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,…, Dn) for certain normal projective variety X and \(\mathbb {Q}\)-divisors D1, …, Dn on X.

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Notes

  1. 1.

    Some people call it the Dolgachev-Pinkham-Demazure construction.

  2. 2.

    Instead of assuming that A is Noetherian, it is sufficient to assume that \(A_{\mathbb {R}_{\ge 0}}\boldsymbol {a}\) is Noetherian. In fact, if \(A_{\mathbb {R}_{\ge 0}}\boldsymbol {a}\) is Noetherian, we can find a Noetherian subring of A with the same situation as A.

  3. 3.

    In many cases, d1/r does not equal d2/s when there exist homogeneous elements f, g satisfying (3.7). In the case (a, b, c) = (1, 1, 1), d1/r = d2/s = 1 holds. The authors do not know any other examples satisfying d1/r = d2/s.

  4. 4.

    The difficulty in this proof lies in that we have to find such a vector b contained in \(\mathbb {Q}^{n-s}\).

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Correspondence to Kazuhiko Kurano.

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Dedicated to Professor Kei-ichi Watanabe on the occasion of his 74th birthday

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Arai, Y., Echizenya, A. & Kurano, K. Demazure Construction for ℤn-Graded Krull Domains. Acta Math Vietnam 44, 173–205 (2019). https://doi.org/10.1007/s40306-018-0281-0

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Keywords

  • Demazure construction
  • Dolgachev-Pinkham-Demazure construction
  • Multi-section ring
  • Mori dream space
  • Krull ring

Mathematics Subject Classification (2010)

  • 13A02
  • 14E99