Quartic Spectrahedra

TL;DR

This paper provides a new, self-contained, and algorithmic proof for classifying quartic spectrahedra in 3-space, advancing the understanding of their geometric structure and stratification.

Contribution

It extends Cayley's characterization of quartic symmetroids and offers a new proof for the stratification of quartic spectrahedra, aiding in their classification.

Findings

Identified 20 maximal strata of quartic spectrahedra

Extended Cayley's characterization for quartic symmetroids

Provided a self-contained, algorithmic proof for stratification

Abstract

Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of its ten nodes. There are twenty maximal strata, identified recently by Degtyarev and Itenberg, via the global Torelli Theorem for real K3 surfaces. We here give a new proof that is self-contained and algorithmic. This involves extending Cayley's characterization of quartic symmetroids, by the property that the branch locus of the projection from a node consists of two cubic curves. This paper represents a first step towards the classification of all spectrahedra of a given degree and dimension.