Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Abstract

For each integer $s\ge 1$, we present a family of curves that are ${\mathbb{F}}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case $s=2$, we give necessary and sufficient conditions for such curves to be ${\mathbb{F}}_q$-Frobenius nonclassical with respect to the linear system of conics. In the ${\mathbb{F}}_q$-Frobenius nonclassical cases, we determine the exact number of ${\mathbb{F}}_q$-rational points. In the remaining cases, an upper bound for the number of ${\mathbb{F}}_q$-rational points will follow from Stöhr-Voloch theory.