The Quantitative Behavior of Asymptotic Syzygies for Hirzebruch Surfaces

TL;DR

This paper investigates the asymptotic syzygies of Hirzebruch surfaces, confirming some conjectures about their behavior and revealing that higher degree syzygies do not follow a normal distribution.

Contribution

It provides a quantitative analysis of asymptotic syzygies for Hirzebruch surfaces and verifies the normality heuristic for linear syzygies.

Findings

Asymptotic linear syzygies conform to the normality heuristic.

Higher degree asymptotic syzygies are not normally distributed.

The study advances understanding of syzygy behavior in toric surfaces.

Abstract

The goal of this note is to quantitatively study the behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $\mathcal{O}(d,2)$ conform to Ein, Erman, and Lazarsfeld's normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed.