#### Vol. 304, No. 2, 2020

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The geometry of the flex locus of a hypersurface

### Laurent Busé, Carlos D’Andrea, Martín Sombra and Martin Weimann

Vol. 304 (2020), No. 2, 419–437
##### 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 in ${¶}^{3}$. 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.

##### Keywords
hypersurfaces, flex locus, multivariate resultants
Primary: 14J70
Secondary: 13P15
##### Milestones
Received: 29 July 2019
Revised: 27 August 2019
Accepted: 27 August 2019
Published: 12 February 2020
##### Authors
 Laurent Busé Université Côte d’Azur, INRIA Sophia Antipolis France Carlos D’Andrea Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain Martín Sombra Departament de Matemàtiques i Informàtica Universitat de Barcelona Barcelona Spain Institució Catalana de Recerca i Estudis Avançats (ICREA) Barcelona Spain Martin Weimann Laboratoire de mathématiques Nicolas Oresme, UMR CNRS 6139 University of Caen Caen France Laboratoire de mathématiques GAATI University of French Polynesia Faaa French Polynesia