Acta Mathematica Vietnamica

, Volume 44, Issue 1, pp 173–205 | Cite as

Demazure Construction for ℤn-Graded Krull Domains

  • Yusuke Arai
  • Ayaka Echizenya
  • Kazuhiko KuranoEmail author


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.


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

Mathematics Subject Classification (2010)

13A02 14E99 


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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.IBM Japan Services Company Ltd.TokyoJapan
  2. 2.Department of Mathematics, Faculty of Science and TechnologyMeiji UniversityKawasakiJapan

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