[Paper Review] Geometric Proofs of Some Results of Morita

Richard Hain, David Reed|ArXiv.org|Oct 8, 1998

Algebraic Geometry and Number Theory21 references24 citations

TL;DR

This paper provides geometric proofs of three cohomological results by Morita concerning moduli spaces of curves, focusing on relations between characteristic classes in $H^2$ of $σ_g$, $σ_g[2]$, and $σ_g[l]$. It establishes that the second cohomology of the mapping class group with coefficients in the homology of a curve is isomorphic to $\mathbb{Z}/(2g-2)\mathbb{Z}$, realized geometrically via the relative Picard group of the universal curve, and proves that this class vanishes precisely when the degree is divisible by $2g-2$. The results are derived using spectral sequences, the universal coefficient theorem, and the solution to the Francetta Conjecture.

ABSTRACT

In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth result of Morita. One motivation of this work is to facilitate the application of these results in our work (in preparation) on the Arakelov geometry of moduli spaces of curves.

Motivation & Objective

• To reprove Morita's results on characteristic classes in the cohomology of moduli spaces of curves using geometric methods.
• To interpret the cohomology class $H^2(\Gamma_g, H_\mathbb{Z})$ geometrically via the relative Picard group of the universal curve.
• To establish the isomorphism $H^2(\Gamma_g, H_\mathbb{Z}) \cong \mathbb{Z}/(2g-2)\mathbb{Z}$ for $g \geq 3$ using the canonical bundle and level structures.
• To analyze the restriction of this cohomology class to level $l$ subgroups, showing it is $\mathbb{Z}/(2g-2)\mathbb{Z}$ for odd $l$ and $\mathbb{Z}/(g-1)\mathbb{Z}$ for even $l > 0$.

Proposed method

• Use the spectral sequence for the extension $1 \to H_\mathbb{Z} \to \Gamma_g^1 \to \Gamma_g \to 1$ and show it degenerates at $E_2$ for level $\geq 4$, implying $H^2(\Gamma_g^1, \mathbb{Q}) \cong \mathbb{Q} \oplus H^2(\Gamma_g, \mathbb{Q}) \oplus H^1(\Gamma_g, H_\mathbb{Q})$.
• Prove $H^1(\Gamma_g, H_\mathbb{Q}) = 0$ for $g \geq 2$ using the center kills trick with $-I \in \Gamma_1$.
• Construct a group homomorphism $\epsilon: \mathbb{Z} \to H^2(\Gamma_g, H_\mathbb{Z})$ via the relative Picard group $\operatorname{Pic}^d_{\mathcal{M}_g} \mathcal{C}_g$.
• Use the canonical bundle as a section of $\operatorname{Pic}^{2g-2}_{\mathcal{M}_g} \mathcal{C}_g$ to show $\epsilon$ factors through $\mathbb{Z}/(2g-2)\mathbb{Z}$.
• Apply the solution to the Francetta Conjecture to show that $\epsilon(d) = 0$ only if $d$ is divisible by $2g-2$, proving injectivity.
• Use the universal coefficient theorem and Johnson's theorems to show $H^1(\Gamma_g, H_\mathbb{Z})$ is trivial, supporting the isomorphism.

Experimental results

Research questions

• RQ1 What is the geometric interpretation of the cohomology class $H^2(\Gamma_g, H_\mathbb{Z})$?
• RQ2 How does the relative Picard group of the universal curve realize the cohomology class $H^2(\Gamma_g, H_\mathbb{Z})$?
• RQ3 Why does the restriction of $H^2(\Gamma_g, H_\mathbb{Z})$ to level $l$ subgroups yield $\mathbb{Z}/(2g-2)\mathbb{Z}$ for odd $l$ and $\mathbb{Z}/(g-1)\mathbb{Z}$ for even $l > 0$?
• RQ4 What is the role of the canonical bundle in determining the order of the cohomology class in $H^2(\Gamma_g, H_\mathbb{Z})$?
• RQ5 How does the degeneration of the spectral sequence for level $\geq 4$ subgroups support the computation of $H^2(\Gamma_g^1, \mathbb{Q})$?

Key findings

• The second cohomology group $H^2(\Gamma_g, H_\mathbb{Z})$ is isomorphic to $\mathbb{Z}/(2g-2)\mathbb{Z}$ for all $g \geq 3$.
• The isomorphism is realized geometrically via the map $\epsilon: \mathbb{Z} \to H^2(\Gamma_g, H_\mathbb{Z})$ defined by the relative Picard group $\operatorname{Pic}^d_{\mathcal{M}_g} \mathcal{C}_g$.
• The class $\epsilon(d)$ vanishes if and only if $d$ is divisible by $2g-2$, which is shown using the Francetta Conjecture and the triviality of $H^1(\Gamma_g, H_\mathbb{Z})$.
• The restriction map $H^2(\Gamma_g, H_\mathbb{Z}) \to H^2(\Gamma_g[l], H_\mathbb{Z})$ has image $\mathbb{Z}/(2g-2)\mathbb{Z}$ when $l$ is odd and $\mathbb{Z}/(g-1)\mathbb{Z}$ when $l$ is even and positive.
• The spectral sequence for the extension $1 \to H_\mathbb{Z} \to \Gamma_g^1 \to \Gamma_g \to 1$ degenerates at $E_2$ for level $\geq 4$, which implies $H^1(\Gamma_g, H_\mathbb{Q}) = 0$ for $g \geq 2$.
• The canonical bundle provides a section of $\operatorname{Pic}^{2g-2}_{\mathcal{M}_g} \mathcal{C}_g$, which forces the homomorphism $\epsilon$ to factor through $\mathbb{Z}/(2g-2)\mathbb{Z}$.