Rational Quartic Spectrahedra

Abstract

Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in ${\mathbb{R}}^3$ of semidefinite, real symmetric $(4\times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in $\mathbb{C}{\mathbb{P}}^3$ of the boundary of a rational spectrahedron is a rational complex symmetroid. We give necessary conditions on the configurations of singularities of the corresponding real symmetroids in $\mathbb{R}{\mathbb{P}}^3$ of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the missing example does not occur.

Updated version to be published in Mathematica Scandinavica. Substantially rewritten, with strengthened results. 19 pages, 8 figures