The 2-Hessian and sextactic points on plane algebraic curves

Paul Aleksander Maugesten, Torgunn Karoline Moe

Published 2017-09-06Version 1

In an article from 1865 Arthur Cayley claims that given a plane algebraic curve there exists an associated 2-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the 2-Hessian of a plane curve. In addition, we provide a formula for the number of sextactic points on cuspidal curves and tie this formula to the 2-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points can be interpreted as the zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.