On the ideals and singularities of secant varieties of Segre varieties

0.23 MB·English

by  J. M. Landsberg

ON THE IDEALS AND SINGULARITIES OF SECANT VARIETIES OF
SEGRE VARIETIES
7
0
0
2 J.M. LANDSBERGAND JERZY WEYMAN

n
a Abstract. We find generators for the ideals of secant varieties of Segre varieties in the cases
J of σk(P1×Pn×Pm) for all k,n,m, σ2(Pn×Pm×Pp×Pr) for all n,m,p,r (GSS conjecture
6 for four factors), and σ3(Pn×Pm×Pp) for all n,m,p and provethey are normal with rational
1 singularities in thefirst case and arithmetically Cohen-Macaulay in thesecond two.

]
G
A
. 1. Introduction
h
at Let V be a vector space over a field K of characteristic zero and let X ⊂ PV be a projective
m variety. Define σ (X), the variety of secant Pr−1’s to X by
r
[
σr(X) = ∪x1,...,xr∈XPx1,...,xr
2
v where P ⊂ PV denotes the linear space spanned by x ,...,x (usually a Pr−1) and the
x1,...,xr 1 r
2 overline denotes Zariski closure.
5 Let A ,...,A be vector spaces over K, with dimA = a . Let Seg(PA∗×···× PA∗) ⊂
4 1 n j j 1 n
1 P(A∗1⊗···⊗ A∗n) denote the Segre variety of decomposable tensors. (We use the dual vector
0 spaces A∗ when discussing varieties because we will mostly be concerned with modules of poly-
j
6
nomials and this convention enables our modules to be ∗-free.)
0
/ For applications to computational complexity, algebraic statistics, and other areas, one would
h like to have the defining equations for secant varieties of Segre varieties σ (PA∗×···× PA∗) =
t r 1 n
a σ (Seg(PA∗×···× PA∗)) and understand their singularities. In computational complexity one
m r 1 n
studies the stratification of A∗⊗A∗⊗A∗ by the secant varieties of the Segre, as given a bilinear
1 2 3
v: map f : A1 × A2 → A∗3 (such as matrix multiplication when each Aj is the space of m × m
i matrices), the smallest r such that f ∈ σ (PA∗ ×PA∗ ×PA∗) is a measure of its complexity.
X r 1 2 3
More generally, in algebraic statistics (see, e.g, [7]), one would like as much information as
r
a possible about different algebraic statistical models, and secant varieties of Segre varieties are
important special classes of such models. The techniques employed in this paper will be useful
for the general study of these models.
Remarkablylittleisknownaboutevenset-theoreticdefiningequationsoftheσ (PA∗×···× PA∗),
r 1 n
let alone generators of the ideals (which is considerably more difficult). The only case well un-
derstood is the case n = 2 where the secant varieties are the classical determinantal varieties.
In the case n = 3, the defining ideal of σ (PA∗ ×PA∗ ×PA∗) was described in [4], using the
2 1 2 3
methods of [9]. Set theoretic generators for σ (PA∗×···×PA∗) were also described in [4].
2 1 n
In the present paper we take the next step in understanding generators of the ideals and
singularities of the varieties σ (PA∗ ×···×PA∗). We make extensive use of the machinery of
r 1 n
[9].
Asignificantrolein ourstudyisplayed byauxiliary varieties thatcontain σ (PA∗×···×PA∗)
r 1 n
and have ideals that are easier to study. The simplest of these is the following:
Supportedrespectively by NSFgrants DMS-0305829 and DMS-0600229.
MSC 13P99,14Q15,15A69.
1
2 J.M.LANDSBERGANDJERZYWEYMAN
Definition 1. Let b ≤ a := dimA be nonnegative integers. Define the subspace varieties
j j j
Sub := {T ∈A∗⊗···⊗ A∗ |∃A′∗ ⊆ A∗, dimA′ = b , T ∈ A′∗⊗···⊗ A′∗}.
b1,...,bn 1 n j j j j 1 n
Subspacevarieties are cousins of the rank varieties in [9]. We use the terminology “subspace”
to avoid confusion with tensor rank. They areusefulbecause σ (PA∗×···× PA∗)⊂ P(Sub )
r 1 n r,...,r
and Sub admits a nice desingularization described in §3.
r,...,r
We first determine generators of the ideals of the subspace varieties using elementary repre-
sentation theory and prove that they are normal, with rational singularities using techniques
from [9] in §3. Subspace varieties enable one to reduce the problem of finding generators of
the ideals of σ (PA ×··· ×PA ) where dimA ≥ r to the cases where dimA = r for all j
r 1 n j j
(Proposition 5.1), which we refer to as the basic cases. In §7 and §6 we respectively resolve the
basic cases of σ (P2×P2×P2) and σ (P1×P1×P1×P1).
