A quick proof of nonvanishing for asymptotic syzygies

TL;DR

This paper introduces a simple, elementary method to prove the nonvanishing of asymptotic syzygies for projective varieties, simplifying previous complex proofs and providing effective results for certain classes.

Contribution

It offers a new, elementary proof approach for asymptotic nonvanishing theorems of syzygies, reducing complex arguments to simple monomial computations.

Findings

Elementary proof of asymptotic nonvanishing of Veronese syzygies

Effective results for arithmetically Cohen-Macaulay varieties

Simplification of previous complex proofs

Abstract

We give a quick new approach to the main cases of the nonvanishing theorems of first and third authors concerning the asymptotic behavior of the syzygies of a projective variety as the positivity of the embedding line bundle grows. Specifically, we present a surprisingly elementary and concrete proof of the asymptotic nonvanishing of Veronese syzygies, and we obtain effective results for arithmetically Cohen-Macaulay varieties. The idea is that one can reduce the statements to some simple computations with monomials.