Determinantal quartic surfaces with a definite Hermitian representation

TL;DR

This paper investigates the singularities of determinantal quartic surfaces in three-dimensional space, providing bounds, examples, and conjectures related to their real singularity configurations and Hermitian representations.

Contribution

It offers new bounds on singularities, examples of real singularity configurations, and extends existing theorems concerning Hermitian determinantal representations.

Findings

Bound on the number of isolated, essential singularities.

Examples of real singularity configurations on quartic surfaces.

Conjecture extending a theorem by Degtyarev and Itenberg.

Abstract

We give a bound on the number of isolated, essential singularities of determinantal quartic surfaces in 3-space. We also provide examples of different configurations of real singularities on quartic surfaces with a definite Hermitian determinantal representation, and conjecture an extension of a theorem by Degtyarev and Itenberg.