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arXiv · 2011.13830

A generalization of the space of complete quadrics

Abeer Al Ahmadieh·Mario Kummer·Miruna-Stefana Sorea
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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.

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Abeer Al Ahmadieh, Mario Kummer, Miruna-Stefana Sorea. 2020-11-27. A generalization of the space of complete quadrics. https://doi.org/10.4418/2021.76.2.9

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