The total coordinate ring of a smooth projective surface

Abstract

In [5], Cox introduced the homogeneous coordinate ring of a toric variety, which is a polynomial ring that allows to show that such a variety behaves like a projective space in many ways. An analogous to that ring can also be defined for a smooth projective variety X over an algebraically closed field k such that linear and numerical equivalence coincide for divisors on X, condition which is assumed for all varieties considered in this paper. Indeed, let us fix {[Li]}i=1 a Z-basis of Pic(X), and set n = (n1, n2, . . . , nr ) ∈ Z and Dn =∑ri=1 niLi . By regarding the vector spaces H 0(X,OX(Dn))= {f ∈K∗ | divX(f )+ Dn 0} ∪ {0} as k-subvector spaces of the function field K of X, the total coordinate ring of X (or the Cox ring of X) is defined as the graded k-subalgebra of K: