Bisector fields and projective duality

TL;DR

This paper introduces bisector fields, arrangements of paired lines with midpoint crossing properties, and classifies them over various fields using algebraic and projective geometry tools.

Contribution

It provides a complete classification of bisector fields over real closed and algebraically closed fields, advancing understanding of their geometric and algebraic structure.

Findings

Complete classification over real closed fields

Complete classification over algebraically closed fields

Partial classification over finite fields

Abstract

Working over a field ${\mathbb{k}}$ of characteristic $\ne 2$, we study what we call bisector fields, which are arrangements of paired lines in the plane that have the property that each line in the arrangement crosses the paired lines in pairs of points that all share the same midpoint. To do so, we use tools from the theory of algebraic curves and projective duality. We obtain a complete classification if ${\mathbb{k}}$ is real closed or algebraically closed, and we obtain a partial classification if ${\mathbb{k}}$ is a finite field. A classification for other fields remains an open question. Ultimately this is a question regarding affine equivalence within a system of certain rational quartic curves.