Algebraic Structure on a Hurwitz space

Our applications are to interpreting points on Hurwitz spaces as a surrogate for finding covers of the projective line with given arithmetic/geometric monodromy groups. For example, interpretation of the R(egular) I(nverse) G(alois) P(roblem) that requires knowing properties of the spaces – their fields of definition, components and cusps – that don't make sense unless there exist – versus you need them at hand – algebraic equations for the spaces.

The spaces have abstract defined definitions, through  Nielsen classes. That brings up how we can get our hands on desirable properties. Also, there are two famous theoretical issues. 

  1. One is considered an unsolved problem: production of branch cycles for a cover without using topology.
  2. The other is psychological:  A disturbing feeling that working  with such abstract spaces without equations is an airy pursuit, unlike combinatorial pursuits yielding pretty formulas or coefficient anomolies that focus attention on a visually clear starting point.
Fortunately, both objections in #2 have answers. The Hurwitz monodromy (a combinatorial quotient of the Artin Braid)  group is the source of tremendous combinatorics.  Acquainting yourself with its use and the permutation representations that produce various versions of Hurwitz spaces is akin to picking up a version of Galois Theory.

As for learning comfort in dealing without equations, we say this. Once mastered, the combinatorial approach to cusps gives a direct line to properties of the spaces by imitating the classical example of modular curves. Still, there are occasional subtle questions for which no one has yet found a work-around to having equations. Usually, however, that means the question has been answered only in special cases, and there is no possibility of writing equations for all relevant Hurwitz spaces.

It takes some doing to acquaint yourself with the use of the Hurwitz monodromy group (a combinatorial quotient of the Artin Braid group) and the permutation representations that produce various versions of Hurwitz spaces, depending on the equivalence one uses on covers. Still, as one's skill grows on that, it becomes a version of using Galois Theory. Here are our topics.

  1. Galois closure of a cover:
  2. Branch cycles and Nielsen classes:
  3. Definition fields of Hurwitz spaces from the B(ranch) C(ycle) L(emma):
  4. Hurwitz space components and their definition fields:
  5. Cusps and the boundary of Hurwitz spaces:

I. Galois closure of a cover: We need the n-fold fiber product of a degree n cover with itself.

II. Branch cycles and Nielsen classes: Given a group G and a collection of conjugacy classes C, the Nielsen class Ni(G, C) has a simple seeming theorem attached to it: G is the monodromy group of a cover, f: X → P1z, in the Nielsen class if and only if the Nielsen class is nonempty.

II.a. A Nielsen class mystery: One‟if part” requires explanation: Its statement requires no topology. That is, suppose you have such an f (defined by algebraic equations, even) and you also know the conjugacy C classes defined by its branch points z={z1,…, zr}. Then, how can you assert there are g=(g1,…, gr) with generation (<g1,…, gr>=G) and product -one (g1… gr=1) without using classical generators?

II.b. Nielsen class as a generalization of genus: Fixing a Nielsen class is an analog of fixing the genus in the theory of curves and their moduli spaces. That is, given a cover f: X → P1z in a Nielsen class for inner or absolute equivalence, you automatically know the genus of X from the R(iemann)-H(urwitz) formula for its genus (see Nielsen-ClassesCont.html).

II.c. Covers of P1z are algebraic: That is equivalent to the following simple phrase, if f has degree nf, then

III. Definition fields of Hurwitz spaces from the B(ranch) C(ycle) L(emma): For each τ ∈ Aut(C), the conjugate space H∞(C)τ is still a Hurwitz space, which r,G only depends on the restriction τ|Qab ∈ Gal(Qab/Q); namely it is H∞(Cχ(τ)) (where χ is r,G the cyclotomic character and Cχ(τ) =(C1 χ(τ),...,Crχ(τ))). Thus the (generally reducible) varieties H∞and H∞(C) can be de�ned over Q and Qab respectively, in the sense that r,G r,G their (geometric) components are permuted transitively by Gal(Q/Q) and Gal(Q/Qab) respectively. Furthermore, the Hurwitz space H∞(C) is itself de�ned over Q if C is a r,G rational union of conjugacy classes of G, i.e., if for every integer m prime to |G|, there exists σ ∈ Sr such that Cm = Cσ(i). More generally, given a �eld k ⊂ Qab, the tuple C is i said to be a k-rational union of conjugacy classes of G if the same property holds for all integers m ≡ χ(τ) modulo |G| with τ ∈ Gal(Qab/k). Under this condition, the Hurwitz space H∞(C) is de�ned over k. For example, the �eld generated by all roots of unity of r,G order |G| is a rationality �eld for C.

IV. Hurwitz space components and their definition fields: 

V. Cusps and the boundary of Hurwitz spaces: S. Wewers has given a general construction of Hurwitz spaces, which leads to a de�nition of Hr,G and of some compacti�cation Hr,G as schemes over Spec(Z[1/|Gp|]). For each prime p not dividing |G|, the corresponding �bers above p are denoted by Hand r,G Hp This includes the case p = ∞ for which one recovers the space H∞ r,G. r,G. There is good reduction of Hr,G at those primes p � ||G|: the �ber Hp is a (reducible) r,G smooth variety de�ned over Fp and its components correspond to those of H∞through r,G the reduction process. Furthermore, each Hp is a moduli space, for covers of P1 with r r,G branch points and monodromy group G, over algebraically closed �elds of characteristic p. The compacti�cation Hr,G is locally the quotient of a smooth variety by a �nite group Components in Hr,G are closures of components in Hr,G. The natural ´etale morphism Ψr : Hr,G →Ur extends to a rami�ed cover Hr,G →Ur. Points on the boundary Ur \Ur represent degenerations of tuples t =(t1,...,tr) when two or more of the ti “coalesce� (i.e. become equal). More formally they correspond to stable marked curves of genus 0 with a root, i.e. trees of curves of genus 0 with a distinguished component T0 — the root <>— equipped with an isomorphism P1 � T0 and at least three marked points (including the double points) on any component but the root. Points on the boundary Hr,G \Hr,G represent admissible covers (in a certain sense) of stable marked curves of genus 0.