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:
- 1st half: If in a (smooth) family of tamely ramified covers of
the sphere, the branch points don't move then the family is
iso-trivial.
- 2nd half: There is no obstruction to the local deformation of a
tamely ramified cover.
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:
X→
P1 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.