New applications of the geometric method of calculating syzygies.

Talk at the 6-th conference on nilpotent orbits and representation theory
Kawaguchi-ko, Japan, September 6-th, 2004.
In this talk I describe two recent applications of the method.
We work over an algebraically closed eld K of characteristic zero.

1. The geometric technique.

Let us consider the projective variety V of dimension m. For the purposes of this talk,
V = G=P for some reductive group G and parabolic subgroup P .
Let X = ANK be the ane space which is a representation of G. The space X 2 V canbe viewed as a total space of trivial vector bundle E of dimension n over V . Let us consider
the subvariety Z in X 2 V which is the total space of a subbundle S in E . We denote by
q the projection q : X 2 V 0! X and by q the restriction of q to Z . Let Y = q(Z ). We
get the basic diagram0

Z  X 2V
#q
#q
Y  X
The projection from X 2 V onto V is denoted by p and the quotient bundle E =S byT . Thus we have the exact sequence of vector bundles on V00 0! S 0! E 0! T 0! 0
The dimensions of S and T will be denoted by s, t respectively. The coordinate ring of X
will be denoted by A. It is a polynomial ring in N variables over K. We will identify thesheaves on X with A-modules.
The locally free resolution of the sheaf OZ as an OX V -module is given by the Koszulcomplex^2^t
K ( ) : 0 ! (p  ) ! : :: ! (p ) ! p ( ) ! OX V
where  = T . The di erentials in this complex are homogeneous of degree 1 in thecoordinate functions on X . The direct image p (OZ ) can be identi ed with the the sheafof algebras Sym() where  = S .
The idea of the geometric technique is to use the Koszul complex K( ) to construct
for each vector bundle V on V the free complex F (V ) of A-modules with the homology
supported in Y . In many cases the complex F(OV ) gives the free resolution of the de ningideal of Y .
For every vector bundle V on V we introduce the complexK(; V ) := K( ) X V p V
This complex is a locally free resolution of the OX V -module M (V ) := OZ p V .
Now we are ready to state the basic theorem (Theorem (5.1.2) in [W]).
23332333

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Theorem 1. For a vector bundle V on V we de ne a free graded A-modules
i^+jM
jF(V )i = H (V;  V ) k A(0i 0 j )j 0a) There exist minimal di erentials
di(V ) : F(V )i ! F(V )i 1
0of degree 0 such that F(V ) is a complex of graded free A-modules withH i(F(V ) ) = Riq M (V )In particular the complex F(V ) is exact in positive degrees.
b) The sheaf Ri q M (V ) is equal to H i (Z; M (V )) and it can be also identi ed with thegraded A-module H i (V; Sym( ) V ).
c) If  : M (V ) ! M (V )(n) is a morphism of graded sheaves then there exists a morphismof complexes
0

3

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f () : F(V ) ! F(V ) (n)




0

Its induced map H i (f ()) can be identi ed with the induced map
H i(Z;M (V )) ! H i(Z; M (V ))(n):0

0If V is a one dimensional trivial bundle on V then the complex F(V ) is denoted
simply by F .
The next theorem gives the criterion for the complex F to be the free resolution of
the coordinate ring of Y .
Theorem 2. Let us assume that the map q : Z 0! Y is a birational isomorphism. Then0

the following properties hold.
a) The module q OZ is the normalization of K[Y ].
b) If Ri q OZ = 0 for i > 0, then F is a nite free resolution of the normalization ofK[Y ] treated as an A-module.
c) If Ri q OZ = 0 for i > 0 and F0 = H 0 (V; 0  ) A = A then Y is normal and it has
rational singularities.
0303V03

This is Theorem (5.1.3) in [W].
This technique was successfully applied to determinantal varieties related to generic,symmetric and skew-symmetric matrices. Also interesting results regarding de ning idealsof nilpotent orbits were obtained. The determinantal expressions for resultants and discriminants were also obtained usiing this method. All these developments are described in[W].
One should point out that the terms of the complex F can be calculated fully only
when the bundle  has convenient form (for example is a tensor product of tautologicalbundles on some Grassmannians). We will refer to this saying that  is in a simple form.
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Notation. In the examples below, for the integers ai with a1  : ::  an  0, we denote
S(a1 ;:::;an )E the highest weight representation of the group GL(E ) corresponding to theweight (a1 ;: :: ; an ), i.e. the Schur functor.

