  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].
