof. Since C is smooth, we have a well-defined morphism ψ : C → T obtained by sending x ∈ C
to the tangent line at x. This is a finite morphism, and by Riemann–Hurwitz, the ramification
divisor is 0. Hence ψ is an isomorphism.

4.5

Chow forms.

Yet another approach is as follows.
∗
• G/P = Gr(3,
V2 A )
∗
• U =R ⊗ VB
• U ′ = C3 ⊗ 2 C5
23

• G′ = (GL3 (C) × GL5 (C))/{(x, x−1 ) | x ∈ C∗ }
The space U ′ contains a G′ -invariant degree 15 hypersurface. The corresponding degeneracy
locus X ′ is a section of O(5).
Proposition 4.9. X ′ is the Chow form of C.
Proof. To obtain the Chow form of C, let Fl(1, 3, A∗ ) be a partial flag variety with projections π1 , π2
to P(A∗ ) and Gr(3, A∗ ). Then Rπ2∗ Lπ1∗ OC is quasi-isomorphic to a complex whose determinant
is the Chow form. A locally free resolution of OC in P(A∗ ) is
∗ 2

∗

0 → (det B ) (−5) → S2,14 B (−3) →

4
^

B ∗ (−2) → OP(A∗ ) .

Applying Rπ2∗ π1∗ to this complex gives a 2-term complex over OGr(3,A∗ ) :
0 → (det B ∗ )2 ⊗ S 2 R(−1) → OGr(3,A∗ ) ⊕ S2,14 B ∗ (−1).

(4.10)

In this case, the determinant is just the actual determinant of a 6 × 6 matrix. This gives a section
of O(5) which is the Chow form of C.
V
We claim that this map is a sheafy version of the following map. For U ′ = C3 ⊗ 2 C5 , we
have (det C5 )−2 ⊗ S3,1,1 (C3 )∗ ⊂ S 5 (U ′∗ ) with multiplicity 1, and also (det C5 )−2 ⊗ S3,1,1 (C3 )∗ ⊂
S 2 (U ′∗ ) ⊗ (det C3 )∗ ⊗ S2,14 (C5 )∗ with multiplicity 1, so this gives a 6 × 6 matrix
(det C5 )−2 ⊗ S3,1,1 (C3 )∗ ⊗ Sym(U ′∗ ) → Sym(U ′∗ )(5) ⊕ [(det C3 )∗ ⊗ S2,14 (C5 )∗ ⊗ Sym(U ′∗ )(2)]
whose determinant is the degree 15 invariant.
Taking sections of (4.10), we see that the process of constructing the Chow form of C is a
4
2
map that takes a section v to an element in [(det B)2 ⊗ S3,1,1 A]
This can be
V2⊕ [∧ B ⊗ S A].
2
5
B) → (det B) ⊗ S3,1,1 A and
interpreted
V as GL(A) × GL(B)-equivariant linear maps S (A ⊗
S 2 (A ⊗ 2 B) → ∧4 B ⊗ S 2 A. But such maps are unique up to scalar (checked with LiE [LiE]), so
the claim follows.

4.6

Projective duality.

Here is another approach.
∗
• G/P = Gr(4,
V2 A ) = P(A)
∗
• U =R ⊗ VB
• U ′ = C4 ⊗ 2 C5
• G′ = (GL4 (C) × GL5 (C))/{(x, x−1 ) | x ∈ C∗ }
The space U ′ contains a G′ -invariant degree 40 hypersurface. The polynomial f is described
as
V
follows: let a1 , . . . , a40 be a basis for the Lie algebra C ⊕ sl4 ⊕ sl5 . Then for x ∈ C4 ⊗ 2 C5 , we
have f (x) = det(A1 x · · · A40 x).
The corresponding degeneracy locus is a degree 10 hypersurface C ′ in P(A). By the Katz–
Kleiman formula [GKZ, §2.3.C], the projective dual of an Abelian variety X ⊂ PN is a hypersurface
of degree (dim X + 1)(deg X). In our case, for X = C from the last section, we get a hypersurface
of degree 10.
Proposition 4.11. For generic v ∈ V , the projective dual of C is C ′ .

24

V
V
Proof. Let f be the equation for the hyperdiscriminant of C4 ⊗ 2 C5 . An element x ∈ C4 ⊗ 2 C5
can be thought of as a 5 × 5 skew-symmetric matrix of linear forms on P3 . Generically, the 4 × 4
Pfaffians give a 0-scheme of degree 5, and f (x) = 0 if andVonly if this 0-scheme is non-reduced.
So for generic v ∈ V , we first write V = H0 (P(A); Q⊗ 2 B). Then as a 4×10 matrix (along the
fibers), v has full rank 4 over each point in P(A). Now
V Then H is a hyperplane in
V pick H ∈ P(A).
P(A∗ ) and we have a canonical identification H0 (H; 2 B(1)) = (Q⊗ 2 B)(H) (where the notation
(H) means “fiber at H”). So H ∩ C is identified with the 5 points mentioned above, and hence
H ∈ C ∨ if and only if f (v(H)) = 0, which proves our claim.

5

V3

C9 .

Let V be aVvector space of dimension 9. The relevant data:
• U = 3V
• G = GL(V )
∗
• G/P V
= P(V ∗ ) = Gr(1,
V2 V ∗ )
2 ∗
∗
∼
Q ⊗ O(1) ∼
• U = VQ ⊗R =
= Ω2 (3)
2
C8
• U′ =
′
• G = GL8 (C)
The ring of invariants Sym(U ∗ )(G,G) is a polynomial ring with generators of degrees 12, 18, 24,
30, and the graded Weyl group W is Shephard–Todd group 32 [Vin1, §9].
Remark 5.1. The invariants for W acting on its reflection representation h were explicitly calculated by Maschke [Mas]. It is known that the GIT quotient U//G ∼
= h/W contains an open set
which is isomorphic to the moduli space of genus 2 curves C with a marked Weierstrass point (i.e.,
ramification point for the hyperelliptic map). This was shown by Burkhardt [Bur]. See also [DL,
§4.3] for a modern treatment and further discussion.
V
Remark 5.2. The orbits in 3 C9 were classified in [EV], but the connection to geometric objects
as treated here is not made.

5.1

Modules over OU ′ .

