Nodal elliptic curves on K3 surfaces

Abstract

Let (X, L) be a general primitively polarized K3 surface with \(c_1(L)^2 = 2g-2\) for some integer \(g \ge 2\). 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 (Math Ann 324:71–104, 2002) has shown that all rational curves in \(|L|\) are nodal, and the number of such rational curves is given by the Yau-Zaslow formula [20]. 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 \(O(e^{C\sqrt{g}})\).