Syzygies of Line Bundles on GIT Quotients

TL;DR

This paper investigates the preservation of syzygy properties of line bundles under GIT quotients, establishing conditions under which the property $N_p$ is maintained, with applications to finite groups and specific cases.

Contribution

It proves that the property $N_p$ of a line bundle descends to the GIT quotient under certain conditions, extending understanding of syzygies in geometric invariant theory.

Findings

$N_p$ property is preserved under descent in GIT quotients given certain conditions.

Finite group actions with $N_p$ and $N_0$ properties imply $N_p$ for the descended line bundle.

Applications to specific cases demonstrate the practical relevance of the theoretical results.

Abstract

Let $k$ be an algebraically closed field. Consider a reductive group $G$ over $k$. Let $X$ be a projective variety over $k$ with a $G$-action and let $L$ be a very ample $G$-linearized line bundle on $X$. Suppose that $L$ descends to the GIT quotient of $X$ by $G$. If $L$ satisfies the property $N_p$ one can ask if its descent also has $N_p$ property. In this article, we show this is the case under certain conditions. We then apply our results to some cases of interest. As a consequence of our results, we show that if $G$ is a finite group and $L$ satisfies $N_p$ property and its descent satisfies $N_0$ property then it satisfies $N_p$ property as well under suitable conditions.