The orbits in U ′ are given by the vanishing of Pfaffians of various sizes. We are only interested in
the vanishing locus of the 8 × 8 Pfaffian and the vanishing locus of the 6 × 6 Pfaffians. The latter
is described in Example 1.11. We denote their global versions in U by Y and X , respectively.

5.2

Geometric data from a section.

V
The subvariety Y has the following resolutions over OU = Sym( 2 Q ⊗ O(−1)):
0 → (det Q) ⊗ OU (−4) → OU → OY → 0.

(5.3)

We can simplify this by noting that det Q = O(1).
From Example 1.11, the subvariety X has this resolution:
0 → OU (−9) →

2
^

2

Q ⊗ OU (−7) → S2,16 Q ⊗ OU (−7)

→ (S Q ⊗ OU (−4)) ⊕ ((S 2 Q)∗ ⊗ OU (−5))

→ S2,16 Q ⊗ OU (−4) →

6
^

Q ⊗ OU (−3) → OU → OX → 0.
25

(5.4)

So for v ∈ U gen , we have that Y = Y ∩ v(P(V ∗ )) and X = X ∩ v(P(V ∗ )) will have codimensions
1 and 6 in P(V ∗ ), respectively.
The self-duality of the resolution for OX shows that ωX = Ext6 (OX , O(−9)) = OX .

Theorem 5.5. For v ∈ U gen , we have hi (X; OX ) = 2i . In particular, for v ∈ U sm , X is a torsor
over an Abelian surface.
(We will only do this calculation once. For the remaining examples, we leave it to the reader.)
Proof. First, we replace the sheaf of algebras OU with the structure sheaf of P(V ∗ ) in (5.4).
In the notation of §2.3, we have P(V ∗ ) = Fl(1; V ∗ ) and Sλ Q ⊗ OP(V ∗ ) (d) = R(µ) where µ =
(d, −λ8 , −λ7 , . . . , −λ1 ), and ρ = (8, 7, . . . , 1, 0). In particular, when we add ρ to the sequence µ for
any term in homological degrees {2, 3, 4, 5} of (5.4), all cohomology vanishes by Theorem 2.9 because there will always be a repeating term. For example, S2,16 Q⊗O(−4) has vanishing cohomology
because µ + ρ = (4, 7, 5, 4, 3, 2, 1, 0, −2).
V
Now consider the remaining terms. For 6 Q ⊗ O(−3), we have λ = (16 ), so µ + ρ =
(5, 7, 6, 4, 3, 2, 1, 0, −1). We can sort this using 2 consecutive swaps, and
V subtracting ρ again leaves
∗ ); 6 Q ⊗ O(−3)) = det V . By
us with a sequence of all −1. Hence Theorem 2.9 says H2 (P(VV
similar considerations (or Serre duality), one can show that h6 ( 2 Q ⊗ O(−7)) = 1. Finally, we
already know that h0 (OP(V ∗ ) ) = h8 (O(−9)) = 1.
Now the result follows from a spectral sequence argument (or equivalently by splicing (5.4) into
short exact sequences). The last statement follows from Theorem 3.1.
From (5.3), we see that Y is a cubic hypersurface. In fact, it is the Coble cubic of X, see [Be2]
and [Cob1] for more information on the Coble cubic.
Proposition 5.6. The polarization on X induced by OX (1) is indecomposable and of type (3, 3).
Proof. We have h0 (OX (1)) = 9, so X is embedded via a complete linear series. So K(O(1)) =
Z/D⊕Z/D where D = (3, 3) or D = (1, 9). Since X is the singular locus of a cubic hypersurface, we
conclude that it is the intersection of the quadrics (partial derivatives) that contain it. However, an
Abelian surface in P8 with a (1, 9)-polarization cannot be the intersection of the quadrics containing
it [GP, Remark 3.2], so we conclude that the polarization is (3, 3).
Furthermore, X is not the product of 2 elliptic curves as a polarized Abelian variety. To see
this, first consider elliptic curves E1 , E2 ⊂ P2 embedded as cubics. Then the product polarization
is given by theVSegre embedding
E1 × E2 ⊂ P2 × P2 ⊂ P8 . The quadratic equations vanishing on
V2 3
2 3
2
2
C and the quotient of S 2 (C3 ⊗ C3 ) by this space is S 2 C3 ⊗ S 2 C3 , none
C ⊗
P × P give
of which vanish on E1 × E2 . Hence E1 × E2 is not the intersection of quadrics in P8 .
Remark 5.7. It could be the case that X is abstractly isomorphic to the product of two elliptic
curves if we ignore the polarizations. See for example [Poo, §12].
Remark 5.8. Given a curve C of genus 2, the moduli space SUC (3) admits a degree 2 map to
P8 which is branched along a degree 6 hypersurface. This hypersurface is projectively dual to the
Coble cubic (see [Ngu] and [Ort] for two different proofs of this, together with some more discussion
of the hypersurfaces).

26

5.3

Macaulay2 code.

Here we give some Macaulay2 code for generating examples of sections. To do this, first consider
the short exact sequence of vector bundles over P(V ∗ ):
0→

2
^

Q∗ ⊗ O(1) →

2
^

V ⊗ O(1) → Q∗ ⊗ O(2) → 0.

V
V
Taking sections, we get the inclusion 3 V ⊂ 2 V ⊗
V3V . Since this map is GL(V )-equivariant, it
V , we can comultiply to get an element of
must be the comultiplication map. Thus given v ∈
V
2
V ⊗ V , which we may think of as a skew-symmetric 9 × 9 matrix ϕv of linear forms on P(V ∗ ).
To restrict back to skew-symmetric matrix on Q∗ , we restrict our attention to an affine open
set. Pick homogeneous coordinates z1 , . . . , z9 on P(V ∗ ) and consider the open set given by z9 = 1.
Then z1 , . . . , z8 give a trivialization for Q over this open set, so the relevant data is the upper-left
8 × 8 submatrix ϕ′v of ϕv . Now our degeneracy loci correspond to the usual Pfaffian loci of this
submatrix. Assuming that there are no components contained in the hyperplane z9 = 0, we get
the homogeneous ideals of these degeneracy loci by saturating with respect to z9 .
The function basicMat below takes as input s = (i,j,k) and computes the matrix ϕ′v for
v = zi ∧ zj ∧ zk . Then we take random coefficients and calculate the Pfaffian ideals.
P=101;
R = ZZ/P[z_1..z_9];
basicMat = s -> (
ans := mutableMatrix(0*id_(R^8));
ans_(s_0-1,s_1-1) = -z_(s_2);
ans_(s_1-1,s_0-1) = z_(s_2);
if not(member(9,s)) then (
ans_(s_0-1,s_2-1) = z_(s_1);
ans_(s_2-1,s_0-1) = -z_(s_1);
ans_(s_1-1,s_2-1) = -z_(s_0);
ans_(s_2-1,s_1-1) = z_(s_0);
);
matrix ans
)
M=0;
for i in subsets(1..9,3) do M = M + random(ZZ/P) * basicMat(i);
I = saturate(pfaffians(8,M),ideal(z_9));
J = saturate(pfaffians(6,M),ideal(z_9));
K = saturate(pfaffians(4,M),ideal(z_9));
Then generically, I is the ideal of the cubic hypersurface, J is the homogeneous ideal of the
Abelian surface, and K is the unit ideal.
Example 5.9. Here is an example calculation we can do with this. Define M by
M=0;
for i from 1 to 3 do
for j from 4 to 6 do
for k from 7 to 9 do M = M + random(ZZ/P) * basicMat({i,j,k});
27

