Kummer surfaces and quadratic line complexes in characteristic two

TL;DR

This paper explores the classical relationship between quadratic line complexes and Kummer surfaces, focusing on how this theory adapts or changes in characteristic two, which differs from the classical case in other characteristics.

Contribution

It provides an analogue of the classical theory of quadratic line complexes and Kummer surfaces specifically in characteristic two, where the classical relationships do not directly apply.

Findings

Classical theory holds in characteristics not equal to two.

The paper establishes a new analogue of the theory in characteristic two.

Differences in the behavior of Kummer surfaces in characteristic two are characterized.

Abstract

In this paper, we study the classical theory of quadratic line complexes and Kummer surfaces. A quadratic line complex is the intersection of the Grassmannian $G(2,4)$ and a quadric hypersurface in ${\bf P}^5$, and a Kummer surface is the quotient of the Jacobian of a curve of genus 2 by the inversion. F. Klein discovered a relationship between a quadratic line complex and a curve of genus 2, its Jacobian and the associated Kummer surface. This theory holds in any characteristic not equal to two. However the situation in characteristic two is entirely different. The purpose of this paper is to give an analogue in characteristic 2 of this classical theory.