Some Koszul Rings from Geometry

TL;DR

This paper explores the conditions under which certain algebraic rings derived from geometric objects are Koszul, focusing on ample line bundles on projective varieties and adjoint line bundles on irregular surfaces of general type.

Contribution

It establishes a general criterion linking the Koszul property to Castelnuovo-Mumford regularity and provides new examples of Koszul rings from algebraic geometry.

Findings

Koszul property depends on regularity of line bundles

Proved a general result relating Koszul property to ample line bundles

Provided new examples of Koszul rings from algebraic geometry

Abstract

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo-Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on irregular surfaces of general type.