# 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 to 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.
The cotangent sheaf is ample if and only if there do not exist fibers in 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 meets the cotangent map at a point. We call such a curve a non-ample curve.
We classify non-ample curves according to their self-intersection.
We then proceed to a classification of surfaces possessing an infinite number of non-ample curves.
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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Contributor : Anne-Marie Plé <>
Submitted on : Thursday, December 11, 2008 - 4:39:54 PM
Last modification on : Monday, March 9, 2020 - 6:15:52 PM
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### Identifiers

• HAL Id : tel-00346502, version 1

### Citation

Xavier Roulleau. L'application cotangente des surfaces de type général. Mathématiques [math]. Université d'Angers, 2007. Français. ⟨tel-00346502⟩

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