### UW Algebra Seminar

Abstracts

**Speaker:** Dan Nakano, University of Georgia

**Title:**
Support varieties for Weyl modules

**Date:** July 2, 2004

**Abstract:**
Support varieties were developed around 25 years ago in
pioneering work of Alperin and Carlson. The varieties allow one
to provide a bridge between the representation and cohomology
theory for restricted Lie algebras. In my talk I will present
recent progress on connecting support varieties for Weyl modules
with the classical theory of nilpotent orbits.

**Speaker:** Koen De Naeghel,
Limburgs Universitair Centrum, Belgium

**Title: **
Hilbert schemes of points on quantum projective planes

**Date:** August 11, 2004

**Abstract:**
In algebraic geometry subschemes of dimension zero and degree n on
P^2 are parameterized by the Hilbert scheme of points
Hilb_n(P^2). Set-theoretically such a
subscheme corresponds to n points in the plane. We replace
P^2 by noncommutative deformations called quantum projective
planes P^2_q. By definition this is a noncommutative
projective scheme which has as coordinate ring a Koszul three dimensional
Artin-Schelter regular algebra A. The Hilbert scheme of points
Hilb_n(P^2_q) for such a noncommutative
plane was recently constructed by Nevins and Stafford. Its objects are
graded, rank one, torsion-free A-modules up to shift of grading. In general
there appear, in stark contrast to the commutative case, reflexive objects
which form an open subset of this Hilbert scheme. We give the possible
Hilbert series and minimal resolutions of the (reflexive) objects of
Hilb_n(P^2_q).

**Speaker:** Koen De Naeghel,
Limburgs Universitair Centrum, Belgium

**Title: **
On the incidence between strata of the Hilbert scheme of points on
the projective plane

**Date:** August 18, 2004

**Abstract:**
The Hilbert scheme of points on P^2 has a natural
stratification given by the Hilbert series of the corresponding ideal
sheaves. This stratification is related to the properties of linear systems
on P^2. Unfortunately the precise inclusion relation between the
closures of the strata is unknown. Under a technical condition this problem
was recently solved by Guerimand in the special case where the Hilbert
series of the strata are ``as close as possible'', i.e. when there is no
intermediate Hilbert series which is numerically possible. We give a new
proof of Guerimand's result based on deformation theory. In our approach
the technical condition is not necessary and furthermore we are able to
treat other incidence
problems as well. Our new proof was found while investigating the
corresponding noncommutative problem, that is, for Hilbert schemes of points
on generic quantum projective planes. Alhough the research for this
noncommutative problem is still in progress,
we compare its solution with the commutative case.

**Speaker:**
Pramathanath Sastry,
Department of Mathematics, University of Toronto

**Title: **
Residues via the Deligne-Verdier approach

**Date:** September 16, 2004

**Abstract:**
The Deligne Verdier approach to duality, while elegant,
has the drawback that explict formulae for traces (or residues)
are hard to deduce from it, even for projective space. Recent
work by the speaker on non-flat base change for Cohen-Macaulay
maps makes it possible to get around this problem, and I plan to
speak a little on the strategy involved in getting an explicit
hold of residues (or equivalently--traces).

*To request disability accommodations, contact the Office of the ADA
Coordinator, ten days in advance of the event or as soon as possible:
543-6450 (voice); 543-6452 (TDD); 685-3885 (FAX); access@u.washington.edu (E-mail).
*

Back to Algebra Seminar