3 2
Recall that for any variety Z ⊂ PV invariant under the action of an algebraic group G,
the generators of the ideal of Z will be grouped into G-modules. In our case G = SL(A )×
1
··· × SL(A ), and the special linear group has the added feature that the decomposition of
n
its various modules is essentially independent of the dimension of the vector space A . For
j
example, when n = 2, the ideal of σ (PA∗ × PB∗) is generated by the irreducible module
r
Λr+1A⊗Λr+1B ⊂ Sr+1(A⊗B) which corresponds to the space of r+1×r+1 minors as long
as dimA,dimB ≥ r+1.
Finally, recall that a flattening of a tensor T ∈ A ⊗···⊗ A is to let I = {i ,...,i } ⊂
1 n 1 p
{1,...,n}, J = {1,...,n}\I, A = A ⊗···⊗ A , A = A ⊗···⊗ A and consider T ∈
I i1 ip J j1 jn−p
A ⊗A . Flattenings are useful because the ideals of secant varieties of Segre products of two
I J
projective spaces are well understood.
Notation. For a partition π = (p ,...,p ) of d, we write l(π) = r, |π| = d, [π] is the irreducible
1 r
S -module associated to π, and S V is the associated irreducible GL(V) module. Sym(V)
d π
denotes the symmetric algebra. For a variety X ⊂ PV, we let Xˆ ⊂ V denote the corresponding
cone in V. A is a vector space of dimension a and we assume a ≥ 2 to avoid trivialities. We
j j j
often write σ =σ (PA∗×···× PA∗).
r r 1 n
Our main results are as follows:
Theorem 1.1. The varieties σ (P1 ×Pb−1 ×Pc−1) = σ (PA∗ ×PB∗ ×PC∗) are normal, with
r r
rational singularities. Their ideal is generated in degree r+1 by the irreducible modules in the
two flattenings:
Λr+1(A⊗B)⊗Λr+1C, and Λr+1(A⊗C)⊗Λr+1B ⊂ Sr+1(A⊗B⊗C)
The redundancy in the above description is the irreducible module Sr+1A⊗Λr+1B⊗Λr+1C.
Theorem 1.2. Fix positive integers a,b,c,d. The variety σ (Pa−1 × Pb−1 × Pc−1 × Pd−1) =
2
σ (PA∗×PB∗×PC∗×PD∗) is arithmetically Cohen-Macaulay. Its ideal is generated in degree
2
three by the modules defining the subspace variety Sub (namely Λ3A⊗Λ3(B⊗C⊗D) plus
2222
permutations minus redundancies) and two copies of the module S A⊗S B⊗S C⊗S D
21 21 21 21
which arise from the flattenings of the form (A⊗B)⊗(C⊗D).
Note that a priori there are three modules obtained from flattenings but they only span two
independent copies of S A⊗S B⊗S C⊗S D, see Equation (2) and Remark 2.3 below.
21 21 21 21
The first set of modules in Theorem 1.2 may be thought of as arising from the flattenings of the
form A⊗(B⊗C⊗D).
TheassertionregardingthegeneratorsoftheidealistheGarcia-Stillman-Sturmfelsconjecture
for four factors [2], discussed further in §4.
ON THE IDEALS AND SINGULARITIES OF SECANT VARIETIES OF SEGRE VARIETIES 3
Theorem 1.3. Fix positive integers a,b,c ≥ 3. The variety σ (Pa−1×Pb−1×Pc−1)= σ (PA∗×
3 3
PB∗×PC∗) is arithmetically Cohen-Macaulay. Its ideal is generated in degree four by the module
S A⊗S B⊗S C which arises from Strassen’s commutation condition.
211 211 211
A priori there are three modules obtained by flattenings but they can only span the unique
copy of S A⊗S B⊗S C in S4(A⊗B⊗C), see Equation (3).
211 211 211
The equations arising from Strassen’s commutation condition originated in [8]. A discussion
of them in language compatible with this paper can be found in [5].
Remarkably, in each of thesecases, theidealis generated in theminimalpossibledegree (k+1
for σ , see [4]).
k
Overview. In §3 we prove all the necessary facts about subspace varieties and we deduce
Theorem 1.1.
In §4 we describe Garcia-Stillman-Sturmfels conjecture from [2], and a reduction of it (The-
orem 4.1).
The remainder of the proofs proceed in two steps. First, in §5 we show that the generators
of the ideal of secant varieties of Segre varieties can be deduced from solving the basic cases
of σ (Pr−1×···× Pr−1) (Proposition 5.1), and moreover the arithmetically Cohen Macaulay
r
(ACM) property holds in any given case if it holds for the relevant basic case plus a technical
hypothesis on modules occurring in the minimal free resolution of the ideal in the basic cases
(Lemma5.3). ToprovetheACMpropertyisinheritedweusearelative versionof themachinery
of [9]. Namely, insidethedesingularization ofthesubspacevariety, weconsiderasubbundlethat
gives a partial desingularization of σ and whose fibers are isomorphic to the basic case, and
r
push down the minimal free resolution of this subbundle. Then we study the “relative version”
of this resolution on the desingularization of Sub . Our results follow from the analysis of
r,...,r
the terms of this complex of sheaves. The methods from [9] allow us to establish two key facts
(Lemma 5.2). First, the higher cohomology of the terms of this complex vanishes. Second, the
sections of the terms are maximal Cohen-Macaulay modules supported in Sub . (The proof
r,...,r
of this second fact is the most subtle point in this paper.) Lemma 5.2 allows us to compute the
length of a minimal free resolution of σ under certain assumptions described in Lemma 5.3.
r
The basic cases for Theorems 1.2 and 1.3 are σ (P1 ×P1 ×P1 ×P1) and σ (P2 ×P2 ×P2).
