[Paper Review] On the Euler numbers of certain moduli spaces of curves and points

Wei-Ping Li, Zhenbo Qin|ArXiv.org|Aug 7, 2005

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper computes the topological Euler numbers of moduli spaces of 1-dimensional closed subschemes in smooth projective varieties that admit a Zariski-locally trivial fibration with 1-dimensional fibers. Using virtual Hodge polynomials and torus actions, it derives a generating function for these Euler numbers in terms of Hilbert schemes and punctual partitions, verifying a conjecture on rationality and modular invariance under q → 1/q for r ≤ 3 or Calabi-Yau 3-folds.

ABSTRACT

We determine the topological Euler number of certain moduli space of 1-dimensional closed subschemes in a smooth projective variety which admits a Zariski-locally trivial fibration with 1-dimensional fibers. The main approach is to use virtual Hodge polynomials and torus actions. The results might shed some light on the corresponding Donaldson-Thomas invariants.

Motivation & Objective

• To determine the topological Euler number of moduli spaces of 1-dimensional closed subschemes in smooth projective varieties with a Zariski-locally trivial fibration structure.
• To investigate the relationship between these Euler numbers and Donaldson-Thomas invariants, particularly in the context of Calabi-Yau 3-folds.
• To verify a conjecture on the rationality and modular invariance (under q → 1/q) of the reduced partition function for Euler numbers.
• To establish a generating function formula for the Euler numbers of these moduli spaces in terms of Hilbert schemes and punctual r-dimensional partitions.

Proposed method

• Decomposes the moduli space $\mathfrak{M}_n$ into locally closed subsets using the fibration structure $\mu: X \to S$ with 1-dimensional fibers.
• Constructs bijective morphisms between these subsets and model spaces over $\mathbb{C}^{r-1} \times C$, reducing the problem to local computations.
• Applies virtual Hodge polynomials to preserve Euler number information under decomposition and bijection.
• Uses Cheah’s combinatorial framework for Hilbert schemes to reduce computations to punctual Hilbert schemes and punctual partitions.
• Employs torus actions on $\mathfrak{M}_{n,L,O}^{\mathbb{C}^r}$ and $\mathrm{Hilb}^n(\mathbb{C}^r, O)$ to compute Euler numbers via punctual r-dimensional partitions.
• Derives generating functions for $\widetilde{P}_r(n)$ and $P_r(n)$, and proves $\sum \widetilde{P}_3(n) q^n = \sum P_3(n) q^n / (1 - q)$ for $r=3$.

Experimental results

Research questions

• RQ1 Is the reduced partition function $\sum \chi(\mathfrak{I}_n(X,\beta)) q^n / \sum \chi(X^{[n]}) q^n$ a rational function of $q$?
• RQ2 Does this partition function satisfy $q \to 1/q$ invariance when $K_X = 0$?
• RQ3 Can the Euler number of $\mathfrak{M}_n = \mathfrak{I}_{(1-g)+n}(X,\beta)$ be expressed in terms of Hilbert schemes and punctual partitions?
• RQ4 Is the generating function $\sum \widetilde{P}_r(n) q^n / \sum P_r(n) q^n$ equal to $1/(1 - q)^{r-2}$ for $r \geq 2$?
• RQ5 What is the precise structure of the moduli space $\mathfrak{M}_n$ when $X$ is a fibration over a smooth base with genus-$g$ fibers?

Key findings

• The generating function for the Euler numbers of $\mathfrak{M}_n$ is given by $\sum \chi(\mathfrak{M}_n) q^n = \left(\sum \chi(X^{[n]}) q^n\right) \cdot \chi(S) \cdot \left(\sum \widetilde{P}_r(n) q^n / \sum P_r(n) q^n\right)^{2-2g}$.
• For $r = 3$, the identity $\sum \widetilde{P}_3(n) q^n = \sum P_3(n) q^n / (1 - q)$ is proven using generating function identities and limits of plane partition generating functions.
• The conjecture on rationality and $q \to 1/q$ invariance holds for $2 \leq r \leq 3$ or when $K_X = 0$ (implying $g = 1$).
• The Euler number of $\mathfrak{M}_n$ is computed via virtual Hodge polynomials and torus actions, reducing to punctual $r$-dimensional partitions.
• The formula $\sum \chi(\mathfrak{M}_n) q^n = \left(\sum \chi(X^{[n]}) q^n\right) \cdot \chi(S) \cdot (1 - q)^{-(r-2)(2-2g)}$ is established for $2 \leq r \leq 3$.
• The proof relies on a decomposition of $\mathfrak{M}_n$ into locally closed subsets, bijective morphisms to model spaces, and combinatorial reduction via Cheah’s theory.