arXiv subjects

Ronan Terpereau

Publications and source records attributed to Ronan Terpereau.

21 records · Page 2Linked to original sources

arXiv2014

Invariant Hilbert schemes and desingularizations of quotients by classical groups

Let $W$ be a finite-dimensional representation of a reductive algebraic group $G$. The invariant Hilbert scheme $\mathcal{H}$ is a moduli space that classifies the $G$-stable closed subschemes $Z$ of $W$ such that the affine algebra $k[Z]$ is the direct sum of simple $G$-modules with prescribed multiplicities. In this article, we consider the case where $G$ is a classical group acting on a classical representation $W$ and $k[Z]$ is isomorphic to the regular representation of $G$ as a $G$-module. We obtain families of examples where $\mathcal{H}$ is a smooth variety, and thus for which the Hilbert-Chow morphism $γ: \mathcal{H} \rightarrow W//G$ is a canonical desingularization of the categorical quotient.

math.AG↗

arXiv2013

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$, and let $μ:\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $μ^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. For these classes of examples, $μ^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert-Chow morphism with the (well-known) symplectic desingularizations of $μ^{-1}(0)/\!/G$.

math.AG↗

arXiv2012

Schémas de Hilbert invariants et théorie classique des invariants

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 γ: 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 μ: W -> End(V). We study the invariant Hilbert scheme H' which parametrizes the subschemes included in μ^{-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 μ^{-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.

math.AG↗