Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T14:18:58.543Z Has data issue: false hasContentIssue false
  • English
  • Français

On Cox rings of K3 surfaces

Part of: Cycles and subschemes Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  25 March 2010

Michela Artebani ,
Jürgen Hausen  and
Antonio Laface
Michela Artebani
Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile (email: martebani@udec.cl)
Jürgen Hausen
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany (email: juergen.hausen@uni-tuebingen.de)
Antonio Laface
Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile (email: alaface@udec.cl)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study Cox rings of K3 surfaces. A first result is that a K3 surface has a finitely generated Cox ring if and only if its effective cone is rational polyhedral. Moreover, we investigate degrees of generators and relations for Cox rings of K3 surfaces of Picard number two, and explicitly compute the Cox rings of generic K3 surfaces with a non-symplectic involution that have Picard number 2 to 5 or occur as double covers of del Pezzo surfaces.

Keywords

Cox ringsK3 surfaces

MSC classification

Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14C20: Divisors, linear systems, invertible sheaves
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 4 , July 2010 , pp. 964 - 998
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Alexeev, V. and Nikulin, V. V., Del Pezzo and K3 surfaces, MSJ Memoirs, vol. 15 (Mathematical Society of Japan, Tokyo, 2006).Google Scholar
[2]Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of algebraic curves, Grundlehren der Mathematischen Wissenschaften, vol. 267 (Springer, New York, 1985).CrossRefGoogle Scholar
[3]Atiyah, M. F. and Macdonald, I. G., Introduction to commutative algebra (Addison-Wesley, Reading, MA, 1969).Google Scholar
[4]Barth, W. P., Hulek, K., Peters, A. M. C. and Van de Ven, A., Compact complex surfaces, in Ergebnisse der Mathematik und ihrer Grenzgebiete (3), second edition, A Series of Modern Surveys in Mathematics, vol. 4 (Springer, Berlin, 2004).Google Scholar
[5]Batyrev, V. and Popov, O., The Cox ring of a del Pezzo surface, in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Progress in Mathematics, vol. 226 (Birkhäuser, Boston, MA, 2004), 85–103.Google Scholar
[6]Berchtold, F. and Hausen, J., Homogeneous coordinates for algebraic varieties, J. Algebra 266 (2003), 636–670.Google Scholar
[7]Berchtold, F. and Hausen, J., Cox rings and combinatorics, Trans. Amer. Math. Soc. 359 (2007), 1205–1252.Google Scholar
[8]Cox, D., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17–50.Google Scholar
[9]Derenthal, U., Singular Del Pezzo surfaces whose universal torsors are hypersurfaces, arXiv:math/0604194.Google Scholar
[10]Fulton, W., Intersection theory, in Ergebnisse der Mathematik und ihrer Grenzgebiete (3), second edition, A Series of Modern Surveys in Mathematics, vol. 2 (Springer, Berlin, 1998).Google Scholar
[11]Galindo, C. and Monserrat, F., The total coordinate ring of a smooth projective surface, J. Algebra 284 (2005), 91–101.Google Scholar
[12]Hausen, J., Cox rings and combinatorics II, Mosc. Math. J. 8 (2008), 711–757.CrossRefGoogle Scholar
[13]Hu, Y. and Keel, S., Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.Google Scholar
[14]Kondō, S., Algebraic K3-surfaces with finite automorphism groups, Nagoya Math. J. 116 (1989), 1–15.CrossRefGoogle Scholar
[15]Kovács, S., The cone of curves of a K3-surface, Math. Ann. 300 (1994), 681–691.Google Scholar
[16]Laface, A. and Velasco, M., Picard-graded Betti numbers and the defining ideals of Cox rings, J. Algebra 322 (2009), 353–372.CrossRefGoogle Scholar
[17]Mayer, A. L., Families of K3-surfaces, Nagoya Math. J. 48 (1972), 1–17.CrossRefGoogle Scholar
[18]Morrison, D., On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105–121.Google Scholar
[19]Nikulin, V. V., On factor groups of automorphism groups of hyperbolic forms modulo subgroups generated by 2-reflections: algebro-geometric applications, Dokl. Acad. Nauk. SSSR 248 (1979), 1307–1309.Google Scholar
[20]Nikulin, V. V., Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, J. Sov. Math. 22 (1983), 1401–1475.Google Scholar
[21]Nikulin, V. V., Surfaces of type K3 with a finite automorphism group and a Picard group of rank three, Tr. Mat. Inst. Steklova 165 (1984), 119–142; Proc. Steklov Inst. Math. 3 (1985), 131–157.Google Scholar
[22]Piatetski-Shapiro, I. and Shafarevich, I. R., A Torelli theorem for algebraic surfaces of type K3, Math. USSR Izv. 5 (1971), 547–588.Google Scholar
[23]Saint-Donat, B., Projective models of K3-surfaces, Amer. J. Math. 96 (1974), 602–639.CrossRefGoogle Scholar
[24]Scheja, G. and Storch, U., Zur Konstruktion faktorieller graduierter Integritätsbereiche, Arch. Math. (Basel) 42 (1984), 45–52.Google Scholar
[25]Stillman, M., Testa, D. and Velasco, M., Groebner bases, monomial group actions, and the Cox rings of Del Pezzo surfaces, J. Algebra 316 (2007), 777–801.CrossRefGoogle Scholar
[26]Sturmfels, B. and Xu, Z., Sagbi bases of Cox–Nagata rings, arXiv:0803.0892, J. Eur. Math. Soc. (JEMS), to appear.Google Scholar
[27]Testa, D., Várilly-Alvarado, A. and Velasco, M., Cox rings of degree one del Pezzo surfaces, arXiv:0803.0353, Algebra & Number Theory, to appear.Google Scholar
[28]Zhang, D.-Q., Quotients of K3-surfaces modulo involutions, Japan. J. Math. (N.S.) 24 (1998), 335–366.Google Scholar