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The cotangent map of surfaces of general type. (L’application cotangente des surfaces de type général.) (French. English summary) Zbl 1180.14041
The author considers complex surfaces of general type whose cotangent bundle \(\Omega_S\) is generated by global sections and whose irregularity \(q\) satisfies \(q>3\).
For such surfaces he studies the ampleness of \(\Omega_S\) by looking for curves \(C\hookrightarrow S\) such that \(\Omega_S\otimes \mathcal O_C\) has a quotient bundle isomorphic to \(\mathcal O_C\). Those curves are called non-ample curves and they are the obstruction to the ampleness of the contangent bundle by Gieseker’s criterion.
Moreover a curve \(C\) on \(S\) in non-ample if and only if there exists a section \(t:C \hookrightarrow\mathbb P(T_S)\) contracted to a point by the cotangent map of the surface \(S\). The point is then to study the image of the cotangent map and its degree. Surfaces with an infinity of non-ample curves are also described.

14J29 Surfaces of general type
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
Full Text: DOI arXiv
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