Skip to Main Content
Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Kummer surfaces for the self-product of the cuspidal rational curve

Author: Stefan Schröer
Journal: J. Algebraic Geom. 16 (2007), 305-346
Published electronically: December 4, 2006
MathSciNet review: 2274516
Full-text PDF

Abstract | References | Additional Information

Abstract: The classical Kummer construction attaches a K3 surface to an abelian surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the self-product of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, we give the correct Kummer-type construction for this situation. We encounter rational double points of type $D_4$ and $D_8$ instead of type $A_1$. It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change.

References [Enhancements On Off] (What's this?)

  • Michael Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962), 485–496. MR 146182, DOI
  • Michael Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136. MR 199191, DOI
  • M. Artin, Supersingular $K3$ surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 371899
  • M. Artin, Algebraic construction of Brieskorn’s resolutions, J. Algebra 29 (1974), 330–348. MR 354665, DOI
  • M. Artin, Coverings of the rational double points in characteristic $p$, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 11–22. MR 0450263
  • Dave Bayer and David Eisenbud, Ribbons and their canonical embeddings, Trans. Amer. Math. Soc. 347 (1995), no. 3, 719–756. MR 1273472, DOI
  • E. Bombieri and D. Mumford, Enriques’ classification of surfaces in char. $p$. III, Invent. Math. 35 (1976), 197–232. MR 491720, DOI
  • N. Bourbaki, Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1285, Hermann, Paris, 1971 (French). Seconde édition. MR 0271276
  • Nicolas Bourbaki, Éléments de mathématique, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR 647314
  • Egbert Brieskorn, Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann. 178 (1968), 255–270 (German). MR 233819, DOI
  • Dan Burns Jr. and Michael Rapoport, On the Torelli problem for kählerian $K-3$ surfaces, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 2, 235–273. MR 447635
  • M. Demazure, P. Gabriel: Groupes algébriques. Masson, Paris, 1970.
  • Torsten Ekedahl, Canonical models of surfaces of general type in positive characteristic, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 97–144. MR 972344
  • Jean Giraud, Improvement of Grauert-Riemenschneider’s theorem for a normal surface, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 13–23 (1983) (English, with French summary). MR 694126
  • G.-M. Greuel and H. Kröning, Simple singularities in positive characteristic, Math. Z. 203 (1990), no. 2, 339–354. MR 1033443, DOI
  • A. Grothendieck: Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 24 (1965).
  • A. Grothendieck: Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 32 (1967).
  • Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
  • Robin Hartshorne, Generalized divisors on Gorenstein schemes, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 287–339. MR 1291023, DOI
  • Hiroyuki Ito, The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic $3$, Math. Z. 211 (1992), no. 1, 1–39. MR 1179777, DOI
  • Sven Toft Jensen, Picard schemes of quotients by finite commutative group schemes, Math. Scand. 42 (1978), no. 2, 197–210. MR 512270, DOI
  • Toshiyuki Katsura, On Kummer surfaces in characteristic $2$, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 525–542. MR 578870
  • Ernst Kunz, Kähler differentials, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1986. MR 864975
  • Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 276239
  • L. Moret-Bailly: Familles de courbes et de varietes abeliennes sur $P^1$. In: L. Szpiro (ed.), Séminaire sur les pinceaux de courbes de genre au moins deux, pp. 109–140. Astérisque 86 (1981).
  • David Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 5–22. MR 153682
  • Arthur Ogus, A crystalline Torelli theorem for supersingular $K3$ surfaces, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 361–394. MR 717616
  • Frans Oort, Which abelian surfaces are products of elliptic curves?, Math. Ann. 214 (1975), 35–47. MR 364264, DOI
  • A. Rudakov, I. Safarevic: Inseparable morphisms of algebraic surfaces. Math. USSR, Izv. 10 (1976), 1205–1237.
  • A. Rudakov, I. Safarevic: Supersingular $K3$ surfaces over fields of characteristic $2$. Math. USSR, Izv. 13 (1979), 147–165.
  • A. Rudakov, I. Shafarevich: Surfaces of type $K3$ over fields of finite characteristic. In: I. Shafarevich, Collected mathematical papers, pp. 657–714. Springer, Berlin, 1989.
  • Stefan Schröer, Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors, Compos. Math. 140 (2004), no. 6, 1579–1592. MR 2098403, DOI
  • C. S. Seshadri, Triviality of vector bundles over the affine space $K^{2}$, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 456–458. MR 102527, DOI
  • Jean-Pierre Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel; Seconde édition, 1965. MR 0201468
  • Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • Tetsuji Shioda, Supersingular $K3$ surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 564–591. MR 555718
  • Tetsuji Shioda, Kummer surfaces in characteristic $2$, Proc. Japan Acad. 50 (1974), 718–722. MR 491728
  • Helmut Strade and Rolf Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. MR 929682
  • Philip Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 92 (1970), 419–454. MR 291170, DOI
  • Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324

Additional Information

Stefan Schröer
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
MR Author ID: 630946

Received by editor(s): May 19, 2005
Received by editor(s) in revised form: August 30, 2005, October 19, 2005, and November 11, 2005
Published electronically: December 4, 2006