Some Homogeneous Rings Associated to Finite Morphisms

Abstract

In this article we present a new technique to handle the study of homogeneous rings of a projective variety endowed with a finite or a generically finite morphism to another variety Y whose geometry is easier to handle. Under these circumstances it is possible to use the information given by the algebra structure of \( {{\mathcal{O}}_{X}} \) over \( {{\mathcal{O}}_{Y}} \) to describe the homogeneous ring associated to line bundles which are pullback of line bundles on Y. In this article we illustrate our technique to study the canonical ring of curves (a well-known ring that we revisit with this new technique) equipped with a suitable finite morphism and homogeneous rings of certain class of Calabi-Yau threefolds.