Counting rational curves on K3 surfaces

Abstract

The aim of these notes is to explain the remarkable formula found by Yau and Zaslow [Y-Z] to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families (Fg)g≥1 ; a surface in Fg admits a gdimensional linear system of curves of genus g . A naive count of constants suggests that such a system will contain a positive number, say n(g) , of rational (highly singular) curves. The formula is∑