Schémas de Hilbert invariants et théorie classique des invariants

Abstract : Let W be an affine variety equipped with an action of a reductive group G. The invariant Hilbert scheme is a moduli space which classifies the G-stable closed subschemes of W such that the affine algebra is the direct sum of simple G-modules with previously fixed finite multiplicities. In this thesis, we first study the invariant Hilbert scheme, denoted H. It parametrizes the GL(V)-stable closed subschemes Z of W=n1 V \oplus n2 V^* such that k[Z] is isomorphic to the regular representation of GL(V) as GL(V)-module. If dim(V)<3, we show that H is a smooth variety, so that the Hilbert-Chow morphism \gamma: H -> W//G is a resolution of singularities of the quotient W//G. However, if dim(V)=3, we show that H is singular. When dim(V)<3, we describe H by equations and also as the total space of a homogeneous vector bundle over the product of two Grassmannians. Then we consider the symplectic setting by letting n1=n2 and replacing W by the zero fiber of the moment map \mu: W -> End(V). We study the invariant Hilbert scheme H' which parametrizes the subschemes included in \mu^{-1}(0). We show that H' is always reducible, but that its main component Hp' is smooth if dim(V)<3. In this case, the Hilbert-Chow morphism is a resolution of singularities (sometimes a symplectic one) of the quotient \mu^{-1}(0)//G. When dim(V)=3, we describe Hp' as the total space of a homogeneous vector bundle over a flag variety. Finally, we get similar results when we replace GL(V) by some other classical group (SL(V), SO(V), O(V), Sp(V)) acting first on W=nV, then on the zero fiber of the moment map.
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Submitted on : Wednesday, November 7, 2012 - 8:05:09 AM
Last modification on : Wednesday, July 18, 2018 - 8:08:03 AM
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  • HAL Id : hal-00749160, version 1
  • ARXIV : 1211.1472

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Ronan Terpereau. Schémas de Hilbert invariants et théorie classique des invariants. Géométrie algébrique [math.AG]. Université Joseph Fourier - Grenoble, 2012. Français. ⟨hal-00749160⟩

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