2 3
Respectively in§6 and§7weprove thesevarieties areACM,determinegenerators of their ideals,
and show the technical hypotheses necessary to apply Lemma 5.3 hold. Unfortunately this step
utilizes a computer calculation.
It is interesting to ask if the the ACM property holds for general secant varieties of Segre
varieties. From our approach it follows that the ACM property for σ (PA∗×···× PA∗), with
r 1 n
dimA ≥ rwouldfollowfromcheckingtheassumptionsofLemma5.3forthevarietyσ (Pr−1×···× Pr−1)
j r
(n-factors).
Since we use results from representation theory, commutative algebra, and the geometric
method of [9] throughout, we begin in §2 with brief remarks from these areas.
Acknowledgment. We thank the anonymous referee for very useful suggestions to improve
the exposition of this paper.
2. Review from representation theory, commutative algebra and the geometric
method of [9]
2.1. Syzygies. We summarize from [9] (5.1.1-3,5.4.1):
Theorem 2.1. [9] Let Y ⊂ PV be a variety and suppose there is a projective variety B and a
vector bundle E → B that is a subbundle of a trivial bundle V → B with V ≃ V for z ∈ B such
z
that E → Yˆ is a desingularization. Write η = E∗ and ξ = (V/E)∗
4 J.M.LANDSBERGANDJERZYWEYMAN
If the sheaf cohomology groups Hi(B,Sdη) are all zero for i > 0 and if the linear maps
H0(B,Sdη)⊗V∗ → H0(B,Sd+1η) are surjective for all d≥ 0, then
(1) Yˆ is normal, with rational singularities
(2) The coordinate ring K[Yˆ] satisfies K[Yˆ] ≃ H0(B,Sdη).
d
(3) The vector space of minimal generators of the ideal of Yˆ in degree d is isomorphic to
Hd(B,Λd+1ξ).
(4) If moreover Y is a G-variety and the desingularization is G-equivariant, then the iden-
tifications above are as G-modules.
More generally, in the situation of Theorem 2.1, ⊕ Hj(Λi+jξ) is isomorphic to the i-th term
j
in the minimal free resolution of Y, and even a “twisted” version of this result holds which we
recall and explain when it is used in §5.
2.2. Representation theory. Let V = A ⊗···⊗ A . Let G = GL(A )×···× GL(A ). The
1 n 1 n
varieties σ are G-varieties so we should study their ideals as G-modules. The first step in
r
doing this is to decompose SdV into G-isotypic components. Recall that to a partition π we
associate a representation [π] of the symmetric group on d letters S and a representation S W
d π
of the general linear group GL(W). Both groups act on W⊗d and each group is the commuting
subgroupoftheother. TheGL(W)-isotypic decompositionofW⊗d isW⊗d = ⊕ [π]⊗S W.
|π|=d π
Proposition2.2. ([4], 4.1)TheG = GL(A )×···× GL(A )isotypicdecomposition ofSd(A ⊗···⊗ A )
1 n 1 n
is
Sd(A ⊗···⊗ A ) = ([π ]⊗···⊗ [π ])Sd⊗S A ⊗···⊗ S A ,
1 n M 1 n π1 1 πk k
|π1|=···=|πk|=d
where ([π1]⊗···⊗ [πk])Sd denotes the space of Sd-invariants (i.e., instances of the trivial rep-
resentation of S ) in [π ]⊗ ··· ⊗[π ].
d 1 n
Note in particular that the decomposition of Sd(A ⊗ ··· ⊗A ) is uniform, i.e. if dimA ≥
1 n i
S
l(πi) (so the corresponding module is non-zero), then the multiplicity ([π1]⊗···⊗ [πk]) d does
not depend on the dimA .
i
ThemultiplicityofSπ1A1⊗···⊗ SπkAk inSd(A1⊗···⊗ An),whichisdim([π1]⊗···⊗ [πk])Sd,
can be computed using characters in low degrees, although there is no general closed form for-
mula. Let χ : S → C denote the character of [π ], then
πj d j
1
S
dim([π ]⊗···⊗ [π ]) d = χ (σ)···χ (σ)
1 n d! X π1 πn
α∈Sd
(see, e.g., [6]).
For example:
S
(1) ([π ]⊗[π ]) d = δ
1 2 π1,π2
i.e. only symmetric terms occur with multiplicity one,
(2) dim([(2,1)],[(2,1)],[(2,1)],[(2,1)])S3 = 2.
and
(3) dim([(2,1,1)],[(2,1,1)],[(2,1,1)])S4 = 1.
