Birational embeddings of the Hermitian, Suzuki and Ree curves with two Galois points

TL;DR

This paper constructs plane curves with two Galois points whose smooth models are Hermitian, Suzuki, and Ree curves, revealing new geometric properties of these classical algebraic curves.

Contribution

It demonstrates the existence of plane curves with two Galois points corresponding to Hermitian, Suzuki, and Ree curves, extending understanding of their embeddings.

Findings

Existence of degree $q^3+1$ plane curves with two Galois points

Smooth models are the Hermitian, Suzuki, and Ree curves

Results hold over fields with characteristic $p>0$

Abstract

We show that there exists a plane curve of degree $q^3+1$ with two inner Galois points whose smooth model is the Hermitian curve of degree $q+1$, where $q$ is a power of the characteristic $p>0$. Similar results hold for the Suzuki and Ree curves respectively.