Curves on Frobenius nonclassical loci of hypersurfaces

TL;DR

This paper studies Frobenius nonclassical loci of hypersurfaces over finite fields, showing how certain curves with inseparable Gauss maps are Frobenius nonclassical, and characterizing the inseparability of Gauss maps in these contexts.

Contribution

It establishes a connection between inseparable Gauss maps and Frobenius nonclassical curves on hypersurfaces, providing new methods to construct such curves and analyzing their properties.

Findings

Curves with inseparable Gauss maps are Frobenius nonclassical.

Frobenius nonclassical hypersurfaces with separated variables have inseparable Gauss maps.

Gauss maps given by p powers are purely inseparable in Frobenius nonclassical hypersurfaces.

Abstract

Let $\mathcal{S} \subset \mathbb{P}^n$ be an absolutely irreducible projective hypersurface defined over a finite field $\mathbb{F}_q$, equipped with the $\mathbb{F}_q$-Frobenius map $\Phi_q$. In this paper, we investigate irreducible curves $\mathcal{X} \subset \mathcal{S}_{\Phi_q}$, where $\mathcal{S}_{\Phi_q}$ is the $\mathbb{F}_q$-Frobenius nonclassical locus of $\mathcal{S}$. In particular, we show that every curve $\mathcal{X} \subset \mathcal{S}_{\Phi_q}$ such that the restriction of the Gauss map of $\mathcal{S}$ to $\mathcal{X}$ is inseparable is $\mathbb{F}_q$-Frobenius nonclassical. This provides a way to construct new Frobenius nonclassical curves, which are curves that tend to have many $\mathbb{F}_q$-rational points. We also prove that a certain type of Frobenius nonclassical hypersurfaces $\mathcal{S}$ defined by separated variables are such that their Gauss maps restricted to any curve contained in $\mathcal{S}$ is inseparable. Finally, in parallel with the plane curve cases, we show that if the strict Gauss map $\Gamma$ of a $\mathbb{F}_q$-Frobenius nonclassical hypersurface $\mathcal{S}$ is given by $p$ powers, then $\Gamma$ is purely inseparable.