Singularities of the dual curve of a certain plane curve in positive characteristic

Abstract

It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree $q^2 + q + 1$ for which the Gauss map is inseparable with inseparable degree $q$. As a special case, we show a relation between the dual curve of the Fermat curve of degree $q^2 + q + 1$ and the Ballico-Hefez curve.