There is a recurrent theme in the theory of algebraic K3 surfaces. The projective geometry of K3 surfaces shows surprising analogies to the theory of linear systems on curves and to a somewhat lesser extent to the theory of line bundles on abelian varieties. This chapter explains the basic aspects of these analogies and in particular Saint-Donat's results on ample linear systems.

We start with a recap of some aspects of the classical theory for curves and state the Kodaira–Ramanujam vanishing theorem in Section 1. The typical features of linear systems on a K3 surface are directly accessible if the linear system is associated with a smooth curve contained in the K3 surface. So, we treat this case first; see Section 2. The general case is then studied in Section 3, where we also give a proof of the Kodaira–Ramanujam vanishing theorem. The last section contains existence results for primitively polarized K3 surfaces of arbitrary even degree.

As we shall not be interested in rationality questions in this section, we assume the ground field k to be algebraically closed. If not mentioned otherwise, its characteristic is arbitrary.

General Results: Linear Systems, Curves, Vanishing

We collect standard results on linear systems on curves and explain first the consequences for the geometry of linear systems on K3 surfaces.

1.1 Recall that with any line bundle L on a variety X one associates the complete linear system |L| which by definition is the projectivization of the space of global sections or, equivalently, the space of all effective divisors linearly equivalent to L. The base locus Bs|L| of |L| is the maximal closed subscheme of X contained in all.

If L has more than one section, i.e., then it induces the rational map

which is regular on the complement of Bs|L|.

For a surface X the base locus Bs|L| can have components of dimension zero and one. The fixed part of |L| is the one-dimensional part of Bs|L|, and we shall denote it by F. Then and the natural inclusion yields an isomorphism. In this sense, on can be identified with and the latter can be extended to a morphism defined on. Here, is the finite set of base points of, which contains the zero-dimensional locus of Bs|L|.