A generalization of the space of complete quadrics

Abstract

To any homogeneous polynomial $h$ we naturally associate a variety $\hat{I}{\text{c◯}}_h$ which maps birationally onto the graph $\hat{I}{“}_h$ of the gradient map $\nabla h$ and which agrees with the space of complete quadrics when $h$ is the determinant of the generic symmetric matrix. We give a sufficient criterion for $\hat{I}{\text{c◯}}_h$ being smooth which applies for example when $h$ is an elementary symmetric polynomial. In this case $\hat{I}{\text{c◯}}_h$ is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when $\hat{I}{\text{c◯}}_h$ is not smooth.

Accepted for publication in Le Matematiche