Seminar in Real and Complex Geometry

Thursday, July 15, 2021, 16:00-17:30, online (via zoom)

Khazhgali Kozhasov
(Universität Osnabrück)

Nodes on quintic spectrahedra


Given generic real symmetric matrices A, B, C of size n x n, it is of interest to study the set S of positive-semidefinite matrices of the form Id + x A + y B + z C, where x, y, z are some real numbers. The set S is a closed convex set in R^3, called a spectrahedron. The Zariski closure of the Eucllidean boundary of S is an algebraic surface {(x,y,z): det(Id+ x A+y B+ z C)=0}, which turns out to be always singular. A natural question in real algebraic geometry is to understand (for a fixed n) possible restrictions on the numbers P, Q of real singular points, respectively, of those singularities that lie on S. In my talk I will discuss this problem for quintic spectrahedra (n=5) and present a complete classification of pairs (P,Q), obtained in a joint work with Taylor Brysiewicz and Mario Kummer.