辛映射与van Geemen-Sarti映射在K3曲面中的特殊性
Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces
AI总结：
本文研究了K3曲面上辛映射与van Geemen-Sarti映射的特殊性质，探讨了其Néron-Severi群的等距性及上同调作用。
摘要：
给定一个K3曲面X上的辛映射ι，X/ι的脱 singular 化Y仍然是一个K3曲面，通常具有不同的Néron-Severi群。然而，如果这个映射是由椭圆纤维上的2- torsion截面的平移诱导的（即是一个van Geemen-Sarti映射），并且Picard数是最小的，那么X和Y的Néron-Severi群是等距的。我们首先确定了无限多个二维子族的射影K3曲面，这些曲面具有辛映射（不是van Geemen-Sarti类型），其一般成员满足NS(X)≈NS(Y)。然后，我们描述了van Geemen-Sarti映射的上同调作用，并且我们刻画了具有van Geemen-Sarti映射的K3曲面的特殊性，使得NS(X)≈NS(Y)仍然成立。存在一个5维的K3曲面家族，其中X≈Y。此类K3曲面允许复乘法，我们描述了其上同调作用。我们简要讨论了由椭圆纤维上的3-torsion截面的平移诱导的阶3辛自同构的类似问题。
英文摘要：
Given a symplectic involution $ι$ on a K3 surface $X$, the desingularization $Y$ of $X/ι$ is still a K3 surface, which in general has a different Néron--Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen--Sarti involution) and the Picard number is minimal, the Néron--Severi groups of $X$ and $Y$ are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen--Sarti type) whose generic members satisfy $NS(X)\simeq NS(Y)$. Then, we describe the cohomological action of a van Geemen--Sarti involution and we characterize specializations of K3 surfaces with a van Geemen--Sarti involution for which it is still true that $NS(X)\simeq NS(Y)$. There is a 5-dimensional family of K3 surfaces with van Geemen--Sarti involution for which $X\simeq Y$. The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration.
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