Commutative Algebra and Algebraic Geometry: Families of ideals of the same size

Seminar | January 31 | 3:10-4 p.m. | 939 Evans Hall

 Irena Peeva, Cornell University

 Department of Mathematics

The Hilbert function is an invariant that measures the size of a homogeneous ideal. It encodes important information (for example, dimension and degree). Grothendieck introduced the Hilbert scheme that parametrizes subschemes of $P^r$ with a fixed Hilbert polynomial. The main general result about the structure of Hilbert schemes is Hartshorne’s Theorem that the Hilbert scheme is connected. The proof of this landmark theorem shows that if two ideals in a polynomial ring have the same Hilbert function then they are connected by a sequence of deformations.

Why only polynomial rings? The theorem of Kruskal-Katona and related combinatorial results suggest that connectedness might hold in other situations, for example, for ideals in an exterior algebra. I will explain why Hartshorne's construction and its variants cannot be extended to cover such cases. My work with Murai and Stillman provides a different proof of the classical theorem that also proves the connectedness of the Hilbert scheme in new cases, including those of an exterior algebra and a Veronese ring.