Nodal elliptic curves on K3 surfaces

Abstract

Let $(X,L)$ be a general primitively polarized K3 surface with $c_1(L{)}^2=2g-2$ for some integer $g\ge 2$. The Severi variety $V^{L,\delta }\subset |L|$ is defined to be the locus of reduced and irreducible curves in $|L|$ with exactly $\delta$ nodes and no other singularities. When $\delta =g$, any curve $C\in V^{L,g}$ is a rational curve; in fact, Chen \cite{Chen02} has shown that all rational curves in $|L|$ are nodal, and the number of such rational curves is given by the Yau-Zaslow formula \cite{YZ96}. In this paper, we consider the next case where $\delta =g-1$ and the Severi variety $V^{L,g-1}$ parametrizing nodal elliptic curves is of dimension 1. Let ${\overline{V}}^{L,g-1}\subset |L|$ denote the Zariski closure. For a reduced curve $C$, we define the geometric genus of $C$ to be the sum of the genera of the irreducible components of the normalization. We prove that the geometric genus of the closure ${\overline{V}}^{L,g-1}\subset |L|$ is bounded from below by $O(e^{C\sqrt{g}})$.

Expanded section 4 into two new sections, corrected a gap in the original Lemma 4.5 which is now Lemma 4.8, removed the appendix. Final version