Negative curves on algebraic surfaces

Bauer, Th. and Harbourne, B. and Küronya, Alex and Knutsen, A. L. and Müller-Stach, S. and Roulleau, x. and Szemberg, T. (2013) Negative curves on algebraic surfaces. DUKE MATHEMATICAL JOURNAL, 162 (10). pp. 1877-1894. ISSN 0012-7094

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Abstract

We study curves of negative self-intersection on algebraic surfaces. In con- trast to what occurs in positive characteristics, it turns out that any smooth complex projective surface X with a surjective non-isomorphic endomorphism has bounded negativity (i.e., that C2 is bounded below for prime divisors C on X). We prove the same statement for Shimura curves on Hilbert modular surfaces. As a byproduct we obtain that there exist only finitely many smooth Shimura curves on a given Hilbert modular surface. We also show that any set of curves of bounded genus on a smooth complex projective surface must have bounded negativity.

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