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 with a surjective nonisomorphic endomorphism has bounded negativity (i.e., that is bounded below for prime divisors on ). We prove the same statement for Shimura curves on quaternionic Shimura surfaces of Hilbert modular type. As a byproduct, we obtain that there exist only finitely many smooth Shimura curves on such a surface. We also show that any set of curves of bounded genus on a smooth complex projective surface must have bounded negativity.
Thomas Bauer. Brian Harbourne. Andreas Leopold Knutsen. Alex Küronya. Stefan Müller-Stach. Xavier Roulleau. Tomasz Szemberg. "Negative curves on algebraic surfaces." Duke Math. J. 162 (10) 1877 - 1894, 15 July 2013. https://doi.org/10.1215/00127094-2335368