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Section: New Results

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

Participant : Laurent Busé.

Given an algebraic variety Sn and a point pS, 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 μp and define Flex(S)={pn|μp>n}.

If n=2, it is known that if C is a plane algebraic curve of degree d then Flex(C) 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(S) 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 Sn of degree d, Flex(S) is obtained by intersecting S with another hypersurface of degree

d k = 1 n n ! k - n !

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).