Rational quartic spectrahedra - Helsø, Martin; Ranestad, Kristian (Mathematica Scandinavica, 2021)
Rational quartic spectrahedra in 3-space are semialgebraic convex subsets in R ^3 of semidefinite, real symmetric (4 × 4)-matrices, whose boundary admits a rational parameterization. The Zariski closure in CP ^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 R 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.