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Contents
The purpose of the mini course is to present the construction of the invariant Hilbert scheme. This object arises in the following situation: given a connected reductive group G acting on an affine scheme X, the invariant Hilbert scheme parametrizes G-stable closed subschemes of X affording a fixed, multiplicity-finite representation of G in their coordinate ring. It was first constructed in [Haiman and Sturmfels, 2004] in the case where G is a torus over a field of any characteristic. This work, quite elementary, provides another construction of Grothendieck's Hilbert scheme. Then, building on this work, the case of an arbitrary connected reductive group G over the complex numbers was obtained in [Alexeev and Brion, 2005]. Both constructions will be presented. The first one is longer, but no previous knowledge is necessary. The second is quite short; some very basic knowledge about affine algebraic groups is recommended. |
Schedule: all sessions are in room 4.35
Tuesday, 27 February: 11am - 12:30pm
Wednesday, 28 February: 11am -12:30pm and 2:30 - 4pm
Thursday, 1 March: 11am - 12:30pm
References
M. Haiman and B. Sturmfels, Multigraded Hilbert
Schemes, J.
Alg. Geom. 13 (2004), 725-769
V. Alexeev and M. Brion, Moduli of
affine schemes with reductive group action,
J.
Alg. Geom. 14 (2005), 83-117
S. Jansou, Deformations
of cones of primitive vectors, to appear in Math. Ann.
Sébastien Jansou is also speaking in the Geometria em Lisboa seminar on Tuesday, 27 February 2007 at 4pm in P3.10.
Departamento de
Matemática
Instituto Superior Técnico
Partially supported by FCT
through program POCTI/FEDER and
the Center
for Mathematical Analysis, Geometry, and Dynamical Systems.