A NOTE ON HURWITZ SCHEMES OF COVERS OF A POSITIVE GENUS CURVE

JOE HARRIS, TOM GRABER, AND JASON STARR

Abstract. Let B be a smooth, connected, projective complex curve of genus h. For w ≥ 2d we prove the irreducibility of the Hurwitz stack H_{S_{d}}^{d,w}(B) parametrizing degree d covers of B simply-branched over w points, and with monodromy group S_{d}.

1. Introduction

Suppose that B is a smooth, connected, projective complex curve of genus h. Let d > 0 and w ≥ 0 be integers such that g := d(h − 1) + \frac{w}{2} + 1 is a nonnegative integer (in particular w is even). We define H^{d,w}(B) to be the open substack of the Kontsevich moduli stack \overline{M}_{g,0}(B, d) parametrizing stable maps f : X → B such that X is smooth and f is finite with only simple branching. Let br(f) ⊂ B denote the branch divisor of f. If we choose a basepoint b_{0} ∈ B−br(f) and an identification ϕ : f^{−1}(b_{0}) → {1, …, d}, there is an induced monodromy homomorphism ϕ̃ : π_{1}(B − br(f), b_{0}) → S_{d} which associates to any loop γ : [0, 1] → B with γ(0) = γ(1) = b_{0}, the permutation of f^{−1}(b_{0}) determined by analytic continuation along γ. In particular, the subgroup image(ϕ̃) ⊂ S_{d} is well-defined up to conjugation independently of ϕ. The corresponding conjugacy class of subgroups determines a locally constant function on H^{d,w}(B). Given a subgroup G ⊂ S_{d}, we define H_{G}^{d,w}(B) to be the open and closed substack of H^{d,w}(B) parametrizing stable maps f : X → B whose corresponding monodromy group is conjugate to G. We are particularly interested in H_{S_{d}}^{d,w}(B), the stack parametrizing Hurwitz covers of B with full monodromy group.

Theorem 1.1. If w ≥ 2d, then H_{S_{d}}^{d,w}(B) is a connected, smooth, finite-type Deligne-Mumford stack over ℂ.

The fact that H_{S_{d}}^{d,w}(B) is a finite-type Deligne-Mumford stack follows from the fact that \overline{M}_{g,0}(B, d) is a finite-type Deligne-Mumford stack. The fact that H_{S_{d}}^{d,w}(B) is smooth follows from a trivial deformation theory computation. So the content of theorem 1.1 is that H_{S_{d}}^{d,w}(B) is connected.

This is a classical fact when h = 0, i.e. for branched covers of ℙ^{1}, (c.f. [1], [5], and for a modern account [3, prop. 1.5]). This fact is well-known to experts, but there seems to be no reference. We used theorem 1.1 in our paper [4], and so we present a proof below. We wish to thank Ravi Vakil for useful discussions.

Date: August 31, 2003.

2. Setup

Our eventual goal is to prove theorem 1.1, but for most of this paper, we shall work with schemes which admit étale maps to H_{S_{d}}^{d,w}(B). Suppose Σ ⊂ B is a finite subset, and suppose b_{0} ∈ Σ is a point. We define M^{d,w}(B, Σ, b_{0}) to be the fine moduli scheme parametrizing pairs (f : X → B, ϕ) where f : X → B is a stable map in \overline{M}_{g,0}(B, d) and where ϕ : f^{−1}(b_{0}) → \{1, …, d\} are such that

(1) f is finite,

(2) f is unramified over Σ, and

(3) ϕ is a bijection.

