Symmetroids and quartic spectrahedra

Cynthia Vinzant
University of Michigan

Intersecting the convex cone of positive semidefinite 4 by 4 matrices with a three-dimensional affine space yields a quartic spectrahedron. It is bounded by a complex quartic hypersurface called a symmetroid. Recently, Degtyarev and Itenberg used the global Torelli Theorem for real K3 surfaces to describe the position of nodes (that is, rank-two matrices) on the real surface and spectrahedron. I will discuss this result in the context of classical algebraic geometry. In particular we will see the special structure of these surfaces and spectrahedra by looking at their projection from a node, as done classically by Cayley. This is joint work with John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels.

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