The curves of elliptic nodes on K3 surfaces

Let ( X , L ) be a polarized K3 surface of genus g and Cₑₙ ⊂ X C en ⊂ X be the curve of singular points of nodal elliptic curves in | L |. When ( X , L ) is generic of genus two, Huybrechts proved that the curve Cₑₙ 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 $$g > 2$$ g > 2