正特征下广义Kummer曲面的分类
The classification of generalised Kummer surfaces in positive characteristic
AI总结：
该研究在Katsura与Rybakov工作基础上，完成了正特征下作用于阿贝尔曲面且商消解为K3曲面的群分类，证明超奇异阿贝尔曲面相关商无法为广义Kummer曲面，并构造了具体例子。
AI中文摘要：
在Katsura与Rybakov前期工作的基础上，我们完成了所有在特征p=2、3、5下，满足p整除群G的阶|G|的群作用分类：这类群G以保持群律的自同构作用在阿贝尔曲面A上，使得商空间A/G的消解为K3曲面。为此，我们研究了p整除|G|时的群作用，利用正特征下的有理二重点理论，证明A/G可能的ADE奇点类型受限于其局部基本群需包含G作为子群的要求，并通过G在A的ℓ进Tate模上的作用确定A/G的奇异轨迹。作为分类的关键步骤，我们证明若A为超奇异阿贝尔曲面且p整除|G|，则A/G永远不可能是广义Kummer曲面。最后，我们构造了这类广义Kummer曲面作为两条椭圆曲线乘积的商的具体例子。
英文摘要：
Building on earlier work of Katsura and Rybakov, we complete the classification, in every characteristic, of all groups $G$ acting on an abelian surface $A$ by automorphisms preserving the group law such that the resolution of the quotient $A/G$ is a K3 surface. In order to do so, we study actions of groups with $p\mid|G|$ in characteristics $p=2,3$ and $5$. Using the theory of rational double points in positive characteristic, we show that the possible ADE singularity types of $A/G$ are constrained by the requirement that their local fundamental group contains $G$ as a subgroup, and we determine the singular locus of $A/G$ via the action of $G$ on the $\ell$-adic Tate module of $A$. As a key step in the classification, we prove that if $A$ is a supersingular abelian surface and $p\mid|G|$, then $A/G$ can never be a generalised Kummer surface. Finally, we construct explicit examples of such generalised Kummer surfaces as quotients of products of two elliptic curves.
