Representation stability for syzygies of line bundles on Segre–Veronese varieties

Abstract

The rational homology groups of the packing complexes are important in algebraic geometry since they control the syzygies of line bundles on projective embeddings of products of projective spaces (Segre-Veronese varieties). These complexes are a common generalization of the multidimensional chessboard complexes and of the matching complexes of complete uniform hypergraphs, whose study has been a topic of interest in combinatorial topology. We prove that the multivariate version of representation stability, a notion recently introduced and studied by Church and Farb, holds for the homology groups of packing complexes. This allows us to deduce stability properties for the syzygies of line bundles on Segre-Veronese varieties. We provide bounds for when stabilization occurs and show that these bounds are sometimes sharp by describing the linear syzygies for a family of line bundles on Segre varieties.As a motivation for our investigation, we show in an appendix that Ein and Lazarsfeld's conjecture on the asymptotic vanishing of syzygies of coherent sheaves on arbitrary projective varieties reduces to the case of line bundles on a product of (at most three) projective spaces.where given a partition δ = (δ 1 , δ 2 , · · · ) of some integer r we write δ[m] for the partition (m − r, δ 1 , δ 2 , · · · ). S δ denotes the Schur functor associated to δ, and we make the convention that S δ[m] is identically zero when m − r < δ 1 .Note that the conclusion of the theorem remains true in the case n = 1 if we replace K p,0 (B a ) with the p-th syzygy module of m a , where m is the homogeneous maximal ideal in the polynomial ring S = Sym(V ): it is well-known (see [BE75, Cor. 3.2] or [Gre84b, (1.a.10)]) that the minimal free resolution of m a is given byTheorem 6.1 was known in the case n = 2 where in fact all the modules K p,q (B a ) can be described explicitly (see [FH98,RR00] or [Wey03, Chapter 6] for a more general story). We will prove Theorem 6.1 by applying the techniques of [FH98] involving combinatorial Laplacians.The description of syzygies in Theorem 6.1 is fairly explicit, the only mystery being the calculation of the multiplicities m λ . This is known to be a complicated plethysm problem, and our theorem is meant to illustrate that the problem of computing syzygies even for simple modules supported on a product of projective spaces is in some sense equally difficult. An asymptotic measure of the complexity of the syzygies in the linear and quadratic strands (K p,0 and K p,1 ) for the Veronese varieties has been obtained by Fulger and Zhou [FZ12] by analyzing the number of distinct irreducible representations appearing in these syzygy modules, as well as the sum of their multiplicities. In Theorem 6.4 we provide a concrete illustration of their theory by describing the linear syzygies of O(1) under a Veronese embedding.We view Theorem 6.1 as a stabilization result in the following way, which we'll be able to generalize further: for a large enough (a ≥ p) the number of irreducible representations (counted with multipli...

A quick proof of nonvanishing for asymptotic syzygies

“…It is tempting to wonder whether there are Gröbner-like techniques that could be used to study the issue systematically. We note that Raicu [9] has reduced the general vanishing conjecture [5, Conjecture 7.1] to the case of Veronese syzygies.…”

Syzygies of projective varieties of large degree: Recent progress and open problems

“…In some situations -for example for the Veronese embeddings discussed in the next section -one can verify that the Conjecture is valid when q = n (see Example 2.4). In general, Raicu [32] shows that knowing the conjecture for X = P n implies its truth for arbitrary varieties. We consider the Conjecture to be the main open problem concerning the rough asymptotics of the K p,q .…”

Asymptotic vanishing of syzygies of algebraic varieties

“…Shortly after the asymptotic vanishing conjecture was proposed, Raicu showed in the appendix of [28] that the general case of the conjecture follows from the case of products of three projective spaces. To prove Theorem 1.1, it is more than enough to establish the following: Theorem 1.2.…”

“…There has been a great deal of attention to the syzygies of Veronese or Segre-Veronese embeddings, e.g. [4,6,7,8,19,25,26,28,30]. The syzygies of these varieties have connections to representation theory and combinatorics.…”

“…The paper is organized as follows. After reviewing basic necessary facts in Section 2, we prove Theorem 1.2 in Section 3, where we also show Theorem 1.1 following Raicu's argument in [28]. Section 4 is devoted to presenting some open problems on asymptotic syzygies of algebraic varieties.…”