Asymptotic syzygies grow exponentially: a remark on a paper of Ein-Lazarsfeld

TL;DR

The paper demonstrates that the dimensions of certain asymptotic Koszul cohomology groups grow exponentially with the degree, revealing new growth patterns in algebraic geometry.

Contribution

It provides the first evidence of exponential growth in asymptotic syzygies, extending previous understanding of their behavior.

Findings

Exponential growth of asymptotic Koszul cohomology groups.

Growth occurs for all relevant q and most p values.

Results apply to a broad class of algebraic varieties.

Abstract

For all $2\leq q\leq \dim(X)$ and most relevant $p$ values, the dimension of the asymptotic Koszul cohomology group $K_{p,q}(X, B;L_d)$ grows exponentially with $d$.