Demazure construction for Z^n-graded Krull domains

TL;DR

This paper develops a Demazure construction for Z^n-graded Krull domains, providing an algebraic proof of cone decompositions for Cox rings of Mori dream spaces and establishing isomorphisms with multi-section rings.

Contribution

It introduces an elementary algebraic proof for cone decompositions and extends the Demazure construction to Z^n-graded Krull domains under certain conditions.

Findings

Elementary algebraic proof of cone decompositions for Cox rings.

Demazure construction extended to Z^n-graded Krull domains.

Isomorphism between Z^n-graded Krull domains and multi-section rings.

Abstract

For a Mori dream space X, the Cox ring Cox(X) is a Noetherian Z^n-graded normal domain for some n > 0. Let C(Cox(X)) be the cone (in R^n) which is spanned by the vectors a \in Z^n such that Cox(X)_a \neq 0. Then C(Cox(X)) is decomposed into a union of chambers. Berchtold and Hausen proved the existence of such decompositions for affine integral domains over an algebraically closed field. We shall give an elementary algebraic proof to this result in the case where the homogeneous component of degree 0 is a field. Using such decompositions, we develop the Demazure construction for Z^n-graded Krull domains. That is, under an assumption, we show that a Z^n-graded Krull domain is isomorphic to the multi-section ring R(X; D_1, \ldots, D_n) for certain normal projective variety X and Q-divisors D_1,...,D_n on X.