On W_2-lifting of Frobenius of algebraic surfaces

• Autores: He Xin
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 67, Fasc. 1, 2016, págs. 69-83
• Idioma: inglés
• DOI: 10.1007/s13348-014-0130-y
• Texto completo no disponible (Saber más ...)
• Resumen

  • The classification of minimal algebraic surfaces in positive characteristics was accomplished by Bombieri and Mumford in the 1970s. In this work we decide completely which minimal algebraic surfaces in positive characteristics allow a lifting of their Frobenius over the truncated Witt rings of length 2. Besides, we show that the Frobenius morphism of many projective rational surfaces cannot be lifted to W_2(k).

• Referencias bibliográficas
  • Beauville, A.: Complex algebraic surfaces, vol. 34. Cambridge University Press (1996)
  • Beauville, A.: Endomorphisms of hypersurfaces and other manifolds. Int. Math. Res. Notices 2001(1), 53–58 (2001)
  • Blass, P.: Unirationality of enriques surfaces in characteristic two. Compos. Math. 45(3), 393–398 (1982)
  • Bogomolov, F., Hassett, B., Tschinkel, Y.: Constructing rational curves on k3 surfaces. Duke Math. J. 157(3), 535–550 (2011)
  • Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char. p, iii. Invent. Math. 35(1), 197–232 (1976)
  • Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char. p ii. Complex analysis and algebraic geometry, Collected Papers dediccated...
  • Buch, A., Thomsen, J.F., Lauritzen, N., Mehta, V.: The frobenius morphism on a toric variety. Tohoku Math. J. 49, 355–366 (1997)
  • Cossec, F.R., Dolgachev, I.: Enriques surfaces, vol. 1. Birkhäuser Boston (1989)
  • Crew, R.: Etale p-covers in characteristic p. Compos. Math. 52(1), 31–45 (1984)
  • Cynk, S., van Straten, D.: Small resolutions and non-liftable calabi-yau threefolds. Manuscr. Math. 130(2), 233–249 (2009)
  • Deligne, P., Illusie, L.: Relèvements modulop 2 et décomposition du complexe de de rham. Invent. Math. 43(2), 247–270 (1987)
  • Dupuy, T.: Positivity and lifts of the frobenius. preprint (2012)
  • Hahn, D.W., Miranda, R.: Quadruple covers of algebraic varieties. Ph.D. thesis, Colorado State University (1996)
  • Hartshorne, R.: Algebraic geometry, vol. 52. Springer (1977)
  • Illusie, L.: Frobenius and hodge degeneration. Introd. Hodge Theory 8, 99–149 (2002)
  • Illusie, L.: Grothendieck’s existence theorem in formal geometry with a letter of jean-pierre serre. Math. Surv. Monogr. 123, 179 (2005)
  • Kunz, E.: Kähler differentials, advanced lectures in mathematics, friedr. Vieweg, Braunschweig (1986)
  • Lang, W.E.: Quasi-elliptic surfaces in characteristic three. Ann. Sci. de l’École Normale Supérieure 12(4), 473–500 (1979)
  • Lange, H., Stuhler, U.: Vektorbündel auf kurven und darstellungen der algebraischen fundamentalgruppe. Math. Z. 156(1), 73–83 (1977)
  • Liedtke, C.: Algebraic surfaces in positive characteristic. Birational geometry, rational curves, and arithmetic. Springer, New York. pp....
  • Liedtke, C., Satriano, M.: On the birational nature of lifting. Adv. Math. 254, 118–137 (2014)
  • Mehta, V., Srinivas, V.: Varieties in positive characteristic with trivial tangent bundle. Compos. Math. 64(2), 191–212 (1987)
  • Miranda, R.: Triple covers in algebraic geometry. Am. J. Math. 107(5), 1123–1158 (1985)
  • Oda, T.: Vector bundles on an elliptic curve. Nagoya Math. J. 43, 41–72 (1971)
  • Paranjape, K.H., Srinivas, V.: Self maps of homogeneous spaces. Invent. Math. 98(2), 425–444 (1989)
  • Raynaud, M.: Around the mordell conjecture for function fields and a conjecture of serge lang in algebraic geometry. In: Proceedings of the...