Negative curves on algebraic surfaces

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 nonisomorphic endomorphism has bounded negativity (i.e., that $C^2$ is bounded below for prime divisors $C$ on $X$). 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.