Skip to main content Accessibility help

On algebraic surfaces of general type with negative $c_{2}$

  • Yi Gu (a1) (a2)


We prove that for any prime number $p\geqslant 3$ , there exists a positive number $\unicode[STIX]{x1D705}_{p}$ such that $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})\geqslant \unicode[STIX]{x1D705}_{p}c_{1}^{2}$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$ . In particular, $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})>0$ . This answers a question of Shepherd-Barron when $p\geqslant 3$ .



Hide All
[Bad01] Badescu, L., Algebraic surfaces, Universitext (Springer, New York, 2001).
[BBT95] Ballico, E., Bertolini, M. and Turrini, C., Projective varieties with degenerate dual variety in char. p , Rend. Semin. Mat. Univ. Politec. Torino 53 (1995), 1318.
[BHPV04] Barth, W., Hulek, K., Peters, C. and Van De Ven, A., Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 4 (Springer, Berlin, 2004).
[Bea79] Beauville, A., L’application canonique pour les surfaces de type général , Invent. Math. 55 (1979), 121140.
[Che14] Chen, X., Xiao’s conjecture on canonically fibered surfaces, Preprint (2014), arXiv:1405.5236v1.
[CD89] Cossec, F. and Dolgachev, I., Enriques surfaces I, Progress in Mathematics, vol. 76 (Birkhäuser, Boston, MA, 1989).
[Eke88] Ekedahl, T., Canonical models of surfaces of general type in positive characteristic , Publ. Math. Inst. Hautes Études Sci. 67 (1988), 97144.
[Igu60] Igusa, J., Betti and Picard numbers of abstract algebraic surfaces , Proc. Natl. Acad. Sci. USA 46 (1960), 724726.
[IS96] Iskoviskikh, V. A. and Shafarevich, I. R., Algebraic surfaces , in Algebraic geometry II, Encyclopaedia of Mathematical Sciences, vol. 35 (Springer, Berlin, 1996).
[Jou83] Jouanolou, J.-P., Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42 (Birkhäuser, Boston, MA, 1983).
[Lie08a] Liedtke, C., Algebraic surfaces of general type with small c 1 2 in positive characteristic , Nagoya Math. J. 191 (2008), 111134.
[Lie08b] Liedtke, C., Uniruled surfaces of general type , Math. Z. 259 (2008), 775797.
[Lie13] Liedtke, C., Algebraic surfaces in positive characteristic , in Birational geometry, rational curves, and arithmetic (Springer, New York, 2013), 229292.
[Liu02] Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford Science Publications–Oxford University Press, Oxford, 2002).
[LLR04] Liu, Q., Lorenzini, D. and Raynaud, M., Néron models, Lie algebras, and reduction of curves of genus one , Invent. Math. 157 (2004), 455518.
[Miy77] Miyaoka, Y., On the Chern numbers of surfaces of general type , Invent. Math. 42 (1977), 225237.
[Per87] Persson, U., An introduction to the geography of surfaces of general type , in Proceedings of symposia in pure mathematics, vol. 46, part 1 (American Mathematical Society, Providence, RI, 1987), 195218.
[Ray78] Raynaud, M., Contre-exemple au ‘vanishing theorem’ en caractéristique p > 0 , in C. P. Ramanujam—a tribute, Tata Institute of Fundamental Research Studies in Mathematics, vol. 8 (Springer, Berlin–New York, 1978), 273278.
[Sal11] Salomão, R., Fibrations by nonsmooth genus three curves in characteristic three , J. Pure Appl. Algebra 215 (2011), 19671979.
[Sch09] Schröer, S., On genus change in algebraic curves over imperfect fields , Proc. Amer. Math. Soc. 137 (2009), 12391243.
[SGA5] Grothendieck, A. et al. , Cohomologie l-adique et fonctions L , in Séminaire de géométrie algébrique du Bois-Marie 1965–1966 (SGA 5), ed. L. Illusie, Lecture Notes in Mathematics vol. 589 (Springer, Berlin–New York, 1977).
[She91a] Shepherd-Barron, N. I., Unstable vector bundles and linear systems on surfaces in characteristic p , Invent. Math. 106 (1991), 243262.
[She91b] Shepherd-Barron, N. I., Geography for surfaces of general type in positive characteristic , Invent. Math. 106 (1991), 263274.
[Shi74] Shioda, T., An example of unirational surfaces in characteristic p , Math. Ann. 211 (1974), 233236.
[Szp79] Szpiro, L., Sur le théorème de rigidité de Parsin et Arakelov , in Journées de géométrie algébrique de Rennes (Rennes 1978), Vol. II, Astérisque, vol. 64 (Société Mathématique de France, Paris, 1979), 169202.
[Szp81] Szpiro, L., Propriétés numériques du dualisant relatif , in Séminaire sur les pinceaux de courbes de genre au moins deux, Astérisque, vol. 86 (Société Mathématique de France, Paris, 1981), 4477.
[Tat52] Tate, J., Genus change in inseparable extensions of function fields , Proc. Amer. Math. Soc. 3 (1952), 400406.
[Yau77] Yau, S., Calabi’s conjecture and some new results in algebraic geometry , Proc. Natl. Acad. Sci. USA 74 (1977), 17981799.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

On algebraic surfaces of general type with negative $c_{2}$

  • Yi Gu (a1) (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed