# L'application cotangente des surfaces de type général

Abstract : We study here surfaces of general type where the cotangent sheaf is generated by his global sections and with an irregularity $$q$$ at least equal to $$4$$.\\ Our approach and the object of this study is the cotangent map that is a morphism of the projectivized cotangent sheaf to the projective space of dimension $$q-1$$. We study the degree of this morphism and the degree of its image. {\selectlanguage{english}} The cotangent sheaf is ample if and only if there do not exist fibres of the cotangent map of strictly positive dimension.\\ If the cotangent sheaf is not ample, then there exists a curve $$C$$ in the surface and there exists a section of $$C$$ in the projectivized cotangent sheaf that is mapped to a point by the cotangent map. We call such a curve a non-ample curve.\\ We classify non-ample curves according to their self-intersection and then proceed to a classification of surfaces possessing an infinite number of non-ample curves. {\selectlanguage{english}} The Fano surfaces provide an example where the cotangent map normally applies. We study the ramification divisor of such surfaces and their non-ample curves.\\ The Fano surface of the Fermat's cubic possesses $$30$$ non-ample curves and we describe their properties in detail.
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https://tel.archives-ouvertes.fr/tel-00271770
Contributor : Xavier Roulleau <>
Submitted on : Thursday, April 10, 2008 - 11:35:23 AM
Last modification on : Monday, August 20, 2018 - 9:44:02 AM
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• HAL Id : tel-00271770, version 1

### Citation

Xavier Roulleau. L'application cotangente des surfaces de type général. Mathématiques [math]. Université d'Angers, 2007. Français. ⟨tel-00271770⟩

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