Duke Mathematical Journal

The cone conjecture for Calabi-Yau pairs in dimension 2

Burt Totaro

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove the Morrison-Kawamata cone conjecture for Kawamata log terminal Calabi-Yau pairs in dimension 2 . For a large class of rational surfaces as well as for K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain.

Article information

Duke Math. J., Volume 154, Number 2 (2010), 241-263.

First available in Project Euclid: 16 August 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J50: Automorphisms of surfaces and higher-dimensional varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays)


Totaro, Burt. The cone conjecture for Calabi-Yau pairs in dimension 2. Duke Math. J. 154 (2010), no. 2, 241--263. doi:10.1215/00127094-2010-039. https://projecteuclid.org/euclid.dmj/1281963650

Export citation


  • V. Alexeev and S. Mori, “Bounding singular surfaces of general type” in Algebra, Arithmetic and Geometry with Applications (West Lafayette, Ind., 2000), Springer, Berlin, 2004, 143–174.
  • L. Bădescu, Algebraic Surfaces, Springer, New York, 2001.
  • W. Barth, C. Peters, and A. Van De Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1984.
  • C. Birkar, P. Cascini, C. Hacon, and J. Mckernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405–468.
  • R. Blache, The structure of l.c. surfaces of Kodaira dimension zero, I, J. Algebraic Geom. 4 (1995), 137–179.
  • N. Bourbaki, Éléments de mathématique, fasc. 34: chapitres 4–6, Actualités Sci. Indust. 1337, Hermann, Paris, 1971.
  • A.-M. Castravet and J. Tevelev, Hilbert's 14th problem and Cox rings, Compos. Math. 142 (2006), 1479–1498.
  • F. R. Cossec and I. V. Dolgachev, Enriques Surfaces I, Progr. Math. 76, Birkhäuser, Boston, 1989.
  • I. V. Dolgachev, “Abstract configurations in algebraic geometry” in The Fano Conference (Turin, 2002), Universitá di Torino, Turin, 2004, 423–462.
  • I. V. Dolgachev and D.-Q. Zhang, Coble rational surfaces, Amer. J. Math. 123 (2001), 79–114.
  • L.-Y. Fong and J. Mckernan, “Log abundance for surfaces” in Flips and Abundance for Algebraic Threefolds (Salt Lake City, 1991), Astérisque 211 Soc. Math. France, Paris, 127–137.
  • T. Fujita, Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), 685–696.
  • C. Galindo and F. Monserrat, The total coordinate ring of a smooth projective surface, J. Algebra 284 (2005), 91–101.
  • M. H. Gizatullin, Rational $G$-surfaces, Math. USSR Izv. 16 (1981), 103–134.
  • A. Grassi and D. R. Morrison, Automorphisms and the Kähler cone of certain Calabi-Yau manifolds, Duke Math. J. 71 (1993), 831–838.
  • B. Harbourne, “Automorphisms of K$3$-like rational surfaces” in Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Symp. Pure Math. 46, Part 2, Amer. Math. Soc., Providence, 1987, 17–28.
  • Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
  • S. Iitaka, Algebraic Geometry, Grad. Texts in Math. 76, Springer, New York, 1982.
  • Y. Kawamata, On the cone of divisors of Calabi-Yau fiber spaces, Internat. J. Math. 8 (1997), 665–687.
  • J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge, 1998.
  • S. Kondo, Enriques surfaces with finite automorphism groups, Japan J. Math. (N. S.) 12 (1986), 191–282.
  • S. J. Kovács, The cone of curves of a K3 surface, Math. Ann. 300 (1994), 681–691.
  • R. Lazarsfeld, Positivity in Algebraic Geometry, I, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 2004.
  • E. Looijenga, Discrete automorphism groups of convex cones of finite type.
  • C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-manifolds, Grad. Texts in Math. 219, Springer, New York, 2003.
  • R. Meyerhoff, Sphere-packing and volume in hyperbolic 3-space, Comment. Math. Helv. 61 (1986), 271–278.
  • J. S. Milne, Étale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), 537–558.
  • D. R. Morrison, “Compactifications of moduli spaces inspired by mirror symmetry” in Journées de géométrie algébrique d'Orsay (Orsay, 1992), Astérisque 218 (1993), 243–271.
  • —, “Beyond the Kähler cone” in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Bar-Ilan Univ., Ramat Gan, 1996, 361–376.
  • S. Mukai, Finite generation of the Nagata invariant rings in A-D-E cases. preprint, 2005.
  • M. Nagata, On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/1961), 271–293.
  • Y. Namikawa, Periods of Enriques surfaces, Math. Ann. 270 (1985), 201–222.
  • V. V. Nikulin, “Discrete reflection groups in Lobachevsky spaces and algebraic surfaces” in Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, 1986), Amer. Math. Soc., Providence, 654–669.
  • —, Algebraic surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces, I, MPI preprint, 1989.
  • —, Basis of the diagram method for generalized reflection groups in Lobachevsky spaces and algebraic surfaces with nef anticanonical class, Internat. J. Math. 7 (1996), 71–108.
  • —, “On algebraic varieties with finite polyhedral Mori cone” in The Fano Conference (Turin, 2002), Università di Torino, Turin, 2004, 573–589.
  • K. Oguiso and T. Peternell, Calabi-Yau threefolds with positive second Chern class, Comm. Anal. Geom. 6 (1998), 153–172.
  • K. Oguiso and J. Sakurai, Calabi-Yau threefolds of quotient type, Asian J. Math. 5 (2001), 43–77.
  • A. Prendergast-Smith, Extremal rational elliptic threefolds, to appear in Michigan Math. J., preprint.
  • H. Sterk, Finiteness results for algebraic K3 surfaces, Math. Z. 189 (1985), 507–513.
  • K. Suzuki, On Morrison's cone conjecture for klt surfaces with $K_X\equiv 0$, Comment. Math. Univ. St. Paul. 50 (2001), 173–180.
  • B. Szendrői, Some finiteness results for Calabi-Yau threefolds, J. London Math. Soc. (2) 60 (1999), 689–699.
  • D. Testa, A. Várilly-Alvarado, and M. Velasco, Big rational surfaces.
  • B. Totaro, Hilbert's 14th problem over finite fields and a conjecture on the cone of curves, Compos. Math. 144 (2008), 1176–1198.
  • H. Uehara, Calabi-Yau threefolds with infinitely many divisorial contractions, J. Math. Kyoto Univ. 44 (2004), 99–118.
  • E. Vinberg, Classification of $2$-reflective hyperbolic lattices of rank $4$, Trans. Moscow. Math. Soc., Amer. Math. Soc., Providence, 2007, 39–66.
  • P. M. H. Wilson, “Minimal models of Calabi-Yau threefolds” in Classification of Algebraic Varieties (L'Aquila, 1992), Contemp. Math. 162, Amer. Math. Soc., Providence, 1994, 403–410.
  • D.-Q. Zhang, Logarithmic Enriques surfaces, J. Math. Kyoto Univ. 31 (1991), 419–466.