V
This gives us a generic vector in C3 ⊗ C3 ⊗ C3 ⊂ 3 V , where the subspace corresponds to the
choice of a decomposition V = C3 ⊕ C3 ⊕ C3 . Using the command primaryDecomposition J
we see that J now defines a reducible variety of degree 12 with 2 components each contained in
z1 = z2 = z3 = 0 and z4 = z5 = z6 = 0. In fact, we know that J is supposed to have degree 18.
There will be a third component of degree 6 inside z7 = z8 = z9 = 0 which we do not see because
of our choice of open affine (one can check this by modifying the definition of basicMat to use a
different affine trivialization). In fact, if we intersect any two of the three ideals in the primary
decomposition of J, we will get an ideal generated by 6 linear equations and a cubic, which is
exactly a plane cubic curve.

V4

6

C8 .

Let V be aVvector space of dimension 8. The relevant data:
• U = 4V
• G = GL(V )
• G/P V
= P(V ∗ ) = Gr(1,
V ∗)
V
3
3 ∗
Q∗ ⊗ O(1) ∼
• U = V Q ⊗ R∗ ∼
= Ω3 (4)
=
3 7
′
C
• U =
• G′ = GL7 (C)
The ring of invariants Sym(U ∗ )(G,G) is a polynomial ring with generators of degrees 2, 6, 8, 10,
12, 14, 18, and the graded Weyl group W is the Weyl group of type E7 [Vin1, §9].
Remark 6.1. Letting h be the 7-dimensional reflection representation of W , it is known that the
GIT quotient U//G ∼
= h/W has an open subset isomorphic to the moduli space of smooth plane
quartics (i.e., non-hyperelliptic genus 3 curves) with a marked flex point (i.e., a point with tangency
of order ≥ 3; there are 24 of them for a generic curve). See for example [DO, §IX.7, Remark 7]
(but note that the 21 mentioned there should be 24) or [Loo, Proposition 1.11].

6.1

Modules over OU ′ .

We are interested in 3 orbits in U ′ . The first is a degree 7 hypersurface Y ′ . The singular locus
of Y ′ is an orbit closure X ′ of codimension 4 in U ′ , and the singular locus of X ′ is
V3an orbit
′
′
′
W . Set
closure Z Vof codimension 7 in U . Let W be a 7-dimensional vector space with U =
A = Sym( 3 W ∗ ) = OU ′ . The minimal free resolutions of these orbit closures are given as follows.
The minimal free resolution of OX ′ is given by
∗ 5

∗

∗ 4

0 → (det W ) ⊗ W (−12) → (det W ) ⊗
(det W ∗ )2 ⊗ S2,15 (W ∗ )(−7) → (det W ∗ )2 ⊗

4
^

2
^

W ∗ (−10) →

(6.2)

W ∗ (−6) → A → OX ′ → 0.

The minimal free resolution for OZ ′ is given by
0 → S76 ,6 W ∗ (−16) → S7,65 ,5 W ∗ (−14) → S62 ,54 ,4 W ∗ (−12) → S53 ,43 ,3 W ∗ (−10)
→ S44 ,32 ,2 W ∗ (−8) → S35 ,2,1 W ∗ (−6) → S26 W ∗ (−4) → A → OZ ′ → 0.

(6.3)

We will also use the fact that the local ring of OX ′ at the generic point of OZ ′ is a Cohen–Macaulay
ring of type 3, i.e., its canonical module is minimally generated by 3 elements. This calculation
was done by Federico Galetto [Gal].
28

We set M to be a certain twist of the cokernel of the dual of the last differential in (6.2). In
particular, it has a presentation of the form
(det W ∗ )2 ⊗

2
^

W (−4) → det W ∗ ⊗ W (−2) → M → 0.

(6.4)

So up to a grading shift, M is the canonical module of OX ′ .
Lemma 6.5. There is a G′ -equivariant OX ′ -linear isomorphism S 2 M ∼
= IZ ′ ,X ′ .
Proof. From (6.4), we get the presentation
(det W ∗ )3 ⊗ W ⊗

2
^

W ⊗ OX ′ (−6) → (det W ∗ )2 ⊗ S 2 W ⊗ OX ′ (−4) → S 2 M → 0.

Also, the presentation of IZ ′ in OU ′ is given by
(det W ∗ )3 ⊗ S2,1 W ⊗ OU ′ (−6) → (det W ∗ )2 ⊗ S 2 W ⊗ OU ′ (−4) → IZ ′ → 0.
V
To get a presentation over OX ′ we have to add in the relations (det W ∗ )3 ⊗ 3 W which come from
the ideal generators of IX ′ . Hence we see that the S 2 M and IZ ′ ,X ′ have the same presentations.
By equivariance, we get an isomorphism of these presentations up to a choice of a scalar, and this
implies the desired isomorphism S 2 M ∼
= IZ ′ ,X ′ .
We define Y, X , Z, and M to be the global versions of Y ′ , X ′ , Z ′ , and M .

6.2

Geometric data from a section.

Now choose a section v ∈ U gen and set Y = v(P(V ∗ ))∩Y, X = v(P(V ∗ ))∩X , and Z = v(P(V ∗ ))∩Z.
To get the locally free resolution of X over OP7 , we replace W with Q∗ and replace (−i) with
O(−i). After using det Q = O(1) to simplify, we get
0 → Q ⊗ O(−7) →

2
^

Q ⊗ O(−6) → S2,15 (Q) ⊗ O(−5) →

4
^

Q ⊗ O(−4) → OP7 → OX → 0.

