Effective Non-vanishing of Asymptotic Adjoint Syzygies

TL;DR

This paper proves an effective non-vanishing theorem for syzygies of adjoint line bundles on smooth varieties, generalizing previous asymptotic results and addressing a specific open problem.

Contribution

It establishes a concrete non-vanishing result for adjoint-type divisors with b ≥ n+1, extending prior asymptotic syzygy theorems to a broader setting.

Findings

Effective non-vanishing theorem for adjoint syzygies

Generalization of asymptotic syzygy results to all smooth varieties

Resolution of Problem 7.9 from prior work

Abstract

The purpose of this paper is to establish an effective non-vanishing theorem for the syzygies of an adjoint-type line bundle on a smooth variety, as the positivity of the embedding increases. Our purpose here is to show that for an adjoint type divisor $B = K_X+ bA$ with $b \geq n+1$, one can obtain an effective statement for arbitrary $X$ which specializes to the statement for Veronese syzygies in the paper "Asymptotic Syzygies of Algebraic Varieties" by Ein and Lazarsfeld. We also give an answer to Problem 7.9 in that paper in this setting.