Components of Hilbert Schemes

This workshop, sponsored by AIM and the NSF, 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 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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