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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 $$\leq 3$$.

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