Properties of Hesse derivatives of cubic curves

Abstract

The Hesse curve or Hesse derivative Hess$(\Gamma_f)$ of a cubic curve $\Gamma_{f}$ given by a homogeneous polynomial $f$ is the set of points $P$ such that $\det \left(H_f (P)\right)=0$, where $H_f (P)$ is the Hesse matrix of $f$ evaluated at $P$. Also Hess$(\Gamma_f)$ is again a cubic curve. We show that for a point $P\in$Hess$(\Gamma_{f})$, all the contact points of tangents from $P$ to the curves $\Gamma_{f}$ and Hess$(\Gamma_{f})$ are intersection points of two straight lines $\ell_1^P$ and $\ell_2^P$ (meeting on Hess$(\Gamma_{f})$) with $\Gamma_{f}$ and Hess$(\Gamma_{f})$, where the product of $\ell_1^P$ and $\ell_2^P$ is the polar conic of $\Gamma_{f}$ at $P$. The operator Hess defines an iterative discrete dynamical system on the set of the cubic curves. We identify the two fixed points of this system, investigate orbits that end in the fixed points, and discuss the closed orbits of the dynamical system.