(6.6)

Furthermore, note that ωX = M ⊗ Ov(P(V ∗ )) . From Lemma 6.5, we get an OX -linear map
e = SpecO (OX ⊕ωX ).
µ : S 2 ωX → OX and hence an OX -algebra structure on OX ⊕ωX . We define X
X

e O e ) = hi (X; OX ⊕ ωX ) = 3 .
As in Theorem 5.5, we can check that ωXe = OXe and that hi (X;
i
X
Define U sm+ to be the subset of U sm where the Cohen–Macaulay type of X along Z remains 3.
e is smooth. In particular, X
e is a torsor over an Abelian
Proposition 6.7. If v ∈ U sm+ , then X
e
3-fold, and X is the Kummer variety of X.

Proof. Let IZ denote the ideal sheaf of Z in OX . We check smoothness locally. Let P be a
nonsingular point of X ′ . Then IZ,P = OX,P and ωX,P = OX,P , and µP is just the multiplication
∼
map. Then OX,P
= OX,P [t]/(t2 − 1) as a ring, so the points over P are nonsingular. Now let P ∈ Z
e
be a singular point. Set R = OX,P . Then m = IZ,P is the maximal ideal of R. Write ωR = ωX,P
and S = OX,P
e . Then S = R ⊕ ωR as an R-module, and n = m ⊕ ωR is the unique maximal
ideal of S: S/n ∼
= R/m, and any element not in n is (r, 0) for r ∈ R \ m, so is a unit. Since the
multiplication map S 2 ωR → m is surjective, we have that n2 = m ⊕ m ·ωR . Hence n/n2 ∼
= ωR /m ·ωR .
29

The dimension of this space over R/m is the Cohen–Macaulay type of R, which is 3, and implies
e is smooth.
that S is a regular local ring. So we have shown that X
e
Using Theorem 3.1, the above facts imply that X is a torsor over an Abelian 3-fold. Temporarily
e so that it acquires a group structure. The map ι
choose a point P ∈ π −1 (Z) to be the origin of X
e Then ι fixes P and there is an
which swaps points in the same fiber of π gives an involution on X.
induced linear map on the tangent space at P . Since ι has order 2, the induced linear map is the
e so we conclude that ι is the
negation map. But this is the derivative of the inversion map on X,
e
e
inversion map on X. So X is the Kummer variety X/hιi.
e
Proposition 6.8. L = π ∗ OX (1) defines an indecomposable (2, 2, 2)-polarization on X.

Proof. Note that

e L) = h0 (X; π∗ L) = h0 (X; (OX ⊕ ωX ) ⊗ OX (1)) = 8,
h0 (X;

e → P(V ∗ ) is given by a complete linear series. Hence L = L(a, b, c) where abc = 8.
so the map X
By [Mum, §2, Corollary 4], each of a, b, c must be even, so we have a = b = c = 2. Furthermore,
e L) is a product of s Abelian varieties,
e L) is indecomposable since by [BL, Theorem 4.8.2], if (X,
(X,
∗
s
e
then the degree of the map X → X ⊂ P(V ) is 2 .
In particular, this Abelian 3-fold is the Jacobian of some genus 3 curve C.

Proposition 6.9. C is not hyperelliptic.
Proof. Using the locally free resolution (6.6), the map H0 (OP(V ∗ ) (n)) → H0 (OX (n)) is surjective
for all n ≥ 0. Hence X is projectively normal, so C is not hyperelliptic by [Kha, §2.9.3].
Remark 6.10. The quartic hypersurface has an interpretation as the embedding of SUC (2) (see
Section 3.2) via its theta divisor [NR2]. This is also known as the Coble quartic, see [Be2] and
[Cob2] for more details.

6.3

Flag variety.

Here is another approach which was pointed out to us by Jack Thorne.
• G/P = Fl(1,
7, V ∗ )
V
3
∗
∗
• U =R
V13⊗ 6 (R7 /R1 )
′
C
• U =
• G′ = GL6 (C).
We are interested in the smallest orbit closure in U ′ , which is the affine cone over Gr(3, 6). This
has codimension
V 10. In characteristic 0, its minimal free resolution is as follows (the coordinate
ring is Sym( 3 C6 ) and we only list the partitions).
F1 = (2, 14 )

F2 = (24 , 1) + (3, 2, 14 )
F3 = (32 , 22 , 12 ) + (35 ) + (5, 25 )
F4 = (43 , 23 ) + (42 , 33 , 1) + (5, 33 , 22 )
F5 = (5, 42 , 32 , 2)⊕2
F6 = (52 , 43 , 2) + (53 , 33 ) + (6, 43 , 32 )
F7 = (7, 45 ) + (55 , 2) + (62 , 52 , 42 )
30

F8 = (64 , 5, 4) + (7, 6, 54 )
F9 = (7, 64 , 5)
F10 = (76 ).
To see the ranks, we include the graded Betti table (this follows Macaulay2 notation, so a term
d in row i and column j represents the free module A(−i − j)⊕d in homological degree j).
total:
0:
1:
2:
3:
4:

0 1
2
3
4
5
6
7
8 9 10
1 35 140 301 735 1080 735 301 140 35 1
1 .
.
.
.
.
.
.
. . .
. 35 140 189
.
.
.
.
. . .
. .
. 112 735 1080 735 112
. . .
. .
.
.
.
.
. 189 140 35 .
. .
.
.
.
.
.
.
. . 1

Remark 6.11. The resolution changes in characteristic 2:
total:
0:
1:
2:
3:
4:

0 1
2
3
4
5
6
7
8 9 10
1 35 141 302 735 1080 735 302 141 35 1
1 .
.
.
.
.
.
.
. . .
. 35 140 190
.
.
.
.
. . .
. .
1 112 735 1080 735 112
1 . .
. .
.
.
.
.
. 190 140 35 .
. .
.
.
.
.
.
.
. . 1

and in characteristic 3:
total:
0:
1:
2:
3:
4:

0 1
2
3
4
5
6
7
8 9 10
1 35 140 321 756 1082 756 321 140 35 1
1 .
.
.
.
.
.
.
. . .
. 35 140 189 20
1
.
.
. . .
. .
. 132 736 1080 736 132
. . .
. .
.
.
.
1 20 189 140 35 .
. .
.
.
.
.
.
.
. . 1

In all other characteristics, the Betti numbers agree with characteristic 0.
The corresponding degeneracy locus X2 is a torsor over an Abelian 3-fold: For each partition
|λ|/3
λ in the resolution for Gr(3, 6), we get the sheaf R1
⊗ Sλ (R7 /R1 ). Also, the canonical sheaf of
that ωX2 = OX2 .
Fl(1, 7, V ∗ ) is (det R7 )7 ⊗ R71 , which is the last term of the resolution. This shows
3
i
As in Theorem 5.5, we can use Borel–Weil–Bott to get that h (X2 ; OX2 ) = i .
e
Proposition 6.12. X2 ∼
= X.

