Global aspects of the geometry of surfaces

Abstract

These notes (prepared for the author's lectures at the Cracow Summer School on Linear Systems organized by S. Mueller-Stach and T. Szemberg, held March 23-27, 2009 at the Pedagogical University of Cracow under the sponsorship of the Deutsche Forschungsgemeinschaft) present a number of open problems on the theory of smooth projective algebraic surfaces, and put into historical context recent work on a range of topics, including Mori dream spaces and the finite generation of the Cox ring, Seshadri constants, and the resurgence of homogeneous ideals and the problem of which ordinary powers of homogeneous ideals contain given symbolic powers of those ideals. These notes include many exercises, with solutions.