2. Representations with nitely many orbits.

The natural scope for application of this method is a orded by the orbit closures in theirreducible representations of simple groups with nitely many orbits. These were classi ed
by Kac [K] and, with some minor corrections added, by Leahy [Le].
The rst type of such representations (Table II in [K]) is the family of representations
parametrized by a Dynkin diagram with the distinguished (black) node. Distinguishingsuch a node determines a grading on the corresponding root system, in which all simple roots except the one corresponding to the black node, have degree 0 and the rootcorresponding to the black node has degree 1. This leads to a gradingg = 8i Z g i
2of the Lie algebra g corresponding to our Dynkin diagram. The representation we areinterested in is the representation of G0 (the group corresponding to the Lie algebra g0 on
g1 . The orbits of such actions were classi ed by Vinberg in [V]. They are the irreducible
components of intersections of nilpotent orbits in g with g1.
For Dynkin diagrams of classical types, most of the representations one gets are of
the type Hom(E; F ) where E is a vector space and F is a symplectic (resp. orthogonalvector space), with the action of the group GL(E ) 2 SP (F ) (resp. GL(E ) 2 SO(F )).
Last year Steve Lovett obtained in his thesis [L] some very interesting results on such
orbit closures. Let me describe his results.
Consider the representation Hom(E;F ). We denote dim E = e, dim F = f . The
orbits in such representations are described by two rank conditions: the rank r1 of themap  2 Hom(E; F ) and the rank r2 of the form on F restricted to m(). We denote thecorresponding orbit Or1 ;r2 and the orbit closure Zr1 ;r2 := Or1 ;r2 .
Lovett found desingularizations of the orbit closures Zr1 ;r2 for which geometric technique is applicable. Unfortunately, the bundles  are not very simple, they are extensionsof two tensor products. Lovett rst classi es the orbit closures for which one can nd adesingularization with  in a simple form.

De nition. An orbit closure Zr ;r is special if one of the following conditions holds.
a) r2 = 0,b) r1 = r2 ,c) r1 = e,

1 23d) 2r1 0 r2 = f .

For special orbits the terms of the complexes F are calculated explicitly thus providing
minimal free resolutions of the coordinate rings.
In particular we getTheorem 3 (Lovett). Let E be a vector space, F - a symplectic or orthogonal vector

space. Let Zr1 ;r2 be a special orbit closure. Then Zr1 ;r2 is normal, with rational singularities (hence Cohen-Macaulay), except in the case when F is orthogonal and case d) aboveoccurs (with r2 6= 0 if f is even). In these bad cases the normalization of the coordinate
ring has rational singularities, but the coordinate ring itself is not Cohen-Macaulay.

One should add that the de ning ideals for normal special orbit closures are given
by natural rank conditions, i.e. by r1 + 1 minors of  and by the corresponding pfaans(resp. minors) of the skew-symmetric (resp. symmetric) matrix given by SP (F ) (resp.
SO(F ))-invariants in A of degree 2.
The next question are the properties of nonspecial orbits.

Theorem 4 (Lovett). Let E be a vector space, F - a symplectic or orthogonal vector

space. Let Zr1 ;r2 be a nonspecial orbit closure. Then the coordinate ring of Zr1 ;r2 is CohenMacaulay. Moreover, the de ning ideal of Zr1 ;r2 is given by natural rank conditions.
Sketch of proof Consider the Grassmannian Grass(e 0 r1 ; E ) and the desingularization of

the determinantal variety
Z = f(; R) 2 X 2 Grass(e 0 r1 ;E ) j jR = 0g:
Now, take the relative version of the resulution of the orbit closure Or1 ;r2 in Hom(E F )with dim E = r1 , and changing E to the tautological factor bundle Q. Notice that
this orbit was special (case c)), so it has rational singularities and the resolution could
be e ectively described. Taking the direct image over Grass(e 0 r1 ; E ) of all the terms,
we get a complex of sheaves of B-modules where B = Sym(Q F ) is a sheaf of algebras.
Moreover, the terms of the resolution are the direct sumes of sheaves of type S Q B. Let
us denote the i-th term of this complex by Fi (0  i  d). Here d is the codimension of
Zr1 ;r2 in Hom(E ;F ).
The key observation is as follows0000Fact.

a) H j (Grass(e 0 r1 ;E ); Fi ) = 0 for j > 0, 0  i  d,b) H 0 (Grass(e 0 r1 ;E ); Fi ) is a maximal Cohen-Macaulay module supported in determinantal variety Zr1 ;r1 .

