The 2-Hessian and sextactic points on plane algebraic curves

TL;DR

This paper corrects historical mathematical formulas related to 2-Hessian and sextactic points on plane algebraic curves, providing new formulas and insights especially for cuspidal and rational curves.

Contribution

The paper corrects Cayley's original calculations, offers a new formula for sextactic points on cuspidal curves, and explores the case of rational curves using the Wronski determinant.

Findings

Corrected the defining polynomial for the 2-Hessian.

Derived a formula for sextactic points on cuspidal curves.

Analyzed sextactic points on rational curves via Wronski determinant.

Abstract

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. In addition, we present 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 appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.