Curves on a Surface

Abstract

In this book, unless otherwise specified, by surface we shall always mean a connected compact complex manifold of complex dimension 2 which is a holomorphic submanifold of ℙ^N for some N. Thus, “surface” is short for smooth (connected) complex algebraic surface. By Chow’s theorem, a surface is also described as the zero set in ℙ^N of a finite number of homogeneous polynomials in N + 1 variables. The study of surfaces is concerned both with the intrinsic geometry of the surface and with the geometry of the possible embeddings of the surface in ℙ^N. Just as with curves, we could organize this study in order of increasing complexity. In terms of the extrinsic (synthetic) geometry of a surface in ℙ^N, we could for instance try to study and eventually classify surfaces in ℙ^N of relatively small degree. Or we could attempt to order surfaces by complexity via some intrinsic invariants, by analogy with the genus of a curve. This is the aim of the Kodaira classification, which orders surfaces by their Kodaira dimension. For this scheme, we have a fairly complete understanding of surfaces except in the case of Kodaira dimension 2, general type surfaces. We will cover the broad outlines of the general theory of surfaces. In this chapter, we will discuss the basic invariants, intersection theory and Riemann-Roch, and the structure of the set of ample divisors. In Chapter 3, we will discuss birational geometry.