Curves on Frobenius nonclassical loci of hypersurfaces

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