Minimal Complex Projective Surfaces

Abstract

We give an exposition of the classification of minimal models of smooth complex projective surfaces, with an emphasis on the interplay between complex analytic and complex algebraic methods. We begin with an introduction to the intersection form for complex algebraic surfaces, and show that the topological intersection form on four-manifolds agrees with the algebraic intersection form on complex projective surfaces. We then discuss the structure of birational maps of smooth surfaces, where we show every birational map can be built out particularly simple birational maps called blowups. This allows us to make precise the notion of a minimal model of a surface S, a "simplest" representative of the birational equivalence class of S, and we work to characterize the structure of these minimal models and determine when a surface has a unique minimal model. In the case of irrational ruled surfaces, we show that every minimal model is a geometrically ruled surface, and in the case of rational surfaces, we show that all minimal models are either projective space or a Hirzebruch surface. When S is not ruled or rational, we show that there is a unique minimal model in its birational equivalence class. Finally, we interpret these classification results through the lens of the Minimal Model Program and give a brief discussion on the birational classification problem in higher dimensions.