Invariant Hilbert schemes and invariant deformation theory

Abstract. Let $G$ be an algebraic group acting on a vector space $W$. In this talk we are interested by the invariant Hilbert scheme $H=Hilb^G(W)$, which is the moduli space of the $G$-stable closed subschemes $X \subset W$ whose affine algebra $\mathbb{C}[X]$ is the direct sum of simple $G$-modules with previously fixed finite multiplicities. Many examples of such $H$ have been determined during the last fifteen years, most of them for $G$ a finite group or a torus. If $G$ is arbitrary and $H$ is singular, then it is generally very difficult to determine whether $H$ is reducible, reduced...

On the other hand, the deformation theory is an old and well-known field of the algebraic geometry, but the $G$-invariant version is quite recent and once again very few is know when $G$ is arbitrary.

The aim of this talk is to show the relation between these two topics, and to explain how the invariant deformation theory can be used to determine new examples of invariant Hilbert schemes. As an application, we will discuss some explicit examples where $G \subset GL_3$ is a classical group acting on a classical representation.