Example problem – rough phrasing: For
each
nonnegative integer
g describe explicitly all of the ways the function field of the generic
curve of
genus g contains C(x) (Section 1).
Specifically, give the dimension of the range ( moduli
dimension) of a Hurwitz family in the moduli space of
appropriate genus. Section 2.2 contains a discussion of the genus zero
problem, new at
the time of the paper. This paper narrows to where
g> 1 and the containment of fields gives a solvable Galois
closure.
This alone illustrates Zariski's most
definitive
conjecture touching on this is wrong (Section 2.3). This is how the
Galois closure group ring gives info on endomorphisms of the
Jacobian, a topic which an example of John Ries anticipates, (though we
beat him into
print). By 2001 the genus 0 problem was completed in
zero characteristic and was rapidly advancing in positive
characteristic, with significant applications, though
Guralnick's conjecture, based on his papers with co-writers and
those of Abhyankar (see Prelude
to
the 1999 MSRI Semester
volume and
to my 1999 Schinzel volume paper.
This paper,
however, concentrates on higher monodromy. This has many recent
applications only
slightly anticipated in this paper. Theorem 3.5 presents
the fundamental group Π1(X) and 1st
homology H1(X,Z) of a Riemann
surface X appearing as a (not necessarily Galois – the
interesting case for this problem) cover of the
sphere in terms of branch cycles for the cover. This gives an
explicit action of the Hurwitz monodromy group
Hr on
H1(X,Z) where r is the
number of branch points of the cover. Section 3 interprets the
dimension of the
image of deformations of the cover in the moduli space of curves of
appropriate genus
g from this group action. Increasing application to
specific Hurwitz spaces, and Modular Towers over them, requires
improved computations within this
approach at monodromy action (as in §5 and §6 in "Alternating Groups and Moduli Space
Invariants")