On the (non-)vanishing of syzygies of Segre embeddings

TL;DR

This paper investigates the conditions under which syzygies vanish or not for Segre embeddings, providing bounds and tight results for products of projective lines, and generalizing previous work on property N_p.

Contribution

It introduces new bounds for the non-vanishing of Betti numbers in Segre embeddings and proves their tightness for products of P^1, extending prior results on property N_p.

Findings

Lower bounds for non-zero Betti table rows.

Bounds are tight for Segre embeddings of P^1.

Generalizes results on property N_p.

Abstract

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening laws.