Irreducible components of the equivariant punctual Hilbert schemes

Abstract

Let H ab be the equivariant Hilbert scheme parametrizing the 0-dimensional subschemes of the affine plane invariant under the natural action of the one-dimensional torus T ab :We compute the irreducible components of H ab : they are in one-one correspondence with a set of Hilbert functions. As a by-product of the proof, we give new proofs of results by Ellingsrud and Strømme, namely the main lemma of the computation of the Betti numbers of the Hilbert scheme H l parametrizing the 0-dimensional subschemes of the affine plane of length l [4], and a description of Bialynicki-Birula cells on H l by means of explicit flat families [5]. In particular, we precise conditions of applications of this last description.

A Simple Proof of the Formula for the Betti Numbers of the Quasihomogeneous Hilbert Schemes: Fig. 1.

“…Formula (5) follows from Statement 2 in [8] and equation (6) follows from Statement 3 in [8]. Also for any s ≥ 1 we have (see [8])…”

“…The quasihomogeneous Hilbert scheme (C 2 ) [n] T α,β is compact and in general has many irreducible components. They were described in [6]. In the case α = 1 the Poincare polynomials of the irreducible components were computed in [3].…”

A Bijective Proof of Loehr-Warrington's Formulas for the Statistics $${\rm{ctot}_\frac{q}{p}}$$ ctot q p and $${\rm{mid}_\frac{q}{p}}$$ mid q p

“…Irreducible components of Hilb n p,q (C 2 ) were studied by Evain in [3]. He showed that two monomial ideals I D and I D ′ belong to the same connected component of Hilb n p,q (C 2 ) if and only if the Young diagrams D and D ′ have the same weighted content, i.e.…”

A syzygetic approach to the smoothability of zero-dimensional schemes

“…[16,17,13,19,23,12,6]). For embedding dimension greater than two, very little is known about how to answer this question.…”