Asymptotic syzygies of algebraic varieties

TL;DR

This paper investigates the long-term behavior of syzygies in algebraic varieties as their embedding line bundles become more positive, revealing a surprisingly uniform asymptotic pattern that challenges previous intuitions.

Contribution

It establishes a new understanding of the asymptotic shape of syzygies for algebraic varieties, including effective results for Veronese embeddings and conjectures for higher dimensions.

Findings

Syzygies exhibit a uniform asymptotic shape as positivity increases.

Generators appear in almost all degrees allowed by regularity.

Effective and optimal results are provided for Veronese embeddings.

Abstract

This paper studies the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that as least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. This suggests in particular that a widely-accepted intuition derived from the case of curves -- namely that syzygies become simpler as the degree of the embedding increases -- may have been misleading. For Veronese embeddings of projective space, we give an effective statement that in some cases is optimal, and conjecturally always is so. Finally, we propose a number of questions and open problems concerning asymptotic syzygies of higher-dimensional varieties.