ON THE IDEALS AND SINGULARITIES OF SECANT VARIETIES OF SEGRE VARIETIES 5
Remark 2.3. Assume A ,A ,A ,A have all dimension 2. Then Λ3(A ⊗A )= S A ⊗S A .
1 2 3 4 i j 2,1 i 2,1 j
ThusanytwoflatteningsΛ3(A ⊗A )⊗Λ3(A ⊗A ),embeddingthisrepresentationintoS3(A ⊗A ⊗A ⊗A )
i j k l 1 2 3 4
via3×3minorsofa4×4matrixspantheisotypiccomponentofS A ⊗S A ⊗S A ⊗S A
21 1 21 2 21 3 21 4
in S3(A ⊗···⊗A ).
1 4
2.3. Commutative algebra. Let V be a K-vector space, let A = Sym(V), which we con-
sider as the algebra of polynomials on V∗. For a graded A-module M, pd (M), the projective
A
dimension of M, denotes the length of a minimal free resolution of M as an A-module.
For a homogeneous ideal I ⊂ A, we let Z ⊂ V∗ denote its associated variety (the zero set of
I
the polynomials in I). Similarly, the support of an A/I-module is Z ⊂ Z ⊂V∗.
Ann(M) I
Definition 2. A/I is a Cohen-Macaulay ring iff
pd (A/I) = codim(Z ,V∗).
A I
An A/I-module M is a maximal Cohen-Macaulay module iff
pd (M) = codim(Z ,V∗).
A I
An affine variety Z ⊂ V∗ is arithmetically Cohen-Macaulay (ACM) if its coordinate ring K[Z]
is a Cohen-Macaulay ring, i.e., the length of a minimal free resolution of K[Z] as an A-module
equals the codimension of Z.
The following classical result follows, e.g., from [1], Theorem 18.15.a.
Theorem 2.4. Notations as above. Let I ⊂ A be a homogeneous ideal, let Z = Z ⊂ V∗
I
and let Z be its singular locus. Assume A/I is Cohen-Macaulay, then A/I is reduced iff
sing
codim(Z ,Z)≥ 1.
sing
We also note the following standard Commutative Algebra result, which essentially says that
a generically reduced irreducible algebraic variety has an non-empty open subset of smooth
points.
Proposition 2.5. If an affine variety Z ⊂ V is generically reduced, then codim(Z ,Z)≥ 1.
sing
3. The subspace varieties and their defining ideals
Theorem 3.1. The subspace varieties Sub are normal, with rational singularities. Their
b1,...,bn
ideal is generated in degrees b +1 for 1≤ j ≤ n by the irreducible modules in
j
Λbj+1A ⊗Λbj+1(A ⊗···⊗ A ⊗A ⊗···⊗ A ),
j 1 j−1 j+1 n
such that (reordering such that b ≤ b ≤ ··· ≤ b ) the partitions S A that occur for i ≤ j have
1 2 n πi i
l(π ) ≤ b , unless b = b , in which case we also allow l(π )= b +1.
i i i j i i
Inparticular, if allthe b = r, the ideal of Sub isgenerated indegree r+1bythe irreducible
i r,...,r
modules appearing in
Λr+1A ⊗Λr+1(A ⊗···⊗ A ⊗A ⊗···⊗ A )
j 1 j−1 j+1 n
for 1 ≤ j ≤ n (minus redundancies).
Proof. First note that the ideal of Sub consists of all modules S A ⊗···⊗ S A occur-
b1,...,bn π1 1 πn n
ring in Sd(A ⊗···⊗ A ) where each π is a partition of d and at least one π has l(π ) > b .
1 n j j j j
Also,notice, thatforeachj theidealconsistingofrepresentationsS A ⊗···⊗ S A occurring
π1 1 πn n
in Sd(A ⊗···⊗ A ) where l(π )> b is generated in degree b +1 by
1 n j j j
Λbj+1A ⊗Λbj+1(A ⊗···⊗ A ⊗A ⊗···⊗ A ),
j 1 j−1 j+1 n
6 J.M.LANDSBERGANDJERZYWEYMAN
because it is just theideal for rank at most b tensors in the tensor productof two vector spaces.
j
After reordering of summands so b ≤ ... ≤ b an elementary induction by degree completes
1 n
the argument regarding generators of the ideal.
To prove the results on the singularities, consider the product of Grassmannians
B = G(b ,A∗)×···× G(b ,A∗)
1 1 n n
and the bundle
(4) p :R ⊗···⊗ R → B
1 n
where R is the tautological rank b subspace bundle over G(b ,A∗). Then the total space
j j j j
Z˜ of R ⊗···⊗ R maps to A∗⊗···⊗ A∗. We let q : Z˜ → A∗⊗···⊗ A∗ denote this map
1 n 1 n 1 n
which gives a desingularization of Sub . (A general element of Sub is of the form
b1,...,bn b1,...,bn
[a1⊗···⊗ a1 + ··· + abn⊗···⊗ abn] where dimha1,...,abni = b , so it has a unique preimage
1 n 1 n j j j
under q.)
