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 q² + 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 q² + q + 1 and the Ballico-Hefez curve.