On Inflection Points of Plane Curves.

Abstract

Curves are a well-known subject in algebraic geometry. In particular this thesis will treat curves in the projective plane. Not only the real or complex projective plane, as generally studied in topology, but even the projective plane over a field of positive characteristic. This is an even less intuitive space, nonetheless it entails compelling consequences. In particular we will investigate some specific points called \textit{inflection points} on our plane curves. These points will be closely examined in relation to the Hessian. The Hessian can be considered as a multivariate generalization of the second derivative. The Hessian is not exclusively considered in algebraic geometry, a basic calculus course is often the first place to get acquainted with this notion. However, we consider it in a formal way.