The Geometry of the Artin-Schreier-Mumford Curves over an Algebraically Closed Field

TL;DR

This paper investigates the automorphism group and geometric properties of Artin-Schreier-Mumford curves over algebraically closed fields, revealing their automorphism group structure and embedding in projective space.

Contribution

It determines the automorphism group of ASM(q) over algebraically closed fields and describes its embedding in projective space, extending previous results from finite fields.

Findings

Automorphism group order is 2q^2(q-1).

Automorphism group is semidirect product Q⋊D_{q-1}.

Curve admits a nonsingular model in PG(3,𝕂) that is neither classical nor Frobenius classical.

Abstract

For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $Aut(\mathcal{X})$ of all automorphisms of $\mathcal{X}$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan. Furthermore, we show that $ASM(q)$ has a nonsingular model $\mathcal{Y}$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$.