On the Canonical Ring of Covers of Surfaces of Minimal Degree

Details On the Canonical Ring of Covers of Surfaces of Minimal Degree On the Canonical Ring of Covers of Surfaces of Minimal Degree

2001-11-06

arxiv.org/abs/math/0111052v1

Mathematics

Algebraic Geometry

22 pages, amstex

Scientific paper

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].

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