Components of Hilbert Schemes

July 19 to July 23, 2010

at the

American Institute of Mathematics, Palo Alto, California

organized by

Robin Hartshorne, Diane Maclagan, and Gregory G. Smith

Original Announcement

This workshop will be devoted to understanding the irreducible component structure of Hilbert schemes.

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 Hilbd(An) of d points in affine n-space,
  2. determine if the Hilbert scheme Hd,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

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:
Upgraded methods for the effective computation of marked schemes on a strongly stable ideal
Borel open covering of Hilbert schemes
The Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after all be connected
Gr�bner strata in the Hilbert scheme of points
Components of Gr�bner strata in the Hilbert scheme of points
An explicit non-smoothable component of the compactified Jacobian
Moduli of Generalized Line Bundles on a Ribbon
Detaching embedded points