Algebra & Number Theory

Cox rings of degree one del Pezzo surfaces

Damiano Testa, Anthony Várilly-Alvarado, and Mauricio Velasco

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Let X be a del Pezzo surface of degree one over an algebraically closed field, and let Cox(X) be its total coordinate ring. We prove the missing case of a conjecture of Batyrev and Popov, which states that Cox(X) is a quadratic algebra. We use a complex of vector spaces whose homology determines part of the structure of the minimal free Pic(X)-graded resolution of Cox(X) over a polynomial ring. We show that sufficiently many Betti numbers of this minimal free resolution vanish to establish the conjecture.

Article information

Algebra Number Theory, Volume 3, Number 7 (2009), 729-761.

Received: 8 March 2008
Revised: 5 June 2009
Accepted: 14 September 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J26: Rational and ruled surfaces

Cox rings total coordinate rings del Pezzo surfaces


Testa, Damiano; Várilly-Alvarado, Anthony; Velasco, Mauricio. Cox rings of degree one del Pezzo surfaces. Algebra Number Theory 3 (2009), no. 7, 729--761. doi:10.2140/ant.2009.3.729.

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  • V. V. Batyrev and O. N. Popov, “The Cox ring of a del Pezzo surface”, pp. 85–103 in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), edited by B. Poonen and Y. Tschinkel, Progr. Math. 226, Birkhäuser, Boston, MA, 2004.
  • C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan, “Existence of minimal models for varieties of log general type”, preprint, 2008.
  • W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language”, J. Symbolic Comput. 24:3-4 (1997), 235–265.
  • R. de la Bretèche and T. D. Browning, “On Manin's conjecture for singular del Pezzo surfaces of degree 4, I”, Michigan Math. J. 55:1 (2007), 51–80.
  • R. de la Bretèche, T. D. Browning, and U. Derenthal, “On Manin's conjecture for a certain singular cubic surface”, Ann. Sci. École Norm. Sup. $(4)$ 40:1 (2007), 1–50.
  • M. Brion, “The total coordinate ring of a wonderful variety”, J. Algebra 313:1 (2007), 61–99.
  • A.-M. Castravet and J. Tevelev, “Hilbert's 14th problem and Cox rings”, Compos. Math. 142:6 (2006), 1479–1498.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, “La descente sur les variétés rationnelles”, pp. 223–237 in Journées de Géometrie Algébrique d'Angers, Juillet 1979 (Alphen aan den Rijn, 1979), Sijthoff & Noordhoff, 1980.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, “La descente sur les variétés rationnelles, II”, Duke Math. J. 54:2 (1987), 375–492.
  • J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer, “Intersections de deux quadriques et surfaces de Châtelet”, C. R. Acad. Sci. Paris Sér. I Math. 298:16 (1984), 377–380.
  • D. A. Cox, “The homogeneous coordinate ring of a toric variety”, J. Algebraic Geom. 4:1 (1995), 17–50.
  • P. Cragnolini and P. A. Oliverio, “Lines on del Pezzo surfaces with $K\sp 2\sb S=1$ in characteristic 2 in the smooth case”, Portugal. Math. 57:1 (2000), 59–95. 2001c:14057
  • O. Debarre, Higher-dimensional algebraic geometry, Universitext 13, Springer, New York, 2001.
  • U. Derenthal, “Universal torsors of del Pezzo surfaces and homogeneous spaces”, Adv. Math. 213:2 (2007), 849–864.
  • E. J. Elizondo, K. Kurano, and K.-i. Watanabe, “The total coordinate ring of a normal projective variety”, J. Algebra 276:2 (2004), 625–637.
  • R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • B. Hassett, “Equations of universal torsors and Cox rings”, pp. 135–143 in Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Summer Term 2004, edited by Y. Tschinkel, Universitätsdrucke Göttingen, Göttingen, 2004.
  • B. Hassett and Y. Tschinkel, “Universal torsors and Cox rings”, pp. 149–173 in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), edited by B. Poonen and Y. Tschinkel, Progr. Math. 226, Birkhäuser, Boston, MA, 2004.
  • A. Hefez and S. L. Kleiman, “Notes on the duality of projective varieties”, pp. 143–183 in Geometry today (Rome, 1984), edited by E. Arbarello et al., Progr. Math. 60, Birkhäuser, Boston, MA, 1985.
  • Y. Hu and S. Keel, “Mori dream spaces and GIT”, Michigan Math. J. 48 (2000), 331–348.
  • H. Kaji, “On the inseparable degrees of the Gauss map and the projection of the conormal variety to the dual of higher order for space curves”, Math. Ann. 292:3 (1992), 529–532.
  • J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Math. (3) 32, Springer, Berlin, 1996.
  • A. Laface and M. Velasco, “Picard-graded Betti numbers and the defining ideals of Cox rings”, J. Algebra 322:2 (2009), 353–372.
  • Y. I. Manin, Cubic forms, 2nd ed., North-Holland Mathematical Library 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel.
  • E. Peyre, “Counting points on varieties using universal torsors”, pp. 61–81 in Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), edited by B. Poonen and Y. Tschinkel, Progr. Math. 226, Birkhäuser, Boston, MA, 2004.
  • O. N. Popov, “The Cox ring of a Del Pezzo surface has rational singularities”, preprint, 2004.
  • M. Reid, “Chapters on algebraic surfaces”, pp. 3–159 in Complex algebraic geometry (Park City, UT, 1993), edited by J. Kollár, IAS/Park City Math. Ser. 3, Amer. Math. Soc., Providence, RI, 1997.
  • P. Salberger, “Tamagawa measures on universal torsors and points of bounded height on Fano varieties”, pp. 91–258 in Nombre et répartition de points de hauteur bornée (Paris, 1996), vol. 251, edited by E. Peyre, 1998.
  • V. V. Serganova and A. N. Skorobogatov, “Del Pezzo surfaces and representation theory”, Algebra Number Theory 1:4 (2007), 393–419.
  • V. Serganova and A. Skorobogatov, “On the equations for universal torsors over del Pezzo surfaces”, preprint, 2008.
  • M. Stillman, D. Testa, and M. Velasco, “Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces”, J. Algebra 316:2 (2007), 777–801.
  • B. Sturmfels and Z. Xu, “Sagbi Bases of Cox–Nagata rings”, preprint, 2008. to appear in J. European Math. Soc.
  • H. Terakawa, “On the Kawamata–Viehweg vanishing theorem for a surface in positive characteristic”, Arch. Math. $($Basel$)$ 71:5 (1998), 370–375.
  • D. Testa, A. Várilly-Alvarado, and M. Velasco, “Big rational surfaces”, preprint, 2009.
  • Q. Xie, “Kawamata–Viehweg vanishing on rational surfaces in positive characteristic”, 2008.