Syzygies of projective varieties of large degree: recent progress and open problems

TL;DR

This survey reviews recent advances and open problems in understanding the asymptotic behavior of syzygies of smooth projective varieties as their embedding line bundles become increasingly positive.

Contribution

It highlights new proofs, non-vanishing theorems, and conjectures about the asymptotics of syzygies and Betti numbers, advancing the theoretical understanding of algebraic geometry.

Findings

Non-vanishing of syzygy modules in the asymptotic regime

New proof of syzygy non-vanishing for Veronese varieties

Discussion of the gonality conjecture for curves of large degree

Abstract

This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results and conjectures about the asymptotics of Betti numbers. Finally we discuss the case of syzygies of weight one, and the gonality conjecture on the syzygies of curves of large degree. The exposition also discusses numerous open questions and conjectures.