论文标题
$ k3 $曲线带索引$ k> 1 $
$K3$ curves with index $k>1$
论文作者
论文摘要
令$ \ mathcal {kc} _g ^k $为$ s $ a $ k3 $ surface和$ c \ subset s $ a属$ g $ g $ g $ g $ k $ in $ k $ in $ \ mathrm {pic}（s）$。在本文中，我们研究了健忘的地图$ c_g ^k：（s，c）\ mapsto c $ co $来自$ \ mathcal {kc} _g ^k $ to $ \ mathcal {m} _g $ for $ k> 1 $。首先，我们通过几何计算是指其一般纤维的尺寸。仅当$ s $是完整的交叉点或mukai品种的一部分时，这才是有趣的。在前一种情况下，我们发现存在有趣的Fano品种，该品种在其规范嵌入中扩展了$ c $。在后一种情况下，这与Mukai品种的微妙模块化特性有关。接下来，我们调查$ c_g^k $是否以$ \ Mathcal {m} _g $ of $ k $ -spin曲线为单位，并具有适当数量的独立部分。我们只有当$ s $是一个完整的交叉点时才能够执行此操作，并且在这些情况下获得了一些分类结果。
Let $\mathcal{KC}_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface and $C\subset S$ a genus $g$ curve with divisibility $k$ in $\mathrm{Pic}(S)$. In this article we study the forgetful map $c_g^k:(S,C) \mapsto C$ from $\mathcal{KC}_g ^k$ to $\mathcal{M}_g$ for $k>1$. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when $S$ is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending $C$ in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $c_g^k$ dominates the locus in $\mathcal{M}_g$ of $k$-spin curves with the appropriate number of independent sections. We are able to do this only when $S$ is a complete intersection, and obtain in these cases some classification results for spin curves.
