Configuration spaces for wildly ramified covers


Motivation from Hurwitz spaces:
For tame covers, a natural configuration space the space of unordered branch points of a cover – allows the construction of Hurwitz spaces, and an often effective theory for families of tame covers. This paper defines invariants generalizing Nielsen classes for Hurwitz families. The result is a configuration space for classifying families of wildly ramified covers. Most importantly for wild ramification, this notion does not assume the covers are Galois.

This generalizes the properties of the tame configuration space given in the first half of a famous theorem of Grothendieck:
It also gives some practical ways, and test cases, for approaching the 2nd half.

Local Ramification Data: Any wildly ramified field extension L=k((y)) (not necessarily Galois) of k((x)) has attached to it two notions: ramification data R (generalizes higher ramification groups in the special case of Galois extensions), and regular ramification data R. Both give Newton polygon diagrams, the latter the convex hull of the former. The former derives from describing explicitly the set of embeddings of L/k((x)) in the tame closure of L.

We then form a space P(R) that parametrizes a family of field extensions, all having ramification data R. Further, every such extension of k((x)) have R as its ramification data appears in this family. The precise multiplicity of appearance of each equals a numerical invariant depending on the number of automorphisms of k((y))/k((x)).

Global Configuration Spaces: These local notions then allow attaching to any wildly ramified cover φ: XP1 of the sphere (branched at r points) similar notions of ramification data Rφ = R* From this comes the configuration space P(R*) of sphere covers of ramification data R* (R* covers). It is a finite type space, with an explicit (not finite) map to the space of r unordered points on the sphere.

Locally in the finite topology any family of R* covers has a map to P(R*), defined uniquely up to an equivalence. Generalizing the 1st half of Grothendieck's famous result, if this map from a family of R* covers is constant, then the family is iso-trivial.

The Major Unsolved Problem: This is deciding which directions in the tangent space of a given cover associated to a point on P(R*) actually correspond to deformation directions for covers of the sphere.