Section: New Results

Resultants, flexes, and the generalization of Salmon's formula

Participant : Laurent Busé.

Given an algebraic variety $S\subset {\mathbb{P}}^n$ and a point $p\in S$, the osculation order of the point $p$ is the maximum of the multiplicity of intersection at $p$ of $S$ with any line through $p$. We denote it by ${\mu }_p$ and define $Flex\left(S\right)=\{p\in {\mathbb{P}}^n|{\mu }_p>n\}$.

If $n=2$, it is known that if $C$ is a plane algebraic curve of degree $d$ then $Flex\left(C\right)$ is the intersection of $C$ with its Hessian, this latter being of degree $3d-6$. A famous generalization of this result to the case $n=3$ has been obtained by Salmon in 1860: for a general variety $S$, $Flex\left(S\right)$ is the intersection of $S$ with another hypersurface of degree $11d-24$. In this work, we are studying the generalization of this formula to arbitrary dimension $n$. We proved that given $S\subset {\mathbb{P}}^n$ of degree $d$, $Flex\left(S\right)$ is obtained by intersecting $S$ with another hypersurface of degree

We are also looking for an explicit expression of an equation of this latter hypersurface.

This is a work in progress which is done in the context of a PICS collaboration funded by CNRS. It is a joint work with M. Chardin (University Paris 6), C. D'Andrea (University of Barcelona), M. Sombra (University of Barcelona) and M. Weiman (University of Caen).