Demazure Construction for ℤⁿ-Graded Krull Domains

Abstract

For a Mori dream space X, the Cox ring Cox(X) is a Noetherian $\mathbb {Z}^{n}$-graded normal domain for some n > 0. Let C(Cox(X)) be the cone (in $\mathbb {R}^{n}$) which is spanned by the vectors $\boldsymbol {a} \in \mathbb {Z}^{n}$ such that Cox(X)_a≠ 0. Then, C(Cox(X)) is decomposed into a union of chambers. Berchtold and Hausen (Michigan Math. J., 54(3) 483–515: 2006) 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 $\mathbb {Z}^{n}$-graded Krull domains. That is, under an assumption, we show that a $\mathbb {Z}^{n}$-graded Krull domain is isomorphic to the multi-section ring R(X;D₁,…, D_n) for certain normal projective variety X and $\mathbb {Q}$-divisors D₁, …, D_n on X.