Asymptotic Syzygies in the Setting of Semi-Ample Growth

TL;DR

This paper investigates the asymptotic behavior of syzygies for products of projective spaces with semi-ample embeddings, providing explicit bounds and revealing complex patterns in their non-vanishing properties.

Contribution

It generalizes monomial methods to give explicit ranges for non-zero Betti numbers in semi-ample settings, a novel insight into asymptotic syzygies of such varieties.

Findings

Explicit bounds for non-vanishing of Betti numbers

First example of asymptotic syzygy behavior in semi-ample embeddings

Reveals nuanced and previously unseen patterns

Abstract

We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of $\mathbb{P}^{n_1}\times \mathbb{P}^{n_2}$ embedded by $\mathcal{O}_{\mathbb{P}^{n_1}\times\mathbb{P}^{n_2}}(d_1,d_2)$ are non-zero. These bounds provide the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.