The curves of elliptic nodes on K3 surfaces

Abstract Let ( X , L ) be a polarized K3 surface of genus g and $$C_{en} \subset X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>en</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> 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 $$C_{en}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>en</mml:mi> </mml:mrow> </mml:msub> </mml:math> 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 $$g &gt; 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math>