at the

American Institute of Mathematics, San Jose, California

organized by

Robin Hartshorne, Diane Maclagan, and Gregory G. Smith

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*of^{d}(A^{n})*d*points in affine*n*-space, - determine if the Hilbert scheme
*H*of locally Cohen-Macaulay curves of degree_{d,g}*d*and genus*g*in projective 3-space is connected, - describe the irreducible components of multigraded Hilbert schemes.

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

by Cristina Bertone, Francesca Cioffi, Paolo Lella, and Margherita Roggero, *J. Symbolic Comput. 50 (2013), 263-290 * MR2996879

A Borel open covering of Hilbert schemes

by Cristina Bertone, Paolo Lella, and Margherita Roggero, *J. Symbolic Comput. 53 (2013), 119-135 * MR3027986

The Hilbert schemes of locally Cohen-Macaulay curves in P^3 may after all be connected

by Paolo Lella and Enrico Schlesinger, *Collect. Math. 64 (2013), no. 3, 363-372 * MR3084402

Gröbner strata in the Hilbert scheme of points

by Mathias Lederer, *Proc. Lond. Math. Soc. (3) 108 (2014), no. 1, 187-224 * MR3162825

Components of Gröbner strata in the Hilbert scheme of points

by Mathias Lederer

An explicit non-smoothable component of the compactified Jacobian

by Jesse Leo Kass

Moduli of Generalized Line Bundles on a Ribbon

by Dawei Chen and Jesse Leo Kass, *J. Pure Appl. Algebra 220 (2016), no. 2, 822-844 * MR3399392

Detaching embedded points

by Dawei Chen and Scott Nollet, *Algebra Number Theory 6 (2012), no. 4, 731-756 * MR2966717