Proof. Let π : Fl(1, 7, V ∗ ) → P(V ∗ ) be the projection map. We claim that π(X2 ) = X and that
π|X2 is a finite morphism of degree 2. Note that π ∗ Q = V ∗ /R1 and π ∗ R = R1 . For x ∈ P(V ∗ ),
pick (x ⊂ H) ∈ π −1 (x). We have a surjection
3
3
^
^
∗
∗
( Q ⊗ R )(x) → ( (R7 /R1 )∗ ⊗ R∗1 )(x ⊂ H)

and so (x ⊂ H) ∈ X2 if and only if v(x) ∈ (V ∗ /x)∗ is a pure tensor when mapped to (H/x)∗ . By
Remark 2.20, there exists such an H if and only if x ∈ X, so π(X2 ) = X. Furthermore, there
exists exactly 2 such H if x ∈ X \ Z and there exists exactly 1 such H if x ∈ Z. So π|X2 is a finite
e
morphism of degree 2. We conclude that X2 ∼
= X.
31

6.4

Projective duality.

We can instead work with the following data:
• G/P V
= Gr(7, V ∗ ) = P(V )
• U = V4 R∗
• U ′ = 4 C7
• G′ = GL7 (C)
V
V
The orbit classifications in 4 C7 and 3 C7 are the same,
V3 ∗ as in §6.2.
V4 so∗ we can proceed
∗
Q ⊗ O(1). Hence,
R becomes
Under the identification Gr(7, V ) = P(V ), the bundle
given a section v ∈ U sm+ , we get a quartic hypersurface Y d and a Kummer 3-fold X d in P(V ).
Proposition 6.13. We have isomorphisms X ∼
= Y . Furthermore, Y and Y d are
= X d and Y d ∼
projectively dual varieties.
Proof. We first show that X ∼
= X d . Consider the variety X2 ⊂ Fl(1, 7, V ∗ ) constructed in the
previous section from v. Let π2 : Fl(1, 7, V ∗ ) → Gr(7, V ∗ ) be the projection map. We claim that
π2 (X2 ) = X d and that π2 is finite of degree 2. This will prove the claim.
Note that π2∗ R = R7 . For H ∈ Gr(7, V ∗ ), pick (x ⊂ H) ∈ π2−1 (H). We have a surjection
3
4
^
^
∗
( R )(H) → ( (R7 /R1 )∗ ⊗ R∗1 )(x ⊂ H)

given by comultiplication. So (x ⊂ H) ∈ X2 ifVand only if v(x) ∈ H ∗ isVa pure tensor
mapped
V3when
4 7∗
7 7
7
∗
C , in which
C and
C , we can identify
to (H/x) . After picking a volume form in
case the rest of the argument is similar to the proof of Proposition 6.12.
So X and X d are embedded by dual linear series. By [Pau, Theorem 3.1], the Coble quartics
of X and X d are isomorphic and projective dual to one another.

6.5

Doing calculations.

We have written some Macaulay2 code for calculating the degeneracy loci in §6.2. However, it is
messy so we do not include it here. We will just comment on how one can practically go about
these calculations and explain the analogue of Example 5.9.
we need to find a way to calculate the ideals of the appropriate low codimension orbits in
V3 First,
7
C . An explicit construction of the equation f for the degree 7 hypersurface was given in [Kim,
Remark 4.4]. Namely, he constructs two symmetric 7 × 7 matrices ϕ(x) and ϕ∗ (x) whose entries
are homogeneous polynomials of degrees 3 and 4, respectively, such that ϕ(x)ϕ∗ (x) = f (x)I7 , i.e.,
the pair (ϕ(x), ϕ∗ (x)) is a matrix factorization for f . Explicit summation formulas are given for
the entries of these matrices.
From (6.2), the Jacobian ideal of f is reduced and gives the equations for the codimension 4
orbit. The ideal for the codimension 7 orbit is given by the entries of the matrix ϕ∗ (x). One
possibility to check this is to use that the representation generating this ideal isVS 2 C7 (up to a
∗
power of determinant) (6.3) and that this representation has multiplicity 1 in S 4 ( 3 C7 ).
In practical computations, the product of the matrices ϕ(x) and ϕ∗ (x) can be calculated in
a few seconds. A possible way to proceed is to evaluate these against a chosen section v (after
choosing an open affine in P7 to work over) and then to take the product in order to calculate the
equation of the codimension 1 degeneracy locus. If the section is generic, then the Jacobian ideal
of this equation will be the same as evaluating the Jacobian ideal of f at v (note that calculating
f and its Jacobian ideal before evaluating may take a long time).

32

Example 6.14. Suppose we choose
a decomposition V = C2 ⊕C2 ⊕C2 ⊕C2 , and we pick a generic
V
4
V . In this case, the codimension 4 degeneracy locus will have 4
vector in C2 ⊗ C2 ⊗ C2 ⊗ C2 ⊂
irreducible components, all of degree 6. Taking any two of these components, the affine cone over
their intersection is a bidegree (2, 2) hypersurface in C[x, y] ⊗ C[s, t] whose multi-graded Proj is a
nonsingular curve of genus 1 (this agrees with the data obtained in [Ho, §3.1.2])

V4

8
0C .

