#
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:

- characterize the smoothable component of the Hilbert scheme
*Hilb*^{d}(A^{n}) of *d* points in affine *n*-space,
- 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,
- 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.