Nested Punctual Hilbert Schemes and Commuting Varieties of Parabolic Subalgebras

Michael Bulois (1) , Laurent Evain (2)

1 AGL - Algèbre, géométrie, logique
2 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of parabolic subalgebras of $M_n(\K)$ and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the numerical conditions under which they are irreducible. In some reducible cases, we describe the irreducible components and their dimension.

Mots clés: Hilbert scheme; Commuting variety; parabolic algebra; GIT

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]

Michael Bulois, Laurent Evain. Nested Punctual Hilbert Schemes and Commuting Varieties of Parabolic Subalgebras. Journal of Lie Theory, 2016, 26 (2), pp.497--533. ⟨hal-00835853v1⟩