Rational Quartic Spectrahedra

Abstract

Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in R3 of semidefinite, real symmetric (4×4)-matrices, whose boundary admits a rational parameterization. The Zariski closure in CP3 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 RP3 of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the missing example does not occur.