UNC Algebraic Geometry Seminar
Tuesday, October 17, 2000, 4:00pm, Phillip Hall 330 (UNC)
Jonathan Wahl/Mirel Caibar/Shrawan Kumar (University of North Carolina)
(At UNC) Jet schemes of canonical singularities and a theorem of B. Kostant
Abstract:- We are interested in a recent preprint by M. Mustata (AG/0008002),
with an appendix by David Eisenbud and Eduard Frenkel. The main
result is the proof of a conjecture of Eisenbud and Frenkel: for a
local complete intersection variety X, then all the "jet schemes" are
irreducible if and only if X has canonical singularities. The
appendix is a generalization of a theorem of Kostant for a complex simple Lie algebra: the ring of functions is free over the ring of
invariant functions.
The appeal of the subject to us is the increased visibility of
jet-schemes, or spaces of arcs, in recent years. For a smooth
variety, Kontsevich introduced motivic measures on the space of jets;
the resulting "motivic integration" has been very influential, and was
the subject of our seminar last Spring. (In fact, this technique is
used in the present proof). It was John Nash's old idea to study
singular varieties using jet schemes; and we view this paper as a good
way to get comfortable with some of the basic results.
The first talk will be an introduction, including the basic
definitions and properties of jet schemes of order m. Then in later
talks we'll talk about canonical and log-canonical singularities, and
introduce the main results. We'll explain how motivic integration
will be used in the proof. Finally, Shrawan will explain the
importance of Kostant's theorem, and we'll present the generalization
of Eisenbud-Frenkel.
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