| We will discuss various questions about existence of curves with negative self-intersections on complex projective surfaces. In particular, we will present a recent result that shows the existence of smooth complex projective surfaces containing reduced, irreducible curves $C$ of arbitrarily negative self-intersection $C^2$. Previously the only known examples of surfaces for which $C^2$ was not bounded below were in positive characteristic, and the Bounded Negativity Conjecture, dating back to Federigo Enriques, predicted that no examples could arise over the complex numbers. Our examples are special types of Hilbert modular surfaces. This is joint work with Th. Bauer, B. Harbourne, A. Kuronya, S. Muller-Stach, and T. Szemberg. |