A generalization of Bring's curve in any characteristic

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 $\text{PG}(m-1,\mathbb{K})$ which is the complete intersection of the projective algebraic hypersurfaces of homogeneous equations $x_1^k+\cdots +x_m^k=0$ with $1\le k\le m-2$. In positive characteristic, we also assume $m\le p-1$. Up to a change of coordinates in $\text{PG}(m-1,\mathbb{K})$, we show that $V$ is a projective, absolutely irreducible, non-singular curve of $\text{PG}(m-2,\mathbb{K})$ 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 ${\text{Sym}}_m$. 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 Rédei about the famous Minkowski conjecture proven by Hajós (1941), as well as with a more recent result of Rodríguez Villegas, Voloch and Zagier (2001) on plane curves attaining the Stöhr-Voloch bound, and the regular sequence problem for systems of diagonal equations introduced by Conca, Krattenthaler and Watanabe (2009).