Supersingular reduction of Kummer surfaces in residue characteristic $2$

TL;DR

This paper proves that if an abelian surface over a discrete valuation field has supersingular reduction in characteristic 2, then its associated Kummer surface also has good reduction, confirming a specific case of a broader question.

Contribution

It establishes that supersingular reduction of abelian surfaces in characteristic 2 guarantees good reduction of the associated Kummer surfaces.

Findings

Supersingular reduction implies good reduction for Kummer surfaces in characteristic 2.

Provides an affirmative answer to a specific case of the reduction problem.

Enhances understanding of reduction behavior in algebraic geometry.

Abstract

Given an abelian surface $A$, defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\mathrm{Km}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$.