A generalization of Bring's curve in any characteristic

TL;DR

This paper generalizes Bring's curve to higher dimensions over any field, analyzing its geometric properties, automorphism group, and special cases in positive characteristic, revealing connections to maximal curves and classical conjectures.

Contribution

It introduces a new family of algebraic varieties generalizing Bring's curve, characterizes their geometric and automorphism properties, and explores their behavior in positive characteristic, including maximality and point counts.

Findings

V is a projective, absolutely irreducible, non-singular curve with degree (m-2)!

Automorphism group of V is isomorphic to the symmetric group S_m

For m=5, V can be an F_{p^2}-maximal curve of genus 4

Abstract

Let $p\ge 7$ be a prime, and $m\ge 5$ an integer. A natural generalization of Bring's curve valid over any field $\mathbb{K}$ of zero characteristic or positive characteristic $p$, is the algebraic variety $V$ of $\textrm{PG}(m-1,\mathbb{K})$ which is the complete intersection of the projective algebraic hypersurfaces of homogeneous equations $x_1^k+\cdots +x_m^{k}=0$ with $1\leq k\leq m-2$. In positive characteristic, we also assume $m\le p-1$. Up to a change of coordinates in $\textrm{PG}(m-1,\mathbb{K})$, we show that $V$ is a projective, absolutely irreducible, non-singular curve of $\textrm{PG}(m-2,\mathbb{K})$ with degree $(m-2)!$, genus $\mathfrak{g}= \frac{1}{4} ((m-2)(m-3)-4)(m-2)!+1$, and tame automorphism group $G$ isomorphic to $\textrm{Sym}_m$. We compute the genera of the quotient curves of $V$ with respect to the stabilizers of one or more coordinates under the action of $G$. In positive characteristic, the two extremal cases, $m=5$ and $m=p-1$ are investigated further. For $m=5$, we show that there exist infinitely many primes $p$ such that $V$ is $\mathbb{F}_{p^2}$-maximal curve of genus $4$. The smallest such primes are $29,59,149,239,839$. For $m=p-1$ we prove that $V$ has as many as $(p-2)!$ points over $\mathbb{F}_p$ and has no further points over $\mathbb{F}_{p^2}$. We also point out a connection with previous work of R\'edei about the famous Minkowski conjecture proven by Haj\'os (1941), as well as with a more recent result of Rodr\'iguez Villegas, Voloch and Zagier (2001) on plane curves attaining the St\"ohr-Voloch bound, and the regular sequence problem for systems of diagonal equations introduced by Conca, Krattenthaler and Watanabe (2009).