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.
By a result of Bogomolov and Zarhin, a K3 surface $X$ over a number field $L$ has ordinary reduction at a density 1 set of primes after passing to a finite extension of $L$. In this paper, we refine this result for non-CM K3 surfaces whose transcendental Hodge structure has endomorphism field $F$ abelian over $\mathbb{...
Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every action of a finite $p$-group on $X$ factors through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer...
Let $\cX$ be a curve of genus $g\ge2$ and zero $p$-rank over an algebraically closed field of odd characteristic $p$. We classify the pairs $(\cX,G)$ for which $G\le\Aut(\cX)$ has no common fixed point and $|G|>24g(g-1)$. For $g\ge4$, the curves are explicit cyclic covers of the projective line, the Hermitian curve, or...
Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Kl\'asz...
Let $G$ be the topological fundamental group of a nonsingular complex projective surface. Troncoso and Urz\'ua proved that the Chern slopes $c_1^2/c_2$ of minimal surfaces of general type $S$ with $\pi_1(S)\simeq G$ are dense in $[1,3]$, and left $[1/3,1)$ open. We prove that they are dense in $[1/2,3]$, an interval th...
If $G$ is a connected reductive algebraic group over a field $k$ of characteristic $\ell$, we show that both the Lie-theoretic and group-theoretic commuting schemes associated to $G$ are normal and Cohen--Macaulay when $\ell = 0$, or $\ell$ is larger than the Coxeter numbers of all simple factors of $G$, and that $\ell...
Jack Sempliner, Xiang-Qian Yang· 1 citation
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