The geometry of the flex locus of a hypersurface

TL;DR

This paper provides a formula for the flex locus of a hypersurface using resultants, computes its dimension and degree bounds, and analyzes the generic case where the flex line is unique with expected contact order.

Contribution

It generalizes Salmon's classical result to higher dimensions and offers explicit formulas and bounds for the flex locus of hypersurfaces.

Findings

Derived a formula for the flex locus using multidimensional resultants.

Computed the dimension and degree bounds of the flex locus.

Showed that for generic hypersurfaces, the flex line is unique with expected contact order.

Abstract

We give a formula in terms of multidimensional resultants for an equation for the flex locus of a projective hypersurface, generalizing a classical result of Salmon for surfaces. Using this formula, we compute the dimension of this flex locus, and an upper bound for the degree of its defining equations. We also show that, when the hypersurface is generic, this bound is reached, and that the generic flex line is unique and has the expected order of contact with the hypersurface.