Components of Hilbert Schemes

Original Announcement

Hilbert schemes, introduced by Grothendieck over fifty years ago, have become the fundamental parameter spaces in algebraic geometry. They provide a natural setting for deformation theory and play a key role in the construction of moduli spaces. Despite their importance, many basic geometric properties of Hilbert schemes remain a mystery.

Pathological examples show that Hilbert schemes can have numerous irreducible components, complicated non-reduced structures, and arbitrarily bad singularities. On the other hand, Hilbert schemes parametrizing subschemes of projective space are always connected and Hilbert schemes of points on a smooth surface are smooth and irreducible. The broad aim of this workshop is to explore the significant gap between the well-understood Hilbert schemes and the pathologies.

Working towards this goal, we will focus on the three specific problems:

1. characterize the smoothable component of the Hilbert scheme Hilb^d(Aⁿ) of d points in affine n-space,
2. determine if the Hilbert scheme H_d,g of locally Cohen-Macaulay curves of degree d and genus g in projective 3-space is connected,
3. describe the irreducible components of multigraded Hilbert schemes.

Material from the workshop

The workshop schedule.

A report on the workshop activities.