  Let $(X,L)$ be a general primitively polarized K3 surface with $c_1(L)^2 =
2g-2$ for some integer $g \geq 2$. 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 $O(e^{C\sqrt{g}})$.
