[Paper Review] The moduli space of curves and its invariants

Mehdi Tavakol|arXiv (Cornell University)|Oct 30, 2016

Algebraic Geometry and Number Theory71 references3 citations

TL;DR

This paper provides a comprehensive review of tautological invariants in the moduli spaces of curves, focusing on intersection theory and cohomology. It establishes foundational results on the integral Chow rings of moduli stacks of curves using equivariant and operational Chow theory, with key results including the computation of $ A^*(\mathcal{M}_{1,1}) = \mathbb{Z}[t]/(12t) $ and $ A^*(\mathcal{M}_2) = \mathbb{Z}[\lambda_1,\lambda_2]/(10\lambda_1, 2\lambda_1^2 - 24\lambda_2) $, under characteristic ≠ 2,3.

ABSTRACT

This note is about invariants of moduli spaces of curves. It includes their intersection theory and cohomology. Our main focus in on the distinguished piece containing the so called tautological classes. These are the most natural classes on the moduli space. We give a review of known results and discuss their conjectural descriptions.

Motivation & Objective

• To review known results and conjectural descriptions of tautological classes in moduli spaces of curves.
• To investigate the intersection theory and cohomology of $\overline{M}_{g,n}$, particularly focusing on tautological rings.
• To establish foundational tools for computing Chow rings of moduli stacks using equivariant and operational Chow theory.
• To compute the integral Chow rings of the moduli stacks $\mathcal{M}_{1,1}$ and $\mathcal{M}_2$ via quotient stack constructions.
• To clarify the relationship between equivariant Chow groups, operational Chow rings, and stack-theoretic Chow groups for quotient stacks.

Proposed method

• Utilizes the framework of algebraic stacks, particularly quotient stacks $[X/G]$, to model moduli spaces of curves.
• Applies equivariant Chow theory to compute Chow rings of stacks by reducing to equivariant Chow rings of smooth varieties.
• Employs the identification $A^*(\mathcal{F}) = A^*_G(X)$ for smooth quotient stacks $\mathcal{F} = [X/G]$, linking operational and equivariant Chow rings.
• Uses the Hodge bundle's Chern classes $\lambda_i$ as generators for the Chow ring of $\mathcal{M}_2$, with relations derived from geometric constraints.
• Applies the theory of $Q$-varieties to handle singularities in $\overline{M}_g$, enabling intersection theory via étale local quotients.
• Relies on the fact that $A^*_G(X)$ is independent of the presentation $X/G$, ensuring invariance under different group actions.

Experimental results

Research questions

• RQ1 What is the structure of the integral Chow ring of the moduli stack $\mathcal{M}_{1,1}$ of elliptic curves?
• RQ2 How can the Chow ring of $\mathcal{M}_2$, the moduli stack of genus 2 curves, be computed using equivariant methods?
• RQ3 What is the relationship between equivariant Chow groups and the operational Chow ring of a quotient stack?
• RQ4 How do tautological classes generate the cohomology and Chow rings of $\overline{M}_{g,n}$, and what are their relations?
• RQ5 What conditions ensure that the Chow ring of a quotient stack $[X/G]$ is well-defined and independent of the group action presentation?

Key findings

• The integral Chow ring of $\mathcal{M}_{1,1}$ is $\mathbb{Z}[t]/(12t)$, where $t$ is the first Chern class of the Hodge bundle.
• The integral Chow ring of $\overline{\mathcal{M}}_{1,1}$ is $\mathbb{Z}[t]/(24t^2)$, reflecting the presence of nodal curves in the compactification.
• The moduli stack $\mathcal{M}_2$ is isomorphic to the quotient $Y/\mathrm{GL}_2$, where $Y$ parametrizes pairs $(\pi, \alpha)$ with trivialized Hodge bundle.
• The Chow ring $A^*(\mathcal{M}_2)$ is isomorphic to $\mathbb{Z}[\lambda_1, \lambda_2]/(10\lambda_1, 2\lambda_1^2 - 24\lambda_2)$, valid in characteristic ≠ 2,3.
• The first equivariant Chow group $A^1_G(X)$ coincides with the Picard group of the moduli problem $[X/G]$, linking geometry to algebraic K-theory.
• The operational Chow ring $A^*(\mathcal{F})$ of a smooth quotient stack $\mathcal{F} = [X/G]$ equals the equivariant Chow ring $A^*_G(X)$, ensuring consistency across theories.