The Curves of Elliptic Nodes on K3 Surfaces

Let $(X,L)$ be a polarized K3 surface of genus $g$ and $C_{en} \subset X$ be the curve of singular points of nodal elliptic curves in $|L|$. When $(X,L)$ is generic of genus two, Huybrechts observed that the curve $C_{en}$ is a constant cycle curve and conjectured that this remains true for higher genus cases. In this note, we show that the conjecture holds true for polarized K3 surfaces $(X,L)$ lying in a locus of codimension one in the moduli space of polarized K3 surfaces of genus $g$ for every $g > 2$.