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

TL;DR

This paper establishes representation stability for the homology groups of packing complexes, leading to new stability results for syzygies of line bundles on Segre--Veronese varieties, with bounds and sharpness results.

Contribution

It proves multivariate representation stability for packing complexes' homology, deriving stability properties for syzygies of line bundles on Segre--Veronese varieties.

Findings

Representation stability holds for packing complexes' homology groups.

Stability bounds for syzygies are provided and shown to be sometimes sharp.

Linear syzygies are explicitly described for certain line bundles.

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.