UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

October 16, 2007

939 Evans Hall

3:45PM: The philosophy of stacks

Brian Osserman

After a brief and hand-wavy introduction to moduli functors and moduli stacks, I'll attempt to address the following philosophical question: if we're really only interested in isomorphism classes of objects for a given moduli problem, how do we know that any particular stack structure we consider is the ``right'' one? Could there be another way of thinking about the problem that gives the same isomorphism classes, but different automorphism groups? I'll discuss by way of concrete examples work of Max Lieblich and myself which in some cases provides a negative answer to the latter question. Put differently, we show that stacks can often be completely recovered from the (a priori) coarser data of the associated functor of isomorphism classes of objects.

5:00PM: The Hilbert scheme of 8 points in 4 space

Dustin Cartwright

It has been known for some time that the Hilbert scheme of n points in affine d space has multiple components for large n and d at least 3. However, the asymptotic arguments used do not give an explicit description of the components. By different arguments, the Hilbert scheme of 8 points in 4 space can be shown to have two components, whose structure I will describe. Hopefully, an understanding of this case will provide a model for more general Hilbert schemes of points. This is a description of work in progress.

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