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Quartic spectrahedra in 3-space form a semialgebraic set of dimension 24. This set is stratified by the location of the ten nodes of the corresponding real quartic surface. 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.
Mathematics Subject Classification14P10 14P25 52B55 90C22
Bernd Sturmfels was supported by the NSF (DMS-0968882) and the Max-Planck Institute für Mathematik in Bonn. Cynthia Vinzant was supported by an NSF postdoc (DMS-1204447).
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