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Negative curves play a prominent role in the geometry of projective surfaces. They occur naturally as the irreducible components of exceptional loci of resolutions of surface singularities, at the same time, they are closely related to the geometry of the effective cone, and thus form an important building block of the Minimal Model Program. In the case of surfaces, classes of negative curves span extremal rays of the Mori cone. Any knowledge about them on a given surface reveals important information on linear series as well.
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Sandra Di Rocco, Alex Küronya, Stefan Müller-Stach, Tomasz Szemberg, Mini-Workshop: Negative Curves on Algebraic Surfaces. Oberwolfach Rep. 11 (2014), no. 1, pp. 555–593DOI 10.4171/OWR/2014/10