## Nathan Ilten

Hilbert schemes provide a useful tool for moduli problems, but are difficult to explicitly describe in most situations. In my talk, I will discuss a specific example, namely the Hilbert scheme of degree 12 Fano threefolds. Among its many irreducible components, there are four special components which correspond to different families of smooth Fano threefolds. I will describe the geometry of these special components and their intersection behavior. Motivation coming from mirror symmetry for studying this particular Hilbert scheme will also be discussed. This project is joint work with J. Christophersen.

## 5:00PM: Determinantal facet ideals

Fatemeh We consider the ideals generated by general sets of m-minors of an $m\times n$-matrix of indeterminates. The generators are identiﬁed with the facets of an $(m−1)$-dimensional pure simplicial complex. The ideal generated by the minors corresponding to the facets of such a complex is called a determinantal facet ideal. Given a pure simplicial complex $\Delta$, we discuss the question when the generating minors of its determinantal facet ideal $J_\Delta$ form a Gr\"obner basis and when $J_\Delta$ is a prime ideal. (This is joint work with V. Ene, J. Herzog and T. Hibi)