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
DOI: https://doi.org/10.1090/S1056-3911-06-00438-3
Published electronically: December 4, 2006
MathSciNet review: 2274516
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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.


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  • 1. M. Artin: Some numerical criteria for contractability of curves on algebraic surfaces. Am. J. Math. 84 (1962), 485-496. MR 0146182 (26:3704)
  • 2. M. Artin: On isolated rational singularities of surfaces. Am. J. Math. 88 (1966), 129-136. MR 0199191 (33:7340)
  • 3. M. Artin: Supersingular $ K3$ surfaces. Ann. Sci. École Norm. Sup. 7 (1974), 543-567. MR 0371899 (51:8116)
  • 4. M. Artin: Algebraic construction of Brieskorn's resolutions. J. Algebra 29 (1974), 330-348. MR 0354665 (50:7143)
  • 5. M. Artin: Coverings of the rational double points in characteristic $ p$. In: W. Baily, T. Shioda (eds.), Complex analysis and algebraic geometry, pp. 11-22. Iwanami Shoten, Tokyo, 1977. MR 0450263 (56:8559)
  • 6. D. Bayer, D. Eisenbud: Ribbons and their canonical embeddings. Trans. Am. Math. Soc. 347 (1995), 719-756. MR 1273472 (95g:14032)
  • 7. E. Bombieri, D. Mumford: Enriques' classification of surfaces in char. p. III. Invent. Math. 35 (1976), 197-232. MR 0491720 (58:10922b)
  • 8. N. Bourbaki: Groupes et algèbres de Lie. Chap. I: Algèbres de Lie. Actualités scientifiques et industrielle 1285, Paris, Hermann, 1971. MR 0271276 (42:6159)
  • 9. N. Bourbaki: Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Masson, Paris, 1981. MR 0647314 (83g:17001)
  • 10. E. Brieskorn: Die Auflösung der rationalen Singularitäten holomorpher Abbildungen. Math. Ann. 178 (1968), 255-270. MR 0233819 (38:2140)
  • 11. D. Burns, M. Rapoport: On the Torelli problem for kählerian K-3 surfaces. Ann. Sci. École Norm. Sup. 8 (1975), 235-273. MR 0447635 (56:5945)
  • 12. M. Demazure, P. Gabriel: Groupes algébriques. Masson, Paris, 1970.
  • 13. T. Ekedahl: Canonical models of surfaces of general type in positive characteristic. Inst. Hautes Études Sci. Publ. Math. 67 (1988), 97-144. MR 0972344 (89k:14069)
  • 14. J. Giraud: Improvement of Grauert-Riemenschneider's theorem for a normal surface. Ann. Inst. Fourier 32 (1982), 13-23. MR 0694126 (84f:14025)
  • 15. G.-M. Greuel, H. Kröning: Simple singularities in positive characteristic. Math. Z. 203 (1990), 339-354. MR 1033443 (90k:14001)
  • 16. A. Grothendieck: Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 24 (1965).
  • 17. A. Grothendieck: Éléments de géométrie algébrique IV: Étude locale des schémas et des morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 32 (1967).
  • 18. A. Grothendieck et al.: Revêtements étales et groupe fondamental. Lect. Notes Math. 224, Springer, Berlin, 1971. MR 0354651 (50:7129)
  • 19. R. Hartshorne: Generalised divisors on Gorenstein schemes. K-Theory 8 (1994), 287-339. MR 1291023 (95k:14008)
  • 20. H. Ito: The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic $ 3$. Math. Z. 211 (1992), 1-39. MR 1179777 (94d:14035)
  • 21. S. Jensen: Picard schemes of quotients by finite commutative group schemes. Math. Scand. 42 (1978), 197-210. MR 0512270 (80a:14015)
  • 22. T. Katsura: On Kummer surfaces in characteristic $ 2$. In: M. Nagata (ed.), Proceedings of the international symposium on algebraic geometry, pp. 525-542. Kinokuniya Book Store, Tokyo, 1978. MR 0578870 (82d:14022)
  • 23. E. Kunz: Kähler differentials. Vieweg, Braunschweig, 1986. MR 0864975 (88e:14025)
  • 24. J. Lipman: Rational singularities, with applications to algebraic surfaces and unique factorization. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195-279. MR 0276239 (43:1986)
  • 25. 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).
  • 26. D. Mumford: The topology of a normal surface singularity of an algebraic variety and criterion for simplicity. Publ. Math., Inst. Hautes Étud. Sci. 9 (1961), 5-22. MR 0153682 (27:3643)
  • 27. A. Ogus: A crystalline Torelli theorem for supersingular $ K3$ surfaces. In: M. Artin, J. Tate (eds.), Arithmetic and geometry, Vol. II, pp. 361-394. Progr. Math. 36. Birkhäuser, Boston, 1983. MR 0717616 (85d:14055)
  • 28. F. Oort: Which abelian surfaces are products of elliptic curves? Math. Ann. 214 (1975), 35-47. MR 0364264 (51:519)
  • 29. A. Rudakov, I. Safarevic: Inseparable morphisms of algebraic surfaces. Math. USSR, Izv. 10 (1976), 1205-1237.
  • 30. A. Rudakov, I. Safarevic: Supersingular $ K3$ surfaces over fields of characteristic $ 2$. Math. USSR, Izv. 13 (1979), 147-165.
  • 31. 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.
  • 32. S. Schröer: Some Calabi-Yau threefolds with obstructed deformations over the Witt vectors. Compositio Math. 140 (2004), 1579-1592. MR 2098403 (2005i:14051)
  • 33. C. Seshadri: Triviality of vector bundles over the affine space $ K^2$. Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 456-458. MR 0102527 (21:1318)
  • 34. J.-P. Serre: Algèbre locale. Multiplicités. Lect. Notes Math. 11. Springer, Berlin, 1965. MR 0201468 (34:1352)
  • 35. J.-P. Serre: Local fields. Grad. Texts Math. 67. Springer, Berlin, 1979. MR 0554237 (82e:12016)
  • 36. T. Shioda: Supersingular $ K3$ surfaces. In: K. Lonsted (ed.), Algebraic geometry, pp. 564-591. Lecture Notes in Math. 732. Springer, Berlin, 1979. MR 0555718 (82c:14030)
  • 37. T. Shioda: Kummer surfaces in characteristic $ 2$. Proc. Japan Acad. 50 (1974), 718-722. MR 0491728 (58:10929)
  • 38. H. Strade, R. Farnsteiner: Modular Lie algebras and their representations. Monographs and Textbooks in Pure and Applied Mathematics 116. Marcel Dekker, New York, 1988. MR 0929682 (89h:17021)
  • 39. P. Wagreich: Elliptic singularities of surfaces. Amer. J. Math. 92 (1970), 419-454. MR 0291170 (45:264)
  • 40. C. Weibel: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38. Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)


Additional Information

Stefan Schröer
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany
Email: schroeer@math.uni-duesseldorf.de

DOI: https://doi.org/10.1090/S1056-3911-06-00438-3
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

American Mathematical Society