4. Two existence theorems

Abstract

Given integers $g \geq 2$ and $0 \leq c \leq \lfloor \frac{g-1}{2} \rfloor$, one may ask whether there actually exists a pair (S,L), where S is a K3 surface, L ² = 2g – 2 and all smooth curves in |L| have Clifford index c.

Theorem 4.1 below gives a positive answer to this question. Theorem 4.4 below answers the same kind of question concerning the possible gonalities of a curve on a K3 surface.

The results in this chapter were first given in [Kn3]. We also include the material here, to obtain a complete exposition.