By Theorem 2.1.1, with η = (R ⊗···⊗ R )∗, we need to show
1 n
(i.) Hi(B,Sdη) = 0 for all i > 0, for all d ≥ 0
(ii.) H0(B,Sdη)⊗(A ⊗···⊗ A )→ H0(B,Sd+1η) is surjective for all d≥ 0.
1 n
To see (i.) holds, note that η = R∗⊗···⊗ R∗, and thus Sd(η), is homogeneous, completely
1 n
reducible,andthefactors aretensorproductsonSchurfunctorsonR∗. Each oftheseirreducible
i
factorsisample(infact, aquotientbundleofatrivialbundle)thustheBott-Borel-Weil Theorem
implies Sd(η) has nohigher cohomology (cohomology of an irreduciblebundlecan occur at most
in one degree).
To see (ii.), the ring of sections of Sym(η) is generated in degree 0 because the descrip-
tion of the ideal of Sub given above shows that, the multiplication map is induced by
b1,...,bn
the multiplication in Sym(A ⊗ ··· ⊗A ) after mod-ing out the span of the representations
1 n
S A ⊗···⊗ S A satisfying l(π ) > b for some j. But the Littlewood-Richardson rule (e.g.
π1 1 πn n j j
[9], Theorem (2.3.4)) implies that in the tensor product of two representations S V ⊗S V we
π1 π2
have only the representations S V with the Young diagram of π containing both the diagrams
π3 3
of π and π as sub-diagrams, so if a representation S A ⊗···⊗ S A satisfies l(π ) ≤ b
1 2 π1 1 πn n j j
for all j, and it appears in (S A ⊗···⊗ S A )⊗(A ⊗···⊗ A ) then l(µ ) ≤ b for all j as
µ1 1 µn n 1 n j j
well. (cid:3)
proof of Theorem 1.1. Theorem3.1andStrassen’sresult[8]thatσ (P1×Pr−1×Pr−1)= P(K2⊗Kr⊗Kr)
r
(which is easily established using Terracini’s lemma) imply P(Sub )= σ (PA∗×PB∗×PC∗)
2,r,r r
when b,c ≥ r. (One always has σ (PA∗ × PB∗ × PC∗) ⊆ P(Sub ) and Strassen’s result
r r,r,r
establishes the reverse inclusion.) (cid:3)
4. The varieties Flata and the GSS conjecture
r
A variant on the subspace varieties is as follows. Let a = (a ,...,a ) and define I to be
1 n Flata
r
the ideal generated by the modules Λr+1A ⊗Λr+1A ⊂ Sr+1(A ⊗···⊗ A ) as I,J range over
I J 1 n
complementary subsets of {1,...,n}. We let Flata denote the corresponding variety. Just as
r
with subspace varieties, we have σ (PA∗×···× PA∗)⊆ Flata.
r 1 n r
Garcia, Stillmann and Sturmfels [2] conjectured that IFlata2 = Iσ2(PA∗1×···×PA∗n). We refer
to this statement as to the GSS conjecture. In [4] the conjecture was proven when a =
(a ,a ,a ), and moreover it was shown that as sets, Flata = σ (PA∗×···× PA∗) for all n.
1 2 3 2 2 1 n
Since σ (PA∗×···×PA∗) is reduced and irreducible, and Flata is irreducible, to prove the
2 1 n 2
conjecture it would be sufficient to show Flata is reduced.
2
The application to the GSS conjecture is
ON THE IDEALS AND SINGULARITIES OF SECANT VARIETIES OF SEGRE VARIETIES 7
Theorem 4.1. If Flata is arithmetically Cohen-Macaulay, then the GSS conjecture holds.
2
Proof. We first show
Proposition 4.2. Flata is generically reduced.
2
Proof. Fix bases (asi) ineach A andlet φ belinear coordinate functionson A∗⊗···⊗ A∗.
i i j1,...,jn 1 n
A general element of Flata is of theform x = a1⊗···⊗ a1+a2⊗···⊗ a2, i.e., it has coordinates
2 1 n 1 n
φ = φ = 1 andall other coordinates zero(this, andtheassertion aboutthecodimension
1,...,1 2,...,2
follows by using the identification as sets of Flata with σˆ ). We show that at x, the differentials
2 2
of a set of generators of I span a subspace of T∗(A∗⊗···⊗ A∗) equal to the codimension
Flatar x 1 n
of Flata. In algebraic language, we show that the localization of Sym(A ⊗···⊗ A )/I at
2 1 n Flata2
x has codimension equal to codim(Flata). T∗σ is spanned by dφ | where n−1 of the
2 x 2 j1,...,jn x
j ,...,j are neither 1 nor 2. Fix some p < n and consider the (a ···a )×(a ···a ) matrix
1 n 1 p p+1 n
corresponding to the flattening (A ⊗···⊗ A )⊗(A ⊗···⊗ A ). Examining the differentials
1 p p+1 n
of its three by three minors at x, all are zero except the differentials of minors containing φ
1,...,1
and φ , which will have a unique nonzero term dφ | . For any splitting we recover all
2,...,2 i1,...,in x
the dφ | where none of the i are 1 or 2. In general we recover all the dφ | that are
i1,...,in x s i1,...,in x
neither in the row or column containing φ or φ . Thus if we want a term with k indices
1,...,1 2,...,2
equal to 1 and l indices equal to 2, then (ignoring order for the moment) as long as k < n−p
and l < p there is clearly no problem. To get a different order, just permute the factors.
