Plane polar Cremona maps of arbitrarily large degree in positive characteristic

TL;DR

This paper extends the classification of homaloidal polynomials and line arrangements with birational polar maps to positive characteristic, surpassing the degree limitations known in characteristic zero.

Contribution

It introduces new examples of homaloidal polynomials of arbitrarily large degree in positive characteristic and classifies related line arrangements using combinatorial methods.

Findings

Existence of homaloidal polynomials of arbitrarily large degree in positive characteristic.

Classification of line arrangements with birational polar maps in positive characteristic.

Contrast with characteristic zero where degrees are bounded by three.

Abstract

A result of I.V.Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of degree at most three. In this note we describe homaloidal polynomials in three variables of arbitrarily large degree in positive characteristic. Using combinatorial arguments, we also classify line arrangements whose polar map is homaloidal in positive characteristic.