[Paper Review] Genus stabilization for moduli of curves with symmetries

Fabrizio Catanese, Michael Lönne+1 more|arXiv (Cornell University)|Jan 18, 2013

Geometric and Algebraic Topology27 references5 citations

TL;DR

This paper establishes genus stabilization for moduli spaces of curves with faithful group actions by proving that, for large quotient genus g′, the moduli space of curves with a fixed finite group G and homological invariant ε is irreducible if and only if ε satisfies an 'admissibility' condition. The key result shows that stable classes of such actions are in bijection with admissible homological invariants, resolving the irreducibility of these moduli components via mapping class group actions and invariants in group cohomology.

ABSTRACT

In a previous paper, arXiv:1206.5498, we introduced a new homological invariant $\e$ for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant $\e$, if the genus g' of the quotient curve is sufficiently large, is irreducible (and non empty iff the class satisfies the condition which we define as 'admissibility'). In the unramified case, a similar result had been proven by Dunfield and Thurston using the classical invariant in the second homology group of G, H_2(G, \ZZ). We achieve our result showing that the stable classes are in bijection with the set of admissible classes $\e$.

Motivation & Objective

• To determine the irreducible components of moduli spaces of curves admitting a faithful action by a finite group G.
• To establish a stabilization phenomenon in the genus of the quotient curve C′ = C/G.
• To characterize the irreducibility of moduli spaces Mg,ρ in terms of a homological invariant ε.
• To prove that the stable classes of group actions correspond bijectively to admissible homological invariants ε.
• To set the foundation for homological stabilization in the study of moduli spaces of curves with symmetries.

Proposed method

• The paper uses the homological invariant ε introduced in [CLP12] to classify group actions on curves up to topological type.
• It analyzes the action of the mapping class group Mapg′,d on Hurwitz generating systems via the orbifold fundamental group πorb1(C′/G).
• The authors employ the Teichmüller space Tg,ρ as a contractible parameter space for curves with given group action, and study its quotient by the normalizer of ρ(G) in Mapg.
• They define the evaluation map ev: HS(G; g′, d) → G/[G, G] to extract the homological invariant ε, and show its invariance modulo commutator subgroup under mapping class group actions.
• Using Fuchsian group presentations and the structure of the fundamental group Πg′,d, the paper constructs explicit relations between monodromy data and the invariant ε.
• The proof relies on invariance lemmas (Lemma 4.8) and commutator identities (Lemma 4.7) to show that ε is preserved under mapping class group transformations up to commutator relations.

Experimental results

Research questions

• RQ1 Under what conditions is the moduli space of curves with a faithful G-action irreducible?
• RQ2 How does the genus of the quotient curve C′ = C/G affect the structure of the moduli space Mg,ρ?
• RQ3 What role does the homological invariant ε play in classifying irreducible components of Mg,ρ?
• RQ4 Is there a bijection between stable classes of group actions and admissible homological invariants ε?
• RQ5 How do mapping class group actions affect the evaluation of Hurwitz generating systems modulo [F, R]?

Key findings

• For large quotient genus g′, the moduli space Mg,ρ of curves with a faithful G-action and fixed homological invariant ε is irreducible if and only if ε is admissible.
• The stable classes of group actions on curves are in bijection with the set of admissible homological invariants ε.
• The evaluation map ev: HS(G; g′, d) → G/[G, G] induces a well-defined invariant ε that classifies the irreducible components of Mg,ρ.
• The invariant ε is preserved under mapping class group actions modulo the commutator subgroup [F, R], ensuring its stability under topological deformations.
• The proof establishes that the action of the mapping class group on Hurwitz generating systems preserves ε up to commutator relations, enabling classification via ε.
• The paper confirms that genus stabilization occurs: for g′ >> 0, the moduli space Mg,ρ depends only on the admissibility of ε, not on the specific monodromy data.