Negative curves on algebraic surfaces

Authors

Thomas Bauer, Philipps-Universit¨at MarburgFollow
Brian Harbourne, University of Nebraska-LincolnFollow
Andreas Leopold Knutsen, University of BergenFollow
Alex Kuronya, University of Technology and Economics & Albert-Ludwigs-Universit¨at FreiburgFollow
Stefan Muller-Stach, Johannes Gutenberg- UniversitatFollow
Xavier Roulleau, Instituto Superior TecnicoFollow
Tomasz Szemberg, InstytutMatematyki UPFollow

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