Asymptotic Syzygies and Higher Order Embeddings

Abstract

Abstract We show that vanishing of asymptotic $p$-th syzygies implies $p$-very ampleness for line bundles on arbitrary projective schemes. For smooth surfaces, we prove that the converse holds when $p$ is small, by studying the Bridgeland–King–Reid–Haiman correspondence for tautological bundles on the Hilbert scheme of points. This extends the previous results by Ein–Lazarsfeld and Ein–Lazarsfeld–Yang and gives a partial answer to some of their questions. As an application of our results, we show how to use syzygies to bound the irrationality of a variety.