Abstract
We study surfaces of general type S whose cotangent sheaf is generated by its global sections. We define a map called the cotangent map of S that enables us to understand the obstructions to the ampleness of the cotangent sheaf of S. These obstructions are curves on S that we call “non-ample”. We classify the surfaces with an infinite number of non-ample curves and we partly classify the non-ample curves.
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Barth, W., Hulek, K., Peters, C., Van De Ven, A.: Compact complex surfaces, Ergeb. Math. Grenzgeb. vol. 4, seconde édition augementée, Springer (2004)
Beauville A.: Annulation du H1 et systèmes paracanoniques sur les surfaces. J. Reine Angew. Math. 388, 149–157 (1988)
Conduche D., Palmeiri E.: On the Chern ratio for surfaces with ample cotangent bundle. Mathematische (Catania) 61(1), 143–156 (2006)
Debarre O.: Varieties with ample cotangent bundle. Compo. Math. 141(6), 1445–1459 (2005)
Gieseker D.: On a theorem of Bogomolov on Chern classes of stables bundle. Am. J. Math. 101, 77–85 (1979)
Griffiths P.: Variations on a theorem of Abel. Inv. Math. 35, 321–390 (1976)
Griffiths, P., Harris, J.: Principes of Algebraic Geometry. Wiley (1978)
Hartshorne, R.: Algebraic Geometry. Springer G.T.M. 52 (1977)
Hartshorne, R.: Ample vector bundles. Publi. Math. I.H.E.S. 319–350 (1966)
Lipman J.: Free derivation modules on algebraic varieties. Am. J. Math. t.87, 874–898 (1965)
Martin-Deschamps M.: Propriétés de descente des variétés à fibré cotangent ample. Ann. Inst. Fourier 33, 39–64 (1984)
Milne, J.: Lectures on algebraic geometry. www.math.sa.umich.edu/jmilne/
Mumford, D.: The Red Book of Varieties and Schemes. L.N.M. 1358, Springer (1999)
Ran Z.: The structure of Gauss-like maps. Comp. Math. 52, 171–177 (1984)
Serrano, F.: Isotrivial fibred surfaces. Annali di Math. Pura et Appli., 171 (1996)
Spurr M.: On the zero set of a holomorphic one-form on a compact complex manifold. Trans. Am. Soc. 308, 329–339 (1988)
Spurr M.: Branched coverings of surfaces with ample cotangent bundle. Pac. J. Math. 164, 129–146 (1994)
Szpiro, L. (éditeur).: Séminaire sur les pinceaux de courbes de genre au moins deux. Astérisque 86, SMF (1986)
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Roulleau, X. L’application cotangente des surfaces de type général. Geom Dedicata 142, 151–171 (2009). https://doi.org/10.1007/s10711-009-9364-3
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DOI: https://doi.org/10.1007/s10711-009-9364-3