Related papers: Inseparable Kummer surfaces

The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by…

We introduce Kummer surfaces X=Km(CxC) with the group scheme G=mu_2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational…

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian…

We construct, on a supersingular K3 surface with Artin invariant 1 in characteristic 2, a set of 21 disjoint smooth rational curves and another set of 21 disjoint smooth rational curves such that each curve in one set intersects exactly 5…

We discuss K3 surfaces in characteristic two that contain the Kummer configuration formed by smooth rational curves on it.

We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint…

We discuss several geometric features of a Kummer surface associated with a (1,2)-polarized abelian surface defined over the field of complex numbers. In particular, we show that any such Kummer surface can be modeled as the double cover of…

A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the…

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such…

The aim of this paper is to describe the geometry of the generic Kummer surface associated to a $(1,2)$-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo…

In this paper we first show that each Kummer quartic surface (a quartic surface $X$ with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution $\gamma$ on the…

We describe the possible 3-divisible $A_2^n$ configurations of smooth rational curves on K3 surfaces in characteristic 3 and fully classify the resulting triple covers.

We present three interesting projective models of the supersingular K3 surface X in characteristic 5 with Artin invariant 1. For each projective model, we determine smooth rational curves on X with the minimal degree and the projective…

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $\mathcal{C}$, then $X$ is a Kummer surface $X=Km(B)$…

We show that K3 surfaces in characteristic 2 can admit sets of $n$ disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each $n=8,12,16,20$. More precisely, all values occur on supersingular K3 surfaces, with…

We calculate the automorphism group of the Kummer surface associated with a curve of genus 2 or the product of two elliptic curves in characteristic two under the assumption that the Kummer surface is a $K3$ surface. Moreover we discuss the…

We study a family of lattice polarized $K3$ surfaces which is an extension of the family of Kummer surfaces derived from principally polarized Abelian surfaces. Our family has two special properties. First, it is coming from a resolution of…

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field $K$ of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be…

We construct Zariski K3 surfaces of Artin invariant 1, 2 and 3 in many characteristics. In particular, we prove that any supersingular Kummer surface is Zariski if the characteristic is not congruent to 1 modulo 12. Our methods combine…

In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them…