Monodromy Groups of Hurwitz-type Problems

2008/03/03 by Allcock, Daniel, Hall, Chris
#14D05 #14H30 #20B25 #57M10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0803.0237

Abstract

We solve the Hurwitz monodromy problem for degree-4 covers. That is, the Hurwitz space H_4,g of all simply branched covers of P¹ of degree 4 and genus g is an unramified cover of the space P_2g+6 of (2g+6)-tuples of distinct points in P¹. We determine the monodromy of pi₁(P_2g+6) on the points of the fiber. This turns out to be the same problem as the action of pi₁(P_2g+6) on a certain local system of Z/2-vector spaces. We generalize our result by treating the analogous local system with Z/N coefficients, gcd(3,N)=1, 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.

Related

• Irreducibility of Hurwitz spaces
• Factorization semigroups and irreducible components of Hurwitz space. II
• Branch Stabilisation for the Components of Hurwitz Moduli Spaces of Galois Covers
• Counting Components of Hurwitz Spaces
• Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group