7

V
Let V be a vector space of dimension 8 equipped
a symplectic form ω ∈V 2 V ∗ .VThe symplectic
V
V2 with
V ∗ → 4 V ∗ and we define 40 V = 40 V ∗ to be the
form gives an injective multiplication map
cokernel. The
V relevant data:
• U = 40 V
• G = Sp(V )
∗)
• G/P V
= P(V ∗ ) = Gr(1, VV
3
⊥
∗
∼
• U = V0 (R /R) ⊗ R = 30 (R⊥ /R) ⊗ O(1)
• U ′ = 30 C6
• G′ = Sp6 (C)
The ring of invariants Sym(U ∗ )(G,G) is a polynomial ring with generators of degrees 2, 5, 6, 8,
9, 12, and the graded Weyl group W is the Weyl group of type E6 [Vin1, §9].
Remark 7.1. Letting h be the 6-dimensional reflection representation of W , it is known that the
GIT quotient U//G ∼
= h/W has an open subset isomorphic to the moduli space of smooth plane
quartics (i.e., non-hyperelliptic genus 3 curves) with a marked hyperflex point (i.e., a point with
tangency of order 4; the existence of such a point is a codimension 1 condition on the space of plane
quartics). See [Loo, Proposition 1.15].

7.1

Modules over OU ′ .

V3
′ =
Let W be a 6-dimensional vector space
equipped
with
a
symplectic
form,
so
we
can
write
U
0W
V3 ∗
′
and G = Sp(W ). Set A = Sym( 0 W ) = OU ′ .
There is a degree 4 G′ -invariant hypersurface in U ′ , which we denote by Y ′ . The other orbits
X ′ and Z ′ have codimension 4 and 7, respectively. The minimal free resolution of OX ′ is
0 → W ∗ ⊗ A(−7) →

2
^
0

W ∗ ⊗ A(−6) → S 2 W ∗ ⊗ A(−4) →

3
^
0

W ∗ ⊗ A(−3) → A → OX ′ → 0,
(7.2)

which we can calculate by Macaulay2. Furthermore, the last matrix has corank 3 when specialized
to a nonzero point in the highest weight orbit, so the localization of OX ′ at such a point is a
Cohen–Macaulay ring of type 3.
We set M to be a certain twist of the cokernel of the dual of the last differential of (7.2). In
particular, it has a presentation of the form
2
^
0

W (−2) → W (−1) → M → 0.

33

(7.3)

The orbit closure Z ′ is the affine cone over the Lagrangian Grassmannian. The minimal free
resolution for OZ ′ is
0 → A(−10) → S 2 W ∗ (−8) → S[2,1] W ∗ (−7) → S[2,1,1] W ∗ (−6)

→ S[2,1,1] W ∗ (−4) → S[2,1] W ∗ (−3) → S 2 W ∗ (−2) → A → OZ ′ → 0

(7.4)

(recall that S[λ] was defined in (2.7)).
Lemma 7.5. There is a G′ -equivariant OX ′ -linear isomorphism S 2 M ∼
= IZ ′ ,X ′ .
Proof. The proof is the same as for Lemma 6.5.
We define Y, X , Z, and M to be the global versions of Y ′ , X ′ , Z ′ , and M .

7.2

Geometric data from a section.

Now choose a section v ∈ U gen and set Y = v(P(V ∗ ))∩Y, X = v(P(V ∗ ))∩X and Z = v(P(V ∗ ))∩Z.
To get the locally free resolution of X over OP7 , we replace W with R⊥ /R and replace (−i)
with O(−i). So the locally free resolution for OX is
0 → R⊥ /R(−7) →

3
2
^
^
(R⊥ /R)(−6) → S 2 (R⊥ /R)(−4) → (R⊥ /R)(−3) → OP7 → OX → 0,
0

0

and the locally free resolution for ωX is
0 → OP7 (−8) →

3
^
0

(R⊥ /R)(−5) → S 2 (R⊥ /R)(−4) →

2
^
(R⊥ /R)(−2) → (R⊥ /R)(−1) → ωX → 0,
0

Using Borel–Weil–Bott as in Theorem 5.5, we get h0 (OX ) = 1 and h2 (OX ) = 3 and all other
cohomology vanishes. So by Serre duality, hi (OX ⊕ ωX ) = 3i .
From Lemma 7.5, we get a OX -linear multiplication map µ : S 2 ωX → OX and hence an OX e → X be the
e = SpecO (OX ⊕ ωX ) and let π : X
algebra structure on OX ⊕ ωX . We define X
X
projection map.
We define U sm+ to be the subset of U sm where the Cohen–Macaulay type of X along Z remains
3.
e is smooth. Furthermore, X
e is a torsor over an Abelian 3-fold,
Theorem 7.6. If v ∈ U sm+ , then X
∗
e
e is not the Jacobian of a
L = π OX (1) is an indecomposable (2, 2, 2)-polarization on X, and X
hyperelliptic curve.
Proof. The proofs are analogous to the ones in §6.2.

8

spin(16).

Let B be a vector space of dimension 16 equipped with a quadratic form q ∈ S 2 (B ∗ ). Let Z(q)
be the zero locus of this quadratic form. We let Spin(B) be the simply-connected double cover of
SO(B). One construction is to realize it as a multiplicative subgroup of a Clifford algebra, and we
set GSpin(B) to be the Clifford group generated by Spin(B) and the scalar matrices. There are
two spin representations spin± (B). The choice of spin representation will not affect our results,
so we pick one and call it spin(B). Furthermore, Z(q) possesses two “spinor bundles” which we
34

