Some Koszul Rings from Geometry

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.