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Bounding curves in algebraic surfaces by genus and Chern numbers. (English) Zbl 0870.14020
An increasing interest in the theory of algebraic surfaces \(S\) is devoted to the understanding of which curves of fixed geometric genus are contained in \(S\). The general philosophy suggests that when \(S\) is of general type, then it may contain only few curves of low genus; in this context, F. A. Bogomolov proved results on the boundedness of families of such curves.
In the present paper, the authors use the Miyaoka-Yau inequality for the log cotangent sheaf \(\Omega_S(\)log\(C\)) to obtain finer results on the genus of curves in \(S\); in particular, their bounds show that in a surface of general type, there are finitely many rational or elliptic curves with fixed number of ordinary singularities of multiplicity \(\leq 3\).

14H25 Arithmetic ground fields for curves
14J29 Surfaces of general type
57R20 Characteristic classes and numbers in differential topology
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