 
Summary: MONODROMY GROUPS OF HURWITZTYPE
PROBLEMS
DANIEL ALLCOCK AND CHRIS HALL
Abstract. We solve the Hurwitz monodromy problem for degree
4 covers. That is, the Hurwitz space H4,g of all simply branched
covers of P1
of degree 4 and genus g is an unramified cover of
the space P2g+6 of (2g + 6)tuples of distinct points in P1
. We
determine the monodromy of 1(P2g+6) on the points of the fiber.
This turns out to be the same problem as the action of 1(P2g+6)
on a certain local system of Z/2vector spaces. We generalize our
result by treating the analogous local system with Z/N coefficients,
3 N, in place of Z/2. This in turn allows us to answer a question
of Ellenberg concerning families of Galois covers of P1
with deck
group (Z/N)2
:S3.
A ramified cover C of P1
of degree d is said to have simple branching if
