[Paper Review] Hurwitz Spaces and Moduli Spaces as Ball Quotients via Pull-back

Brent Doran|ArXiv.org|Apr 20, 2004

Algebraic Geometry and Number Theory25 references3 citations

TL;DR

This paper constructs subball quotients within Deligne-Mostow moduli spaces $DM(n,\mu)$ using a topological pull-back construction based on intersection homology and monodromy. By analyzing hypergeometric functions via local systems and exploiting the Hermitian form from intersection pairing, the authors identify conditions under which Hurwitz spaces associated to inhomogeneous binary forms yield ball quotient structures, recovering known results such as the rational elliptic surface moduli space as a ball quotient.

ABSTRACT

We define hypergeometric functions using intersection homology valued in a local system. Topology is emphasized; analysis enters only once, via the Hodge decomposition. By a pull-back procedure we construct special subsets S_{pi}, derived from Hurwitz spaces, of Deligne-Mostow moduli spaces DM(n,mu). Certain DM(n,mu) are known to be ball quotients, uniformized by hypergeometric functions valued in a complex ball (i.e., complex hyperbolic space). We give sufficient conditions for S_{pi} to be a subball quotient. Analyzing the simplest examples in detail, we describe ball quotient structures attached to some moduli spaces of inhomogeneous binary forms. This recovers in particular the structure on the moduli space of rational elliptic surfaces given by Heckman and Looijenga. We make use of a natural partial ordering on the Deligne-Mostow examples (which gives an easy way to see that the original list of Mostow, eventually corrected by Thurston, is in error), and so highlight two key examples, which we call the Gaussian and Eisenstein ancestral examples.

Motivation & Objective

• To establish a topological method for constructing subball quotients within Deligne-Mostow moduli spaces $DM(n,\mu)$ using pull-backs from Hurwitz spaces.
• To clarify the geometric and topological origin of ball quotient structures in moduli spaces of inhomogeneous binary forms.
• To provide a systematic characterization of when such moduli spaces are subball quotients, using intersection homology and monodromy invariants.
• To recover and re-interpret the ball quotient structure on the moduli space of rational elliptic surfaces via the Eisenstein ancestral example.
• To highlight the Gaussian and Eisenstein ancestral examples as foundational cases with rich subball quotient structures.

Proposed method

• Uses intersection homology valued in a local system to define a Hermitian form $\Psi$ that is preserved under monodromy, yielding a lattice $\Lambda$ with $\Gamma$-action.
• Applies a pull-back procedure to rank 1 local systems via branched covers $\pi: \mathbb{P}^1 \to \mathbb{P}^1$, restricting attention to points with nontrivial monodromy.
• Leverages the Hodge decomposition of intersection homology to extract holomorphic 1-forms, which define the hypergeometric map $HG_\mu$ into a complex ball.
• Constructs a pseudo-discriminant map $\Delta$ from bidegree $(a,b)$ inhomogeneous binary forms to $N$-point configurations on $\mathbb{P}^1$, with $N = ad_1 = bd_2$.
• Analyzes the degree and fiber structure of $\Delta$, showing it is generically $\gcd(a,b)$-to-1 or an embedding under specific conditions.
• Uses dimension counting and contradiction arguments to prove that $\Delta$ is injective or finite-to-one when $d_2 > d_1 + 1$, leading to hypersurface images and subball quotients.

Experimental results

Research questions

• RQ1 Under what conditions does a Hurwitz space associated to a branched cover $\pi$ yield a subball quotient inside a Deligne-Mostow moduli space $DM(n,\mu)$?
• RQ2 How can intersection homology and monodromy invariants be used to construct hypergeometric functions that uniformize subball quotients?
• RQ3 What is the geometric significance of the pseudo-discriminant map $\Delta$ from inhomogeneous binary forms to $N$-point moduli spaces?
• RQ4 Why do the Gaussian and Eisenstein ancestral examples support a rich family of subball quotients, and how do they relate to classical moduli spaces?
• RQ5 In what cases is the map $\Delta$ an embedding or finite-to-one, and how does this affect the moduli space structure?

Key findings

• The moduli space of rational elliptic surfaces, described via rational Weierstrass fibrations, is a hyperball quotient isomorphic to the Eisenstein ancestral example.
• For bidegree $(3,2,12)$, the pseudo-discriminant $\Delta$ is an embedding on a suitable open subset, yielding a subball quotient structure.
• For bidegrees $(6,2,6)$ and $(4,2,8)$, the map $\Delta$ is generically 2-to-1, and the image is a hypersurface of codimension 1 in $DM(n,\mu)$, leading to subball quotients.
• The condition $d_2 > d_1 + 1$ ensures that $\Delta$ is injective up to roots of unity, and the degree of $\Delta$ is $\gcd(a,b)$.
• The classification of subball quotients via property $G$ matches exactly the classification of pseudo-discriminant images that are hypersurfaces of codimension 1.
• The paper identifies a natural partial order on Deligne-Mostow examples, which helps correct and clarify the original list of Mostow, eventually refined by Thurston.