Compactifications of Hurwitz spaces (1206.4535v1)

Abstract: We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus $g$ curves expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These compactifications are obtained by allowing the branch points of the covers to collide to a variable extent. They are very well-behaved if $d = 2, 3$, or if relatively few collisions are allowed. We recover as special cases the spaces of twisted admissible covers of Abramovich, Corti and Vistoli and the spaces of hyperelliptic curves of Fedorchuk.

• The paper introduces the construction of modular compactifications by allowing controlled collisions of branch points via epsilon-admissible covers.
• The methodology employs a crimping technique to analyze local deformations and manage singularities in algebraic curve covers.
• It establishes the properness and projectivity of the branch morphism by proving that the Hodge line bundle is relatively anti-ample.

Compactifications of Hurwitz Spaces

Introduction to Hurwitz Spaces

Hurwitz spaces are moduli spaces that parameterize coverings of algebraic curves. Specifically, the Hurwitz space $H^d_{g/h}$ is concerned with genus $g$ curves that are expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These spaces are not only central to understanding curve moduli but also play a pivotal role in the birational geometry of moduli spaces. This is particularly relevant in the context of the Hassett--Keel program, which focuses on constructing compactifications with desirable geometric properties.

Modular Compactifications

In this work, we construct several modular compactifications of the Hurwitz space $H^d_{g/h}$. The main idea is to allow the branch points of the covers to collide to a controlled extent. This generalization includes the spaces of twisted admissible covers and hyperelliptic curves as special cases.

Technical Setup

We consider a map $H^d_{g/h} = \{(\phi \from C \to P)\}$ where $\phi$ denotes a simply branched cover. The non-separated Artin stack $#1 M_{h;b}$ consists of nodal curves $P$ and divisors $\Sigma$. A crucial technical result demonstrates that these stacks, equipped with branch morphisms, are proper and of Deligne--Mumford type.

Epsilon-Admissible Covers

For a given rational number $\epsilon$, we define $\epsilon$-admissible covers through a proper Deligne--Mumford stack $\o{ H}^d_{g/h}(\epsilon)$. This allows us to handle admissible covers with $\lfloor 1/\epsilon \rfloor$ coinciding branch points.

Points of Interest

• Twisted Admissible Covers: The case $\epsilon=1$, where covers are forced to remain distinct, reproduces previously recognized spaces like those in \citet*{acv:03}.
• Fedorchuk's Hyperelliptic Spaces: For $d=2$ and $h=0$, these spaces account for hyperelliptic curves with $A_n$ singularities.
• Singularities: With smaller $\epsilon$, $C$ can develop more complex singularities, such as cusps or ramified nodes, which are not Gorenstein.

Properness and Projectivity

A major achievement is proving that the branch morphism $\br \from \o{ H}^d_{g/h}(\epsilon) \to \o{ M}_{h;b}(\epsilon)$ is projective by showing that the Hodge line bundle is relatively anti-ample. This ensures that the resulting coarse space is projective.

Crimping Technique

We introduce a method of "crimping," which emphasizes local deformations over discrete global data. This technique describes how to crimp a given cover by analyzing its behavior near the branch points. The approach is formalized by considering the quotient of algebras, allowing for computations within projective schemes.

Structural Advantages

• The theoretical framework for crimps over disks lends itself to constructing projective schemes from the crimping process.
• Using this framework, the fiber of the branch morphism is decomposed into crimping operations across branch points, illustrating the compactness of the closure in the moduli problem.

Conclusion

This paper extends the understanding of Hurwitz spaces through carefully constructed compactifications. It unites pre-existing concepts like admissible covers and hyperelliptic curves under a broader framework, while the crimping method enriches the toolbox for handling singularities in algebraic geometry. This work sets the stage for new explorations into moduli space compactifications, optimizing not only the parametrization but also the understanding of complex curve coverings.