Nodal elliptic curves on K3 surfaces.

The purpose of this note is to study the (total) geometric genus of the compactified Severi curve $\overline{V}^{L,g-1}$ consisting of nodal elliptic curves on a general primitively polarized K3 surface $(X,L)$ of genus $g$. We prove that the genus is bounded below by $O(e^{C\sqrt{g}})$ and conjecture an asymptotic of $O(e^{Cg})$.