Abstract
Simple questions in linear algebra often give rise to interesting objects in algebraic geometry. For example, the space of singular $n \times n$ matrices is the classical hypersurface given by the vanishing of the $n\times n$ determinant. This hypersurface has interesting geometric properties. In particular, it contains many linear subspaces, whose study is related to (among other things) geometric complexity theory.
In this talk, I will introduce a geometric object, called a Fano scheme, parameterizing these linear subspaces. We'll look at some basic examples of Fano schemes, and then see a characterization of exactly when our Fano schemes are connected. Motivated by complexity theory, we'll also compare linear spaces contained in the hypersurface cut out by the determinant to those contained in the hypersurface cut out by the permanent. This is joint work with Melody Chan.