论文标题
K3表面上的节点椭圆曲线
Nodal elliptic curves on K3 surfaces
论文作者
论文摘要
令$（x，l）$为普通的原始偏光K3表面，对于某些整数$ g \ geq 2 $，$ c_1（l）^2 = 2G-2 $。 Severi品种$ v^{l，δ} \ subset | l | $被定义为$ | l | $的减少和不可约曲线的轨迹，恰好是$Δ$节点，没有其他奇异性。当$δ= g $时，v^{l，g} $中的任何曲线$ c \ in是一个理性曲线；实际上，Chen \ cite {Chen02}表明，$ | l | $中的所有理性曲线都是节点，并且这种理性曲线的数量由Yau-Zaslow公式\ cite {yz96}给出。 在本文中，我们考虑了下一个情况，即$δ= g-1 $和severi品种$ v^{l，g-1} $参数化nodal椭圆曲线是尺寸1。令$ \ overline {v}^{l，g-1}^{l，g-1} \ subset | l | $表示zariski封闭。对于减少的曲线$ c $，我们将$ c $的几何属定义为标准化不可约组件的属的总和。我们证明，闭合$ \ overline {v}^{l，g-1} \ subset | l | $的几何属由$ o（e^{c \ sqrt {g}}）$从下面界定。
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 $V^{L,δ} \subset |L|$ 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 $C \in V^{L,g}$ 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 $δ= g-1$ 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 $O(e^{C\sqrt{g}})$.