The second fact is proved by geometric method for twisted modules supported in thedeterminantal variety.
4This result implies that we can construct the resolution of the cokerel of the exact
sequence0 ! H 0 (Grass(e 0 r1 ; E ); Fd ) ! : :: !

! H 0 (Grass(e 0 r1 ;E ); F1) ! H 0 (Grass(e 0 r1; E ); F0 )by iterated cone construction from the terms of resolutions of H 0 (Grass(e 0 r1; E ); Fi ).
This gives a nonminimal resolution of the coordinate ring of Zr1;r2 of the length equal toits codimension.
The claim about the equations comes from analysing the low degree terms in thecomplex. After we establish that the generators correspond to natural equations giving
rank conditions, we notice that since the ideal is perfect, in order to prove it is reduced, it isenough to show it is generically reduced. This can be then done explicitly by localization.

3. The secant varieties.

I am collaborating with Joe Landsberg and Laurent Manivel on secant varieties for multiple
Segre embeddings. These were studied in the past only regarding their dimensions. Otherproperties and de ning equations were not investigated. We are dealing here with thetensor product F1 : :: Fn and the subvariety X of totally decomposable tensors. Thevariety Sec(X ) is the closure of the set of tensors that can be written as a sum of two
totally decomposable tensors.
Landsberg and Manivel observed in [LM] that the geometric method can be applied.
Using the fact that the secant variety Sec(X ) in F1 F2 F3 , dim Fi = 2 for 1  i  3
is the whole space F1 F2 F3 they proved

Theorem 5 (Landsberg-Manivel). Let us consider the secant variety Sec(X ) in F1F2 F3 . It is normal, with rational singularities. The de ning ideal of Sec(X ) is generated
in degree 3. The generators are 3 2 3 minors of linear maps i;j : Fi Fj A(01) ! Fk A
(here fi; j; kg = f1; 2; 3g) induced by  2 F1 F2 F3 .
3This case might seem special because of the special property of the secant variety for3 copies of K 2 we used.
We analyzed the case of four copies.

Theorem 6 (Landsberg-Manivel-JW). Let us consider the secant variety Sec(X ) inF1 F2 F3 F4. It's coordinate ring is Cohen-Macaulay. The de ning ideal of Sec(X )is generated in degree 3. The generators are 3 2 3 minors of linear maps i;j : Fi Fj
A(01) ! Fk Fl A (here fi; j; k;lg = f1; 2; 3; 4g) induced by  2 F1 F2 F3 .
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3Sketch of proof The proof is very similar to the proof of theorem 4.

5QConsider the product of Grassmannians 4i=1 Grass(fi 0 2; Fi ) and the desingularization of the corresponding variety Z , which hasp (OZ = Sym( 4i=1Qi)
3

where0 ! Ri ! Fi Grass(fi 0 2;Fi ) ! Qi ! 0is a tautological sequence on Grass(fi 0 2; Fi ), with fi = dim Fi . Now, take the relative
version of the resulution of the variety Sec(X ) in product of 4 copies of two dimensional
spaces. This is a variety of codimension 6 and the resolution of its coordinate ring can becalculated using Macaulay 2.
We get a complex of sheaves of B-modules with the terms being the direct sumes ofsheaves of type 4i=1 Sai ;bi ) Qi B. Let us denote the i-th term of this complex by Fi(0  i  6).
The key observation is again

Fact.

a) H j (Grass(e 0 r1 ;E ); Fi ) = 0 for j > 0, 0  i  6,b) H 0 (Grass(e 0 r1 ;E ); Fi ) is a maximal Cohen-Macaulay module supported in thevariety Z .

Now we can repeat the reasoning used to prove theorem 4.
We have similar results for the second secant variety in the Segre embedding of a triple
product of projective spaces.

References.

[K Kac, V. Some remarks on nilpotent orbits, J. Algebra, 64(1980), 190-213,[LM] Landsberg, J., Manivel L., On the ideals of secant varieties of Segre varieties, preprint,
math.AG/0311388[Le] Leahy, A.,, A classi cation of multiplicity free representations, J. Lie Theory 8, No.2,367-391 (1998),[L] Lovett, S., Orbits of orthogonal and symplectic representations of symmetric quivers,
Ph.D. Thesis, Northeastern University, 2003,[V] Vinberg, The Weyl group of a graded Lie algebra, Math. USSR-Izv. 10(1976), 463495,[W] J. Weyman, \Cohomology of vector bundles and syzygies", Cambridge Tracts in Mathematics, No. 149, Cambridge University Press, 2003.

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