Abstract for the talk on 12.10.2023 (15:00 h)
Seminar on Nonlinear AlgebraDante Luber (TU Berlin)
Singular matroid realization spaces
12.10.2023, 15:00 h, MPI für Mathematik in den Naturwissenschaften Leipzig, G3 10 (Lecture hall)
A matroid is realizable if we can obtain its bases from the indices of linearly independent columns of some matrix. For a given matroid \(M\), this matrix is not unique. The space of all such matrices can be given the structure of an affine scheme, known as the realization space of \(M\). It is known that representation spaces of matroids can be arbitrarily singular, although there are few concrete examples. We use software to study smoothness and irreducibility of representation spaces of rank 3 and rank 4 matroids, isolating examples of singular spaces for \((3,12)\)-matroids. As an application, we show that singular initial degenerations exist for the \((3,12)\)-Grassmannian.