(cid:3)
To conclude the proof of Theorem 4.1 we use Theorem 2.4 and Proposition 2.5.
(cid:3)
Example 3. Consider the case n = 4 and each a = 2. Here are matrices respectively for the
i
splittings (A ⊗A )⊗(A ⊗A ) and (A ⊗A )⊗(A ⊗A ).
1 2 3 4 1 3 2 4
φ φ φ φ
1,1,1,1 1,2,1,1 2,1,1,1 2,2,1,1
 
φ φ φ φ
1,1,1,2 1,2,1,2 2,1,1,2 2,2,1,2
φ1,1,2,1 φ1,2,2,1 φ2,1,2,1 φ2,2,2,1
 
φ1,1,2,2 φ1,2,2,2 φ2,1,2,2 φ2,2,2,2
φ φ φ φ
1,1,1,1 1,1,2,1 2,1,1,1 2,1,2,1
 
φ φ φ φ
1,1,1,2 1,1,2,2 2,1,1,2 2,1,2,2
φ1,2,1,1 φ1,2,2,1 φ2,2,1,1 φ2,2,2,1
 
φ1,2,1,2 φ1,2,2,2 φ2,2,1,2 φ2,2,2,2
Thedφ | where {i,j,k,l} = {1,1,2,2} each appear in the differentials of the eight relevant
ijkl x
(i.e., those containing both φ and φ ) 3×3 minors.
1111 2222
We resolve the four factor case of the GSS conjecture as a consequence of Lemma 5.3 and
Proposition 6.1.
5. Ideals and the ACM property are inherited
Definition4. GivenvectorspacesA′ ⊂ A andamoduleS A′⊗···⊗ S A′ ⊂Sd(A′⊗···⊗ A′ ),
j j π1 1 πn n 1 n
wesaythemoduleS A ⊗···⊗ S A correspondinglyrealizedasasubmoduleofSd(A ⊗···⊗ A )
π1 1 πn n 1 n
is inherited from S A′⊗···⊗ S A′ .
π1 1 πn n
NotethatifS A′⊗···⊗ S A′ isnonzero,wehaveS A′⊗···⊗ S A′ ⊂ I(σ (Seg(PA′×···× PA′ ))
π1 1 πn n π1 1 πn n r 1 n
iffS A ⊗···⊗ S A ⊂ I(σ (Seg(PA ×···× PA )). Thispropertyiscalled inheritance in[4].
π1 1 πn n r 1 n
8 J.M.LANDSBERGANDJERZYWEYMAN
Proposition5.1. LetdimA ,...,dimA ≥ r. Thegeneratorsoftheidealofσ (PA∗×···× PA∗)
1 n r 1 n
are givenby the modules generating the ideal of Sub and the modules inherited from the mod-
r,...,r
ules generating the ideal of σ (Pr−1×···× Pr−1) (n-factors).
r
Proof. The irreducible modules generating the ideal of Sub are all in degree r+1 and are
r,...,r
theirreduciblesubmodulesofΛr+1A ⊗Λr+1(A ⊗···⊗ A ⊗A ⊗···⊗ A ),soinparticular
j 1 j−1 j+1 n
they all contain a partition with r +1 parts. The irreducible modules generating the ideal of
σ (Pr−1×···× Pr−1) cannot contain a partition with more than r parts.
r
Now say some module S A ⊗···⊗ S A is in I(σ (Seg(PA ×···× PA )). We must show
π1 1 πn n r 1 n
it is generated from our candidate generators. If any π has more than r parts, then it is already
j
in the ideal generated by Sub so we are done. But now if each π has length at most r, then
r,...,r j
the same module must also be in the ideal of σ (Pr−1×···× Pr−1). (cid:3)
r
OvertheGrassmannianG(r,A∗),weletR ,Q respectivelydenotetherankr(resp. ranka −
j j j j
r) tautological subspace (resp. quotent) vector bundles. Recall the bundle η = R∗⊗···⊗ R∗.
1 n
Let B = Sym(η).
Lemma 5.2. Let π = (p ,...,p ) be partitions. Consider the sheaf
j j,1 j,r
M := ⊗n S R∗⊗B.
j=1 πj j
(1) Assume that p ≥ −a +1 for 1 ≤ j ≤ n. Then M is acyclic.
j,1 j
(2) Assume that p ≥ 0 and p ≤ rn−1−r for 1 ≤ j ≤ n. Then the Sym(A ⊗···⊗A )-
j,1 j,1 1 n
module H0(B,M), whichissupported inSub , isamaximalCohen-Macaulaymodule.
r,...,r
Proof. The first assertion is a straightforward application of the Bott-Borel-Weil theorem.
