Duality for certain multi-Frobenius nonclassical curves in higher dimensional spaces

TL;DR

This paper explores the duality properties of multi-Frobenius nonclassical curves over finite fields, revealing geometric insights and constructing nonreflexive space curves with specific Frobenius-related properties.

Contribution

It establishes a connection between multi-Frobenius nonclassicality and the geometry of dual curves, providing new constructions and generalizations in finite field algebraic geometry.

Findings

Characterization of intersection multiplicities of dual curves with hyperplanes

Construction of nonreflexive space curves with Frobenius tangent properties

Generalizations of existing results on nonclassical curves

Abstract

We show how a type of multi-Frobenius nonclassicality of a curve defined over a finite field $\mathbb{F}_q$ of characteristic $p$ reflects on the geometry of its strict dual curve. In particular, in such cases we may describe all the possible intersection multiplicities of its strict dual curve with the linear system of hyperplanes. Among other consequence, using a result by Homma, we are able to construct nonreflexive space curves such that their tangent surfaces are nonreflexive as well, and the image of a generic point by a Frobenius map is in its osculating hyperplane. We also obtain generalizations and improvements of some known results of the literature.