call S + and S − . These can be constructed as pushforwards of line bundles from the flag variety of
Spin(B), or more direct geometric means (see [Ott] for details). There is a perfect pairing
S + ⊗ S − → O(−1)
[Ott, Theorem 2.8] and the sections of S + (1) and S − (1) give the two half-spin representations
(which follows from their descriptions as pushforwards of line bundles).
The relevant data:
• U = spin(B)
• G = GSpin(B)
• G/P = Z(q) (quadric hypersurface)
• U = S + (1)
• U ′ = spin+ (14)
• G′ = GSpin14 (C)
The ring of invariants Sym(U ∗ )(G,G) is a polynomial ring with generators of degrees 2, 8, 12,
14, 18, 20, 24, 30, and the graded Weyl group is the Weyl group of type E8 [Vin1, §9].
Remark 8.1. A non-hyperelliptic genus 4 curve lies on a unique quadric in its canonical embedding,
and the locus of curves C where this quadric is singular (i.e., C has a vanishing θ-characteristic) has
codimension 1. This condition also implies that C has a unique degree 3 map to P1 . If we further
impose that this map has a point with non-simple ramification, the locus loses another dimension.
Letting h be the 8-dimensional reflection representation of W , the GIT quotient U//G ∼
= h/W has
an open subset which should be isomorphic to this moduli space. We could not find any mention
of this in the existing literature.
To get the interpretation for C7 /W (E7 ) (Remark 6.1), one first realizes C as the smooth locus
of a cuspidal cubic. Then (P1 , . . . , P7 ) becomes 7 points in the plane and blowing them up gives
a del Pezzo surface. Its anticanonical divisor gives a map to P2 branched along the plane quartic
mentioned in Remark 6.1.
When we do the same thing for 8 points, the corresponding del Pezzo surface is mapped to a
quadric cone in P3 via twice its anticanonical divisor, and it is branched along a genus 4 curve.
However, 8 points in the plane generically do not lie on a cuspidal cubic; this condition on the
points corresponds to the ramification condition on the genus 4 curve mentioned above. This was
brought to our attention by Igor Dolgachev.
The representation spin(14) has four orbits Z(f )′ , Y ′ , X ′ , Z ′ , of interest, which are of codimensions 1, 5, 10, and 14, respectively. All of them are Gorenstein and have rational singularities,
but we have not yet calculated the minimal free resolutions for X ′ and Z ′ due to certain extension
problems. The hypersurface Z(f ) has degree 8.
The results for this example are incomplete, so we will just state what we expect to be true.
For a generic section v ∈ U , the above four orbits give degeneracy loci Z(f ), Y, X, Z. What should
happen is that there is a curve C of genus 4 such that X is the Kummer variety of Jac(C) and Z
is its singular locus consisting of 256 points. Furthermore, we should have Y = SUC (2) (see §3.2
for the definition of SUC (2)).
One can check that Z(f ) is a quartic hypersurface in Z(q), and it should be an analogue
of a Coble hypersurface. The curve C should be a non-hyperelliptic curve with vanishing theta
characteristic: every non-hyperelliptic curve C can be written as a complete intersection of a
quadric and cubic in its canonical embedding, and having a vanishing theta characteristic means
that this quadric is singular. For comparison, the analogues of Coble hypersurfaces for genus 4
curves without a vanishing theta characteristic were studied in [OP].
35

We remark that having the minimal free resolution of X ′ (and of an additional auxiliary module
M ′ ) will allow one to prove that X is the Kummer variety of an Abelian 4-fold, but there do not
seem to be any cohomological characterizations of Jacobians amongst Abelian 4-folds.

C4 ⊗ spin(10).

9

We can write C4 ⊗ spin(10) = spin(6) ⊗ spin(10), so this can be considered as a subcase of
spin(16) in the previous section.
Let A be a vector space of dimension 4 and B be a vector space of dimension 10 equipped with
an orthogonal form ω ∈ S 2 (B ∗ ). The relevant data:
• U = A ⊗ spin(B)
• G = (GL(A) × GSpin(B))/{(x, x−1 ) | x ∈ C∗ }
• G/P = P(A) = Gr(1, A)
• U = Q ⊗ spin(B)
• U ′ = C3 ⊗ spin(10)
• G′ = (GL3 (C) × Spin10 (C))/{(x, x−1 ) | x ∈ C∗ }
The ring of invariants Sym(U ∗ )(G,G) is a polynomial ring with generators of degrees 8, 12, 20,
24, and the graded Weyl group is Shephard–Todd group 31 [Vin1, §9].

9.1

Modules over OU ′ .

Let A′ = C3 and let G′ = (GL(A′ )×Spin(B))/{(x, x−1 ) | x ∈ C∗ }. Let R = Sym((A′ ⊗spin(B))∗ ).
There is a degree 12 G′ -invariant hypersurface X ′ ⊂ U ′ , whose equation is described in [Gyo,
§3]. One can interpret the construction there as taking the determinant of a certain 3×3 symmetric
matrix whose entries are quartic forms. More precisely, the representation S2,2 (spin(B)) contains
a Spin(B)-invariant, so we get polynomials P which span the representation S2,2 (A′∗ ) ⊂ S 4 ((A′ ⊗
spin(B))∗ ). Then S2,2 (A′∗ ) ∼
= S 2 (A′ ) ⊗ (det A′∗ )2 , so we can interpret the space of P as the space
of linear functions on symmetric 3 × 3 matrices of the form
ϕ : A′ → A′∗ ⊗ (det A′ )2 ,

(9.1)

and the equation for X ′ is the determinant of this matrix.
Proposition 9.2. The 2 × 2 minors of ϕ define a radical ideal of codimension 3, and the corresponding variety is the singular locus Z ′ of X ′ .
Proof. The maximal codimension of the submaximal minors of a symmetric matrix is 3. Since this
ideal is G′ -equivariant, it must cut out a union of orbit closures. Since they do not vanish on X ′
and there are no orbit closures of codimension 2, this ideal has codimension 3, and in particular
defines a Cohen–Macaulay variety.
There are 2 orbits of codimension 3. We can check on representatives that the 2 × 2 minors
vanish on only one of these orbits, and that the defined scheme is generically reduced on the other
orbit. This gives that the ideal of minors is radical of codimension 3. To check the statement
about the singular locus, it is enough to check the Jacobian matrix of the ideal of minors on orbit
representatives.
As a corollary, the resolution of OZ ′ is given as follows (see Example 1.12):
0 → A′∗ ⊗ (det A′∗ )5 ⊗ R(−16) → S2,1 (A′∗ ) ⊗ (det A′∗ )3 ⊗ R(−12) →
S 2 (A′∗ ) ⊗ (det A′∗ )2 ⊗ R(−8) → R → OZ ′ → 0.
36

We set M to be a certain twist of the cokernel of ϕ, namely, we have the presentation
A′ ⊗ (det A′∗ )3 ⊗ R(−8) → A′∗ ⊗ (det A′∗ ) ⊗ R(−4) → M → 0.
For P ∈ X ′ \ Z ′ , ϕP has corank 1, so MP ∼
= OX ′ ,P . For P in the open orbit of Z ′ , then ϕP has
corank 2, so MP is minimally generated by 2 elements.
Lemma 9.3. There is a G′ -equivariant OX ′ -linear isomorphism S 2 M ∼
= IZ ′ ,X ′ .
Proof. The proof is similar to the proof of Lemma 6.5.
The other codimension 3 orbit closure Z2′ can be described as maps spin(B)∗ → A′ whose kernel
contains a nonzero pure spinor. This variety fails to be normal, and the minimal free resolution of
its normalization is
0 → det A′ ⊗ S 5 A′ ⊗ R(−8) → det A′ ⊗ S 3 A′ ⊗ B ⊗ R(−6)
eZ ′ → 0.
→ det A′ ⊗ S 2 A′ ⊗ spin(B)∗ ⊗ R(−5) → R ⊕ det A′ ⊗ spin(B) ⊗ R(−3) → O
2

We define X , Z, Z2 , and M to be the global versions of

9.2

X ′,

Z ′,

Z2′

and M .

