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Mat. Sb., 2010, Volume 201, Number 1, Pages 3–24 (Mi msb7370)  

This article is cited in 9 scientific papers (total in 9 papers)

Cox rings, semigroups and automorphisms of affine algebraic varieties

I. V. Arzhantsev, S. A. Gaifullin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study the Cox realization of an affine variety, that is, a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results in the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of an affine toric variety of dimension $\ge2$ without nonconstant invertible regular functions has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to the Anick automorphism of the polynomial algebra in four variables.
Bibliography: 22 titles.

Keywords: affine variety, quotient, divisor theory of a semigroup, toric variety, wild automorphism.

DOI: https://doi.org/10.4213/sm7370

Full text: PDF file (603 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2010, 201:1, 1–21

Bibliographic databases:

UDC: 512.711+512.745
MSC: Primary 14R20; Secondary 14L30, 14J50
Received: 10.10.2008 and 06.06.2009

Citation: I. V. Arzhantsev, S. A. Gaifullin, “Cox rings, semigroups and automorphisms of affine algebraic varieties”, Mat. Sb., 201:1 (2010), 3–24; Sb. Math., 201:1 (2010), 1–21

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    This publication is cited in the following articles:
    1. A. Anisimov, “Spherical subgroups and double coset varieties”, J. Lie Theory, 22:2 (2012), 505–522  mathscinet  zmath  isi  elib
    2. S. Lamy, S. Vénéreau, “The tame and the wild automorphisms of an affine quadric threefold”, J. Math. Soc. Japan, 65:1 (2013), 299–320  crossref  mathscinet  zmath  isi  scopus
    3. I. Arzhantsev, M. Zaidenberg, “Acyclic curves and group actions on affine toric surfaces”, Affine algebraic geometry, eds. Masuda K., Kojima H., Kishimoto T., World Sci. Publ., Hackensack, NJ, 2013, 1–41  mathscinet  zmath  isi
    4. J. Hausen, S. Keicher, “A software package for Mori dream spaces”, LMS J. Comput. Math., 18:1 (2015), 647–659  crossref  mathscinet  zmath  isi  scopus
    5. Donten-Bury M., “Cox Rings of Minimal Resolutions of Surface Quotient Singularities”, Glasg. Math. J., 58:2 (2016), 325–355  crossref  mathscinet  zmath  isi  scopus
    6. Donten-Bury M., Keicher S., “Computing resolutions of quotient singularities”, J. Algebra, 472 (2017), 546–572  crossref  mathscinet  zmath  isi  scopus
    7. Donten-Bury M., Wisniewski J.A., “on 81 Symplectic Resolutions of a 4-Dimensional Quotient By a Group of Order 32”, Kyoto J. Math., 57:2 (2017), 395–434  crossref  mathscinet  zmath  isi  scopus
    8. Yamagishi R., “On Smoothness of Minimal Models of Quotient Singularities By Finite Subgroups of Sln(C)”, Glasg. Math. J., 60:3 (2018), 603–634  crossref  mathscinet  zmath  isi  scopus
    9. Arzhantsev I., Braun L., Hausen J., Wrobel M., “Log Terminal Singularities, Platonic Tuples and Iteration of Cox Rings”, Eur. J. Math., 4:1, 1, SI (2018), 242–312  crossref  mathscinet  zmath  isi  scopus
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