Mathematics, Department of

 

Document Type

Article

Date of this Version

4-3-2012

Citation

Authors 2012

Comments

arXiv:1109.1881v2 [math.AG] 4 Apr 2012

Abstract

We study curves of negative self-intersection on algebraic surfaces. In contrast 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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