The second assertion is the most subtle point of this paper. To prove it, we use the duality
theorem [9], Theorem 5.1.4, which we now recall.
For any vector bundle V → B, following [9], Theorem 5.1.4, define the twisted dual vector
bundle
Vˇ = K ⊗Λrankξξ∗⊗V∗
B
where V∗ denotes the ordinary dual vector bundle, K is the canonical bundle of B, and ξ =
B
(A∗⊗···⊗ A∗ ⊗O /R ⊗···⊗ R )∗. Then [9], Theorem 5.1.4, asserts that
1 n B 1 n
(5) F(Vˇ) = F(V)∗ .
j j+dimB−rankξ
We claim that under the hypotheses of the lemma, the rightmost nonzero term in F(Mˇ) is
•
the zero-th. To see this note that K = S R⊗S Q∗, which up to tensoring with
G(r,a) a−r,...,a−r r,...,r
a trivial bundle (powers of the bundle (Λa1A ⊗···⊗ ΛanA )⊗O ) is isomorphic to S R,
1 n B a,...,a
and, up to tensoring with a trivial bundle,
Λrankξξ∗ ≃ Srn−1,...,rn−1R∗1⊗···⊗ Srn−1,...,rn−1R∗n
Write π = (p ,...,p ). So up to tensoring with a trivial line bundle,
i i,1 i,r
Mˇ ≃ S(rn−1−a1−p1,r),...,(rn−1−a1−p1,1)R∗1⊗···⊗ S(rn−1−an−pn,r),...,(rn−1−an−pn,1)R∗n
Thus, if for each i we have p ≤ rn−1−r, then, applying (1), Mˇ ⊗B has no higher cohomology,
i,1
and the complexes F(M) and F(Mˇ) have length equal to the codimension of the subspace
variety Sub which equals (rankξ−dimB). (cid:3)
r,...,r
Lemma 5.3. If σ (Pr−1×···× Pr−1) (n-factors, with n ≥ 3) is arithmetically Cohen-Macaulay
r
with the property that no module occurring in its minimal free resolution contains a partition
whose first part is greater than rn−1 − r, then σ (PA∗×···× PA∗) is arithmetically Cohen-
r 1 n
Macaulay when dimA ≥r for 1≤ i ≤ n,.
i
ON THE IDEALS AND SINGULARITIES OF SECANT VARIETIES OF SEGRE VARIETIES 9
Proof. Notatations as above. Consider the desingularization of the subspace variety Sub
r,...,r
and the resulting vector bundle E = R ⊗···⊗ R as in Equation (4) with each b = r. Each
1 n j
fiber (R ⊗···⊗ R ) of E over x ∈ B = G(r,A∗)×···× G(r,A∗) is just Cr⊗···⊗ Cr and we
1 n x 1 n
may consider the subvariety Z ⊂ Z˜ such that Z = σˆ (P(R ) ×···× P(R ) ). (Recall that Z˜
x r 1 x n x
is the total space of the bundle E.) Z gives a partial desingularization of σˆ .
r
Under our hypotheses, there is a minimal free resolution G of Z where G = A :=
• x 0
Sym(A ⊗···⊗ A ), G is a sum of modules S A ⊗···⊗ S A ⊗A(−k) where k ≥ r+1, and
1 n 1 π1 1 πn n
thelengthoftheresolutionofG isthecodimensionofZ inPE ,namelyL := rn−r2n+r(n−1),
• x x
as σ (Pr−1×···× Pr−1) is of the expected dimension (rn+1)(r−1) as long as n ≥ 3.
r
By [9] Proposition (5.1.1), part b), B = Sym(η) is a sheaf of algebras isomorphic to p (O ).
∗ Z˜
We form a complex of sheaves of B-modules from G by replacing each G with the sheaf G
• i i
obtained by replacing the Schur functors of the vector spaces A ,...,A with the corresponding
1 n
Schur functors of the sheaves R∗,...,R∗.
1 n
We have projections q : Z → σˆ and p : Z → B. We have p (O ) = B/d(G ) as d(G ) is the
r ∗ Z 1 1
subsheaf of B consisting of the local functions on Z˜ that vanish on Z.
Our complex of sheaves of B-modules G is such that each term is a sum of terms of the form
•
S R∗⊗···⊗ S R∗ ⊗B.
π1 1 πn n
Each term is homogeneous and completely reducible, with each irreducible summand having
nonzero H0, so in particular no term has any higher cohomology.
Define a complex M of A-modules by letting M := H0(B,G ).
• j j
The minimal free resolution of the ideal of σ is the minimal resolution of the cokernel of the
r
complex M . Indeed, by Proposition 5.1, the cokernel M /Image(M ) is exactly K[σ ] because
• 0 1 r
M consists of functions on the subspace variety and M the ideal of the secant variety inside
0 1
the subspace variety.
