# Asymptotic syzygies and higher order embeddings

## TL;DR

This paper explores the relationship between asymptotic syzygies and higher order embeddings, establishing new links and applications in algebraic geometry, especially for smooth surfaces and projective schemes.

## Contribution

It proves that vanishing of asymptotic p-th syzygies implies p-very ampleness, and for smooth surfaces, the converse holds for small p, extending previous work.

## Key findings

- Vanishing of asymptotic p-th syzygies implies p-very ampleness.
- For smooth surfaces, the converse holds when p is small.
- Syzygies can be used to bound the irrationality of a variety.

## 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 previous results of Ein-Lazarsfeld, 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.