Nodal elliptic curves on K3 surfaces

2020/01/15 by Chen, Nathan, Greer, François, Yang, Ruijie
#14D23 #14H10 #14J28 #14N15 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.05104

Abstract

Let (X,L) be a general primitively polarized K3 surface with c₁(L)² = 2g-2 for some integer g ≥ 2. The Severi variety V^L,δ ⊂ |L| is defined to be the locus of reduced and irreducible curves in |L| with exactly δ nodes and no other singularities. When δ=g, any curve C ∈ V^L,g is a rational curve; in fact, Chen \citeChen02 has shown that all rational curves in |L| are nodal, and the number of such rational curves is given by the Yau-Zaslow formula \citeYZ96. In this paper, we consider the next case where δ= g-1 and the Severi variety V^L,g-1 parametrizing nodal elliptic curves is of dimension 1. Let V^L,g-1 ⊂ |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 V^L,g-1 ⊂ |L| is bounded from below by O(e^C√(g)).

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