Using known results on the Kontsevich moduli space \overline{M}_{g,0}(B, d), it is easy to show that M^{d,w}(B, Σ, b_{0}) is a nonempty, smooth, quasi-projective scheme of dimension w. By [2], there is a branch morphism br : M^{d,w}(B, Σ, b_{0}) → (B − Σ)_{w} where (B − Σ)_{w} is the wth symmetric power parametrizing effective degree w divisors on B −Σ. It is clear that br is quasi-finite, and thus br : M^{d,w}(B, Σ, b_{0}) → (B −Σ)_{w} is dominant. We denote by (B −Σ)_{w}^{o} ⊂ (B −Σ)_{w} the Zariski open subset parametrizing reduced effective divisors of degree w in B − Σ. We define H^{d,w}(B, Σ, b_{0}) ⊂ M^{d,w}(B, Σ, b_{0}) to be the preimage under br of (B − Σ)_{w}^{o}.

For each pair (f : X → B, ϕ) in H^{d,w}(B, Σ, b_{0}) with branch divisor br(f), there is an induced monodromy homomorphism ϕ̃ : π_{1}(B − br(f), b_{0}) → S_{d} where S_{d} is the symmetric group of permutations of {1, …, d}. The image of ϕ̃ determines a locally constant function on H^{d,w}(B, Σ, b_{0}). Because M^{d,w}(B, Σ, b_{0}) is smooth and H^{d,w}(B, Σ, b_{0}) is dense in M^{d,w}(B, Σ, b_{0}), this locally constant function extends to all of M^{d,w}(B, Σ, b_{0}). Given a subgroup G ⊂ S_{d} we define M_{G}^{d,w}(B, Σ, b_{0}) (resp. H_{G}^{d,w}(B, Σ, b_{0})) to be the open and closed subscheme of M^{d,w}(B, Σ, b_{0}) (resp. H^{d,w}(B, Σ, b_{0})) on which the image of ϕ̃ equals G.

Let F : X (w) → H_{S_{d}}^{d,w}(B, Σ, b_{0}) × B be the pullback of the universal stable map, i.e. X (w) parametrizes data (f : X → B, ϕ, x) where (f : X → B, ϕ) ∈ H_{S_{d}}^{d,w}(B, Σ, b_{0}) and x ∈ X, and F (f : X → B, ϕ, x) = (f : X → B, ϕ, f(x)). We denote by U ⊂ X (w) ×_{F,F} X (w) the open subscheme of the fiber product of X (w) with itself over H_{S_{d}}^{d,w}(B, Σ, b_{0}) × B parametrizing data (f : X → B, ϕ, x_{1}, x_{2}) such that x_{1} ≠ x_{2}, and such that f(x_{1}) = f(x_{2}) is neither in S nor equal to any branch point of f. We define X_{2}(w) to be the quotient of U by the obvious involution (f : X → B, ϕ, x_{1}, x_{2}) ∼ (f : X → B, ϕ, x_{2}, x_{1}). We denote by X_{2}^{e}(w) the open subscheme of the e-fold fiber product of X_{2}(w) with itself over H_{S_{d}}^{d,w}(B, Σ, b_{0}) parametrizing data (f : X → B, ϕ, {x_{1}^{1}, x_{2}^{2}}, …, {x_{1}^{e}, x_{2}^{e}}) such that f(x_{1}^{1}), …, f(x_{1}^{e}) are all distinct points in B − Σ. Notice that the projection X_{2}^{e}(w) → H_{S_{d}}^{d,w}(B, Σ, b_{0}) is flat. The condition that the image of ϕ̃ be all of S_{d}, and therefore doubly-transitive, implies that X_{2}(w) → H_{S_{d}}^{d,w}(B, Σ, b_{0}) has irreducible fibers. Therefore also X_{2}^{e}(w) → H_{S_{d}}^{d,w}(B, Σ, b_{0}) has irreducible fibers.

For each (f : X → B, ϕ, {x_{1}^{1}, x_{2}^{1}}, …, {x_{1}^{e}, x_{2}^{e}}) in X_{2}^{e}(w) we can associate a pair (f_{a} : X_{a} → B, ϕ_{a}) in H_{S_{d}}^{d,w+2e}(B, Σ, b_{0}) as follows: