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Matematicheskie Zametki, 2009, Volume 85, Issue 5, Pages 643–651
DOI: https://doi.org/10.4213/mzm6907 (Mi mzm6907) 		 
This article is cited in 12 scientific papers (total in 12 papers)

On the Factoriality of Cox rings

I. V. Arzhantsev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Full-text PDF (474 kB) Citations (12)
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DOI: https://doi.org/10.4213/mzm6907
Abstract: The generalized Cox construction associates with an algebraic variety a remarkable invariant — its total coordinate ring, or Cox ring. In this note, we give a new proof of the factoriality of the Cox ring when the divisor class group of the variety is finitely generated and free. The proof is based on the notion of graded factoriality. We show that if the divisor class group has torsion, then the Cox ring is again factorially graded, but factoriality may be lost.
Keywords: total coordinate ring, Cox ring, algebraic variety, factorial ring, graded factoriality, divisor class group, torsion, Weil divisor, Cartier divisor.
Received: 05.02.2008
English version:
Mathematical Notes, 2009, Volume 85, Issue 5, Pages 623–629
DOI: https://doi.org/10.1134/S0001434609050022
Bibliographic databases:
UDC: 512.71
Language: Russian
Citation: I. V. Arzhantsev, “On the Factoriality of Cox rings”, Mat. Zametki, 85:5 (2009), 643–651; Math. Notes, 85:5 (2009), 623–629
Citation in format AMSBIB
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    • This publication is cited in the following 12 articles:
    1. Joaquin Moraga, Roberto Svaldi, “A geometric characterization of toric singularities”, Journal de Mathématiques Pures et Appliquées, 2024, 103620  crossref
    2. Euisung Park, “On rank 3 quadratic equations of projective varieties”, Trans. Amer. Math. Soc., 2023  crossref
    3. Braun L., “Gorensteinness and Iteration of Cox Rings For Fano Type Varieties”, Math. Z., 301:1 (2022), 1047–1061  crossref  mathscinet  isi
    4. Cecil J., Dutta N., Manon Ch., Riley B., Vichitbandha A., “Well-Poised Hypersurfaces”, Commun. Algebr., 49:6 (2021), 2645–2654  crossref  mathscinet  isi
    5. Hausen J., Hische Ch., Wrobel M., “On Torus Actions of Higher Complexity”, Forum Math. Sigma, 7 (2019), e38  crossref  mathscinet  isi
    6. Bechtold B., “Valuative and Geometric Characterizations of Cox Sheaves”, J. Commut. Algebr., 10:1 (2018), 1–43  crossref  mathscinet  zmath  isi  scopus
    7. Arzhantsev I., Kotenkova P., “Equivariant Embeddings of Commutative Linear Algebraic Groups of Corank One”, Doc. Math., 20 (2015), 1039–1053  mathscinet  zmath  isi
    8. Hausen J., Keicher S., Laface A., “Computing Cox Rings”, Math. Comput., 85:297 (2015), 467–502  crossref  mathscinet  isi  scopus
    9. G. Gagliardi, “The Cox ring of a spherical embedding”, J. Algebra, 397 (2014), 548–569  crossref  mathscinet  zmath  isi  scopus
    10. Bechtold B., “Factorially graded rings and Cox rings”, J. Algebra, 369 (2012), 351–359  crossref  mathscinet  zmath  isi  elib  scopus
    11. I. V. Arzhantsev, S. A. Gaifullin, “Cox rings, semigroups and automorphisms of affine algebraic varieties”, Sb. Math., 201:1 (2010), 1–21  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. Arzhantsev I., Gaifullin S., “Homogeneous toric varieties”, J. Lie Theory, 20:2 (2010), 283–293  mathscinet  zmath  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
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