On the Canonical Ring of Covers of Surfaces of Minimal Degree

Abstract

Let f be a generically finite morphism from X to Y. The purpose of this paper is to show how the O_Y algebra structure on the push forward of O_X controls algebro-geometric aspects of X like the ring generation of graded rings associated to X and the very ampleness of line bundles on X. As the main application of this we prove some new results for certain regular surfaces X of general type. Precisely, we find the degrees of the generators of the canonical ring of X when the canonical morphism of X is a finite cover of a surface of minimal degree. These results complement results of Ciliberto [Ci] and Green [G]. The techniques of this paper also yield different proofs of some earlier results, such as Noether's theorem for certain kinds of curves and some results on Calabi-Yau threefolds that had appeared in [GP2].