[Paper Review] On generating series of classes of equivariant Hilbert schemes of fat points

S. M. Gusein‐Zade, I. Luengo+1 more|ArXiv.org|May 12, 2009

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper extends generating series formulae for equivariant Hilbert schemes of zero-dimensional subschemes on smooth varieties to the case of finite group actions, introducing a power structure over the Grothendieck ring of complex varieties. It derives local generating series for cyclic group actions on smooth surfaces, conjectures closed-form products for the logarithmic series, and provides explicit computations for small group orders, revealing stabilization patterns in coefficients.

ABSTRACT

In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms of certain local data and the, so called, power structure over the corresponding ring. Here we give an analogue of these formulae for equivariant (with respect to an action of a finite group on a smooth variety) Hilbert schemes of zero-dimensional subschemes and compute some local generating series for an action of the cyclic group on a smooth surface.

Motivation & Objective

• To generalize previous generating series formulae for Hilbert schemes to equivariant settings under finite group actions.
• To compute local generating series for equivariant Hilbert schemes of zero-dimensional subschemes on smooth surfaces with cyclic group actions.
• To investigate structural patterns in the logarithmic series of these generating functions and conjecture closed-form expressions.
• To explore stabilization phenomena in coefficients of the logarithmic series as group order increases.

Proposed method

• Uses the power structure over the Grothendieck ring $K_0(\mathcal{V}_{\mathbb{C}})$ to define and manipulate generating series of equivariant Hilbert schemes.
• Applies the geometric interpretation of the power structure via configuration spaces of pairs $(K, \varphi)$ with cardinality constraints.
• Reduces global generating series to local data via an integral formula involving the generalized Euler characteristic: $\mathbb{H}_X(T) = \int_X \mathbb{H}_{X,x}(T)^{d\chi_g}$.
• Computes explicit series using symbolic computation (Maple) for specific group actions, such as $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ on $\mathbb{C}^2$.
• Analyzes the logarithmic series $\mathrm{Log}^{(1)}$ and $\mathrm{Log}^{(2)}$ to detect patterns and conjecture product structures.
• Proposes a conjectural product formula for $\mathbb{H}^{M,1}_{\mathbb{C}^2,0}(T)$ based on observed coefficient stabilization.

Experimental results

Research questions

• RQ1 Can the generating series of equivariant Hilbert schemes under finite group actions be expressed in terms of local data using the power structure on the Grothendieck ring?
• RQ2 What is the structure of the logarithmic series $\mathrm{Log}^{(1)}\mathbb{H}^{M,1}_{\mathbb{C}^2,0}(T)$ for cyclic group actions on the plane?
• RQ3 Does the coefficient sequence $p_i^{M,1}(\mathbb{L})$ in $\mathrm{Log}^{(2)}\mathbb{H}^{M,1}_{\mathbb{C}^2,0}(T)$ stabilize for large $M$ relative to $i$?
• RQ4 Is there a closed-form product formula for $\mathbb{H}^{M,1}_{\mathbb{C}^2,0}(T)$, analogous to the non-equivariant case?
• RQ5 What is the geometric meaning of the observed stabilization in the coefficients of the logarithmic series?

Key findings

• For the $\mathbb{Z}/3\mathbb{Z}$-action on $\mathbb{C}^2$, the logarithmic series $\mathrm{Log}^{(1)}\mathbb{H}^{3,1}_{\mathbb{C}^2,0}(T)$ begins with $T + \mathbb{L}T^2 + T^3 + \mathbb{L}T^4 + \mathbb{L}^2T^5 + \cdots$, showing a structured pattern in degrees.
• A conjectural product formula is proposed: $\mathbb{H}^{3,1}_{\mathbb{C}^2,0}(T) = \prod_{i=1}^\infty \frac{1}{(1 - \mathbb{L}^{i-1}T^{3i-2})(1 - \mathbb{L}^i T^{3i-1})(1 - \mathbb{L}^{i-1} T^{3i})}$.
• For the $\mathbb{Z}/4\mathbb{Z}$-action, $\mathrm{Log}^{(1)}\mathbb{H}^{4,1}_{\mathbb{C}^2,0}(T)$ contains terms like $(1+\mathbb{L})T^4 + (2\mathbb{L} + 2\mathbb{L}^2 + \mathbb{L}^3)T^8 + \cdots$, indicating non-trivial coefficient growth.
• The $\mathrm{Log}^{(2)}$ series for $\mathbb{H}^{M,1}_{\mathbb{C}^2,0}(T)$ exhibits negative coefficients, such as $(-\mathbb{L}^2 + \mathbb{L}^3 - \mathbb{L}^6)T^{12}$, suggesting non-trivial algebraic structure.
• Computations suggest that for $M'' > M' > i$, the coefficients $p_i^{M',1}(\mathbb{L})$ and $p_i^{M'',1}(\mathbb{L})$ in $\mathrm{Log}^{(2)}\mathbb{H}^{M,1}_{\mathbb{C}^2,0}(T)$ stabilize, indicating a potential limit in the $M \to \infty$ regime.
• The $\mathrm{Log}^{(1)}$ series for $\mathbb{H}^{5,2}_{\mathbb{C}^2,0}(T)$ shows a repeating pattern in degrees $5k$, with terms like $T^5 + \mathbb{L}T^6 + \cdots + \mathbb{L}^4T^{24} + \cdots$, supporting the idea of periodicity or symmetry.