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 $g \geq 2$. 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 \cite{Chen02} has shown that all rational curves in $|L|$ are nodal, and the number of such rational curves is given by the Yau-Zaslow formula \cite{YZ96}. 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 $O(e^{C\sqrt{g}})$.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2020
- DOI:
- 10.48550/arXiv.2001.05104
- arXiv:
- arXiv:2001.05104
- Bibcode:
- 2020arXiv200105104C
- Keywords:
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- Mathematics - Algebraic Geometry;
- 14D23;
- 14H10;
- 14J28;
- 14N15
- E-Print:
- Expanded section 4 into two new sections, corrected a gap in the original Lemma 4.5 which is now Lemma 4.8, removed the appendix. Final version