The Classes of the Quasihomogeneous Hilbert Schemes of Points on the Plane

Abstract

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasi-projective varieties) of irreducible components of (1, k)-quasi-homogeneous Hilbert schemes of points on the plane. We find a new simple geometric interpretation of the q, t-Catalan numbers. Finally, we investigate a connection between (1, k)-quasi-homogeneous Hilbert schemes and homogeneous nested Hilbert schemes.is smooth and parameterizes quasi-homogeneous ideals of colength n 2010 Mathematics Subject Classification. 14C05, 05A17.

Compactified Jacobians and q,t-Catalan numbers, I

“…Therefore, we can merge the sequences in a unique way. By repeating this procedure inductively, we can reconstruct the remainders of all a i modulo n. The family of varieties M p,q,h for (p, q) = (n, kn + 1) was considered by A. Buryak (Theorem 1.5 in [5]). By the construction, for all h the varieties M p,q,h are smooth subvarieties of Hilb h (C 2 ).…”

“…In Section 3 we explore a relation between our combinatorial description of the Piontkowski's cell decomposition and the q, t-Catalan numbers and their generalizations. In Section 4 we relate Piontkowski's cell decomposition to a cell decomposition of an open subvariety in the Hilbert scheme of points in C 2 , introduced in [5]. Finally, in Section 5 we mention a possible relation of our results to HOMFLY homology of torus knots.…”

“…Recently A. Buryak [5] provided an example of a smooth subvariety of the Hilbert scheme of points in C 2 which admits a decomposition into affine cells enumerated by Dyck paths, such that the dimensions of these cells are related to the h + statistic. In Section 4, we construct an explicit bijection between the cell decompositions from [25] and [5].…”

Generating Series of the Poincaré Polynomials of Quasihomogeneous Hilbert Schemes

“…Let P q (X) = i≥0 dim H i (X)q i 2 . The main result of this paper is the following theorem (it was conjectured in [3]):…”

“…Irreducible components of (C 2 ) [n] T α,β were described in [7]. Poincaré polynomials of irreducible components in the case α = 1 were computed in [3]. For α = β = 1 it was done in [12].…”