A generalization of the space of complete quadrics

TL;DR

This paper introduces a generalized construction of the space of complete quadrics associated with homogeneous polynomials, providing criteria for smoothness and exploring specific cases like elementary symmetric polynomials.

Contribution

It generalizes the space of complete quadrics to a broader class of polynomials and establishes conditions for smoothness, including connections to toric varieties.

Findings

$ Omega_h$ maps birationally onto the graph of the gradient map of $h$

$ Omega_h$ is smooth for elementary symmetric polynomials

Examples of non-smooth $ Omega_h$ are provided

Abstract

To any homogeneous polynomial $h$ we naturally associate a variety $\Omega_h$ which maps birationally onto the graph $\Gamma_h$ 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 $\Omega_h$ being smooth which applies for example when $h$ is an elementary symmetric polynomial. In this case $\Omega_h$ is a smooth toric variety associated to a certain generalized permutohedron. We also give examples when $\Omega_h$ is not smooth.