Geometric data from a section.

Now choose a section v ∈ U gen and set X = v(P(A)) ∩ X and Z = v(P(A)) ∩ Z.
Lemma 9.4. v(P(A)) ∩ Z2 = ∅.
eZ ′ is a perfect module, we can specialize its resolution by replacing A′ by Q, so that
Proof. Since O
2
we get a complex
0 → S 5 Q∗ (−1) → S 3 Q∗ ⊗ B(−1) → S 2 Q∗ ⊗ spin(B)∗ (−1) → OP3 ⊕ spin(B)(−1).
If v(P(A)) ∩ Z2 is nonempty, then it consists of finitely many points. But taking sections above,
we get an exact complex. Hence the intersection is empty.
From Lemma 9.3, we get a OX -linear map µ : S 2 M → OX , which gives an OX -algebra structure
e → X be the structure map.
on OX ⊕ M. We set OXe = SpecOX (OX ⊕ M) and let π : X
sm+
sm
Define U
to be the subset of U where M is minimally generated by 2 elements along Z.
e is an Abelian surface and L = π ∗ OX (1) is an indecomposable
Theorem 9.5. If v ∈ U sm+ , then X
(2, 2)-polarization.
Proof. The proof is similar to the proofs in §6.2.
From (9.1), we also get a symmetric matrix
ϕv : Q → Q∗ ⊗ (det Q)2 .

37

(9.6)

9.3

Quadratic complexes.

Here is another approach:
• G/P = Gr(2, A)
• U = Q2 ⊗ spin(B)
• U ′ = C2 ⊗ spin(10)
• G′ = (GL2 (C) × Spin10 (C))/{(x, x−1 ) | x ∈ C∗ }
There is a G′ -invariant quartic hypersurface in U ′ . Given v ∈ U sm , the corresponding degeneracy
locus is a smooth section Q of O(2).
Proposition 9.7. Q is the quadratic complex associated to the symmetric matrix (9.6) constructed
in the previous section.
Proof. There is a GL(A)-equivariant isomorphism
k : H0 (Gr(2, A); (det Q2 )2 ) → H0 (P(A); S 2 (Q∗ ) ⊗ (det Q)2 )
constructed in [NR1, §8] which associates to a quadratic complex (i.e., section in the domain above)
to a symmetric matrix Q → Q∗ ⊗ (det Q)2 whose determinant is the equation of a Kummer surface.
Since both sides are isomorphic to the irreducible GL(A)-representation S2,2 A, this map is uniquely
determined up to scalar. In the previous section and in the construction above, we have constructed
two GL(A)-equivariant maps S 4 (A ⊗ spin(B)) → S2,2 A. Since S2,2 A appears in S 4 (A ⊗ spin(B))
with multiplicity 1, this map is also unique up to scalar. Hence the diagram
S 4 (A ⊗ spin(B))

❱❱❱❱
❱❱❱❱
❱❱❱❱
❱❱❱❱
❱❱❱+

k / 0
0
2
H (Gr(2, A); (det Q2 ) )
H (P(A); S 2 (Q∗ ) ⊗ (det Q)2 )

commutes (up to possibly nonzero scalar ambiguity), which proves our claim.

9.4

Doing calculations.

Again, we have written code in Macaulay2 for calculating the quadratic complex in affine trivializations in P3 , but it is too messy to include. We instead explain the concepts behind the
calculation.
We will explicitly calculate the symmetric matrix Q → Q∗ ⊗ (det Q)2 starting with v ∈ A ⊗
spin(B). First, note that this symmetric matrix is a section of S 2 Q∗ ⊗ (det Q)2 ⊂ S 2 A∗ ⊗ (det Q)2 ,
and the space of sections of the latter is S 2 A∗ ⊗ S2,2,2 A = S 2 A∗ ⊗ S 2 A∗ ⊗ (det A)2 .
First, we have S2,2 A∗ ⊂ S 4 (A∗ ⊗ spin(B)∗ ). This is a 20-dimensional space of quartics which is
described explicitly in [Gyo, §3] by the polynomials P (x, y, z, w). More specifically, a basis is given
by those x, y, z, w such that x ≤ y, x < z, y < w, z ≤ w, and 1 ≤ x, y, z, w ≤ 4, i.e., such that
x y
T =
is a semistandard Young tableau. Abbreviate this polynomial by PT . Let QT ∈ S2,2 A
z w
be the dual basis vectors.
P
Then given v, we can produce the element pv = T PT (v)QT ∈ S2,2 A = S2,2 A∗ ⊗ (det A)2 . The
inclusion ι : S2,2 A∗ → S 2 A∗ ⊗ S 2 A∗ is defined by
x y
7→ 2(xy ⊗ zw + zw ⊗ xy) − (xz ⊗ yw + xw ⊗ yz + yw ⊗ xz + yz ⊗ xw).
z w
38

So ι(pv ) ∈ S 2 A∗ ⊗S 2 A∗ ⊗(det A)2 . Identifying the latter S 2 A∗ ⊗(det A)2 as sections of (det Q)2 , this
can be interpreted as a 4×4 symmetric matrix whose entries are quadrics on P3 . Let z1 , z2 , z3 , z4 be
homogeneous coordinates. Delete the last row and column of this matrix. This gives the symmetric
matrix Q → Q∗ ⊗ (det Q)2 over the affine open set z4 = 1. Take the ideals defined by the i × i
minors of this matrix for i = 3, 2 and saturate with respect to z4 to get homogeneous ideals for the
degeneracy loci.

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Laurent Gruson, Université de Versailles Saint-Quentin-en-Yvelines, Versailles, France
laurent.gruson@math.uvsq.fr
Steven V Sam, University of California, Berkeley, CA, USA
svs@math.berkeley.edu, http://math.berkeley.edu/~svs/
Jerzy Weyman, Northeastern University, Boston, MA, USA
j.weyman@neu.edu, http://www.math.neu.edu/~weyman/

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