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Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group

Vassil Kanev

TL;DR

The paper extends Fulton’s universal Hurwitz construction to pointed degree $d$ covers of an arbitrary smooth projective curve $Y$ with a fixed transitive monodromy group $G\subset S_d$, thereby circumventing automorphism obstructions. It introduces monodromy invariants $(D,m)$ and constructs a Hurwitz parameter space $H^{\Lambda,G}_{n,\lambda_0}(Y,y_0)$ for pointed $(\Lambda,G)$-covers branched in $n$ points, proving the existence of an explicit universal family over this space. It then analyzes smooth, proper families of such covers, relating pointed $(\Lambda,G)$-covers to pointed $G$-covers via étale base changes and quotients, and establishes a moduli framework via functors. Finally, it provides a fine moduli space with a universal family $(\mathcal{X}(y_0,\lambda_0),\phi,\xi)$, proving universality and detailing the local analytic behavior near branch loci, thereby giving a complete parameterization of pointed covers with a fixed monodromy group.

Abstract

Given a smooth, projective curve $Y$, a point $y_0 \in Y$, a positive integer $n$, and a transitive subgroup $G$ of the symmetric group $S_{d}$ we study smooth, proper families, parameterized by algebraic varieties, of pointed degree $d$ covers of $(Y,y_0)$, $(X,x_{0})\to (Y,y_0)$, branched in $n$ points of $Y\setminus y_{0}$, whose monodromy group equals $G$. We construct a Hurwitz space $H$, an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of $(Y,y_0)$ of this type. We construct explicitly a family parameterized by $H$, whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry.

Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group

TL;DR

The paper extends Fulton’s universal Hurwitz construction to pointed degree covers of an arbitrary smooth projective curve with a fixed transitive monodromy group , thereby circumventing automorphism obstructions. It introduces monodromy invariants and constructs a Hurwitz parameter space for pointed -covers branched in points, proving the existence of an explicit universal family over this space. It then analyzes smooth, proper families of such covers, relating pointed -covers to pointed -covers via étale base changes and quotients, and establishes a moduli framework via functors. Finally, it provides a fine moduli space with a universal family , proving universality and detailing the local analytic behavior near branch loci, thereby giving a complete parameterization of pointed covers with a fixed monodromy group.

Abstract

Given a smooth, projective curve , a point , a positive integer , and a transitive subgroup of the symmetric group we study smooth, proper families, parameterized by algebraic varieties, of pointed degree covers of , , branched in points of , whose monodromy group equals . We construct a Hurwitz space , an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of of this type. We construct explicitly a family parameterized by , whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry.
Paper Structure (4 sections, 27 theorems, 52 equations)

This paper contains 4 sections, 27 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Parameterization of covers with a fixed monodromy group
  3. Smooth, proper families of covers with a fixed monodromy group
  4. Universal families of pointed covers with fixed monodromy group

Key Result

Lemma 2.3

Let $G\subset S(\Lambda)$ be as in §2.0. Let $N_{S(\Lambda)}(G) = \{\sigma \in S(\Lambda)|\sigma G\sigma^{-1}=G\}$. Let $f:X\to Y$ and $f_{1}:X_{1}\to Y$ be two covers with the same branch locus $D$. Let $y_0\in Y\setminus D$. Suppose that there are bijections $\varepsilon : \Lambda \to f^{-1}(y_0)$

Theorems & Definitions (65)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 55 more