Ranks and singularities of cubic surfaces
Abstract
We explore the connection between the rank of a polynomial and the singularities of its vanishing locus. We first describe the singularity of generic polynomials of fixed rank. We then focus on cubic surfaces. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a cubic surface with singularities that are not ordinary double points lies on the Hessian discriminant, and that the Hessian discriminant is the closure of the rank six cubic surfaces.
Downloads
Published
Issue
Section
License
The authors retain all rights to the original work without any restrictions.
License for Published Contents
"Le Matematiche" published articlesa are distribuited with Creative Commons Attribution 4.0 International. You are free to copy, distribute and transmit the work, and to adapt the work. You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
License for Metadata
"Le Matematiche" published articles metadata are dedicated to the public domain by waiving all publisher's rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
No Fee Charging
No fee is required to complete the submission/review/publishing process of authors paper.
