论文标题
双盖和扩展
Double covers and extensions
论文作者
论文摘要
在本文中，我们考虑了与种类扩展的问题相关的投影空间的双重覆盖，特别是规范曲线扩展到$ k3 $表面和Fano 3倍。特别是我们考虑$ k3 $的表面，这是在一般六边形上分支的飞机的双层盖：我们证明，线性系统中的一般曲线拉回了$ k \ geq 7 $的平面曲线，位于独特的$ k3 $表面上。如果$ k \ leq 6 $，则一般曲线可扩展到更高的尺寸品种。在$ k = 4,5,6 $的情况下，这给出了尺寸8、5、3和属17、26、37的单数指数$ k $ fano品种的存在。对于$ k = 6 $，我们恢复了fano品种$ \ mathbf {p}（3，1，1，1，1）$，这是Prokhorov发现的最大属37的只有两个带有典型的Gorenstein奇异性的Fano三倍之一。我们表明，后者不能进一步扩展。对于$ k = 4 $，$ 5 $这些狂热品种已被Totaro确定。我们还研究了第3属$ k3 $表面的平滑学位2部分的扩展。在所有这些情况下，我们计算了所考虑的曲线的高斯的共同层。最后，我们观察到，射影平面双层盖上的线性系统提供了超生的对数Severi品种。
In this paper we consider double covers of the projective space in relation with the problem of extensions of varieties, specifically of extensions of canonical curves to $K3$ surfaces and Fano 3-folds. In particular we consider $K3$ surfaces which are double covers of the plane branched over a general sextic: we prove that the general curve in the linear system pull back of plane curves of degree $k\geq 7$ lies on a unique $K3$ surface. If $k\leq 6$ the general such curve is instead extendable to a higher dimensional variety. In the cases $k=4,5,6$, this gives the existence of singular index $k$ Fano varieties of dimensions 8, 5, 3, and genera 17, 26, 37 respectively. For $k = 6$ we recover the Fano variety $\mathbf{P}(3, 1, 1, 1)$, one of only two Fano threefolds with canonical Gorenstein singularities with the maximal genus 37, found by Prokhorov. We show that the latter variety is no further extendable. For $k=4$ and $5$ these Fano varieties have been identified by Totaro. We also study the extensions of smooth degree 2 sections of $K3$ surfaces of genus 3. In all these cases, we compute the co-rank of the Gauss--Wahl maps of the curves under consideration. Finally we observe that linear systems on double covers of the projective plane provide superabundant logarithmic Severi varieties.