To obtain a not necessarily minimal resolution of the cokernel K[σ ] of the map M → M ,
r 1 0
one can proceed by iterating the mapping cone construction as follows. Let F be a resolution
j•
of M for each j. We obtain a double complex, the tail of which is
j
↓ ↓
F → F →
L,1 L−1,1
↓ ↓
F → F →
L,0 L−1,0
↓ ↓
M → M →
L L−1
We replace this tail by using the mapping cone construction (e.g. [1]), where we replace F
L−1,j
by modules F˜ = F ⊕F , and F˜ becomes the last column of the new complex.
L−1,j L,j−1 L−1,j L−1,•
We iterate this procedure until we end up with a picture
↓ ↓
F˜ → F
11 01
↓ ↓
F˜ → F
10 00
↓ ↓
M˜ → M → C → 0
1 0
where the F is a resolution of M and M˜ is the term replacing M after having iterated the
0• 0 1 1
mapping cone construction, and F˜ its resolution.
1•
The final product of this procedure is a possibly nonminimal resolution F˜ of K[σ ], whose
0• r
j-th term is F˜ = ⊕ F .
0j a+b=j a,b
10 J.M.LANDSBERGANDJERZYWEYMAN
But by Lemma 5.2, the modules M are maximal Cohen-Macaulay, hence the lengths of their
i
minimal free resolutions all equal codimSub = (rankξ−dimB).
r,...,r
But now the complexes F(G ) give the resolutions of the M , so when we apply the iterated
i • i
cone construction, the longest possible length of the possibly nonminimal resolution of K[σ ] is
r
rankξ−dimB +codimσ (Pr−1×···× Pr−1)
r
= (a ···a −rn)−r(a +···+a −nr)+(rn−r2n+r(n−1))
1 n 1 n
= a ···a −r(a +···+a )+rn−r,
1 n 1 n
but
codimσ (PA∗×···× PA∗)= a ···a −1−[r(a +···+a −n)+(r−1)].
r 1 n 1 n 1 n
We see that the (possibly non-minimal) resolution is of minimal length and that length equals
the codimension of σ (PA∗×···× PA∗), hence σ (PA∗×···× PA∗) is Cohen-Macaulay. (cid:3)
r 1 n r 1 n
6. Case of σ (P1×P1×P1×P1)
2
Proposition 6.1. The variety σ (P1×P1×P1×P1) = σ (PA∗×PB∗×PC∗×PD∗) is arith-
2 2
metically Cohen-Macaulay. Its ideal is generated in degree three by two copies of the module
S A⊗S B⊗S C⊗S D which arise from the flattenings of the form (A⊗B)⊗(C⊗D).
21 21 21 21
Proof. Let A = Sym(A⊗B⊗C⊗D) and let I denote the ideal generated by the relevant two
copies S A⊗S B⊗S C⊗S D (see Remark 2.3). We thank Anurag Singh for calculating
21 21 21 21
the minimal free resolution of A/I, which we denote G with terms as follows
•
G : 0→ A12(−10) → A48(−9) → A57(−8) → A20(−6)⊕A48(−5) →
•
→ A78(−4) → A32(−3) → A.
Note that 6 = codimσ (P1 ×P1 ×P1×P1) = codimFlat2222 and since this coincides with the
2 2
length of the minimal free resolution we conclude that A/I is Cohen-Macaulay. But we know
that Flat2222 is Cohen-Macaulay, soby Theorem 4.1 theGSS conjecture follows in this case. (cid:3)
2
Theorem 1.2 follows because if we express the resolution in terms of modules, each module
S A⊗S B⊗S C⊗S D
π1 π2 π3 π4
that occurs in some G indeed satisfies the property that the first part of each π is less or equal
j i
to 6. This can be calculated directly by examining the maps produced by Macaulay2 and then
finding the equivariant form of the resolution explicitly (which we reproduce below). To see it
more directly, note that since the coordinate ring is Cohen-Macaulay, the dual of this resolution
is also an acyclic complex. This means that every representation S A⊗S B⊗S C⊗S D
π1 π2 π3 π4
appearing in the resolution has to have partitions π , π , π , π that are contained in partitions
1 2 3 4
of some representation occurring in the top of the resolution. But in the top piece, dimension
considerations show immediately that the partitions are S[(6,4)(5,5)(5,5)(5,5)] and thus all
the partitions π have all parts ≤ 6 as required.
j
Remark 6.2. Theresolution G expressed as adirect sumof GL(A)×GL(B)×GL(C)×GL(D)-
•
modules is as follows. Denote the i-th term in this resolution by G . Let (a,b)(c,d)(e,f)(g,h)
i
denoteS A⊗S B⊗S C⊗S D. Thetermsintheresolution G havetobesymmetric
(a,b) (c,d) (e,f) (g,h) •
underpermutingthespacessoweletS[(a,b)(c,d)(e,f)(g,h)] denotethedirectsumofalldistinct