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 surfaceX 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.
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