Curves on Frobenius nonclassical loci of 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.