Linear Syzygies Research Articles

On syzygies of projective varieties

Let X be a projective variety. The conditions that guarantee projective normality and quadratic generations of the ideal defining the embedding of X were studied classically. These results of Mumford et al. were considered by Mark Green as the first step towards understanding the higher syzygies. We say that a line bundle L on X satisfies Property Np if the ideal defining the embedding of X in the complete linear system of L is generated by quadratic equations and has linear syzygies till p stage (see [2]). In [3], Green introduced the notion of Koszul cohomology and proved that vanishing of certain higher Koszul cohomology is equivalent to the Property Np. In [4], it was proved that the required higher Koszul cohomology groups vanish for sufficiently ample line bundles ([4, Theorem 3.2]). However, the proof given in [4] makes a tacit assumption that the ideal sheaf of the diagonal embedding of X in its two-fold product X is locally free which is not true. If X is a smooth projective variety, using different methods, L. Ein and R. Lazarsfeld obtained an effective bound on the power of L required for satisfying Property Np (see [2]). In this article, we give a criterion for a line bundle L to have Property Np in terms of vanishing of certain first cohomology. Our method does not assume the smoothness of X. Using this criterion, we see that sufficiently high power of L will have property Np for fixed p. Therefore, we also see that higher Koszul cohomology of a sufficiently high power of an ample line bundle vanish. This answers the question 5.13 of [3]. Here, we would like to mention that when X is a smooth variety, a vanishing theorem of M. Nori can also be used to answer this question ([8, Proposition 3.4]).

Inverse System of a Symbolic Power II. The Waring Problem for Forms

The classical Waring problem for forms is to determine the smallest length s of an additive decomposition of a general degree d homogeneous polynomial or form f in r variables as sum of sdth powers of linear forms. We show that its solution is implied by a result of J. Alexander and A. Hirschowitz, concerning the Hilbert functions of the ideal of functions Vanishing to order two at a generic set of s points in Pr − 1. Using Macaulay′s inverse systems, we show that the Alexander-Hirschowitz result is equivalent to determining the number of linear syzygies of s homogeneous forms in r variables that are dth powers of a given set of general linear forms. We also determine the dimension of the family of degree d forms that have additive decompositions of length s. We then study several notions of length for forms f, having to do with the kind of length-s, zero-dimensional schemes Z in Pr − 1 whose defining ideal I(Z) annihilates the inverse system of f. When Z is to consist of distinct points, we obtain the above length of additive decomposition of f. When Z is smoothable we obtain the "smoothable length" of f; when Z is arbitrary, we obtain a "scheme length" of f. All these lengths are at least as large as the dimension of the vector space of all order-i partial derivates of f, for each i. The above-mentioned length functions are distinct. Using results about the existence of nonsmoothable Gorenstein point singularities in codimension 4, we show that when r = 5 there are forms f of scheme length s, which are not in the closure of the family of forms having additive decompositions of length s. Finally, we propose a new set of Waring problems for forms, using these lengths.

Frobenius Splitting of Schubert Varieties and Linear Syzygies

Introduction. Let G be a connected, simply connected, semi-simple alge? braic group. Let B be a Borel subgroup of G. For an ample line bundle L on G/B, the questions about projective normality of the embedding of G/B in the complete linear system of L and the equations defining the embedding were extensively studied. Ramanathan in [13] proved that the ideal defining the embedding of G/B in the complete linear system of L is generated by quadratic equations. Kempf and Ramanathan proved that the equations defining the embedding of G/B in the complete linear system of L\,... ,Lr are quadratic and that a Schubert variety X is linearly defined with respect to L\,... ,Lr. Ramanathan had conjectured that the method of Frobenius splitting (used in [13]) can be used to study higher syzygies. Let R (BH?(G/B,?Ljj), where the vector (mj) ranges in Nr, be the multigraded coordinate ring of G/B with respect to L\,... ,Lr. In this paper, we use Frobenius splittings to study the higher syzygies of the irrelevant maximal ideal m of R. We prove that m as a /?-module has linear syzygies. Our methods also give simpler proofs of the results obtained in [6] and [13]. Let Z be a projective variety defined over an algebraically closed field. Let L\,... ,L, be ample line bundles on Z and let m denote the irrelevant maximal ideal of the multi-graded ring R = @H?(Z,?Ljj). In the first section we give a criterion for m to have linear syzigies over R. We show that if for every n > 2, the first cohomology of the n-fold product Zn with values in certain ideal sheaves tensored with powers of Lj vanishes then m has linear syzygies (Propositions 1.9 and 1.10). \ye also give a similar criterion for a homogeneous quotient ring of R to have linear syzygies over R (Proposition 1.13). In the second section, we show that any r ample line bundles L\,...,L, on G/B, and their restrictions to a Schubert variety X C G/B over an algebracally closed field of positive characteristic satisfy the above criteria. To prove this result, we use the method of Frobenius splittings. If Z is a Frobenius split variety, and if L is an ample line bundle on Z, then we have Hl(Z,L) = 0 for / > 1. Further if Y is a compatibly split subvariety of Z, then the restriction

A Note on Frobenius Splitting of Schubert Varities and Linear Syzygies

Let G be a connected, simply connected group and B be a Borel subgroup of G. Let X be a Schubert variety in G/B. Let Lj,... ,Lr be invertible sheaves on G/B. For a e Nr, let LQ = L^1 ? ? ? ? ? L?r. The multi-graded coordinate rings R = ?aH?(G/B,La) and A = 0Q//?(X,LQ), were shown to be wonderful in [1] assuming each of L/ to be ample. In this note we extend the result obtained in [1] to effective invertible sheaves on G/B. An effective invertible sheaf on G/B is a pull-back of an ample invertible sheaf on G/P, for some parabolic subgroup P D B of G. In this note, we use this fact to obtain the necessary vanishing conditions for effective invertible sheaves on G/B. This will prove that R and A are multi-wonderful.

Nets of Alternating Matrices and the Linear Syzygy Conjectures

In this paper we classify the 3-dimensional vector spaces (nets) of 4 × 4 and of 5 × 5 alternating matrices over an algebraically closed field. We apply the second classification to check our "linear syzygy conjectures" for modules over the polynomial ring of dimension 5.

Linear syzygies and line bundles on an algebraic curve

Linear syzygies and line bundles on an algebraic curve