The Representation Theory, Geometry, and Combinatorics of Branched Covers

The study of branched covers of the Riemann sphere has connections to many fields. We recall the classical relationship between branched covers and group theory via the Riemann existence theorem, which then leads to represention-theoretic formulas for Hurwitz numbers, counting the number of branched covers of prescribed types. We also review the Hurwitz spaces parametrizing branched covers as we allow the branch points to move, and the relationship between the components of a Hurwitz space and the orbits of a certain braid group action. Finally, we present two new results in this field: a connectedness theorem for Hurwitz spaces in the classical setting, joint with Fu Liu, and a result analogous to the Riemann existence theorem describing certain tamely branched cover of the projective line in positive characteristic. 1. Branched covers of the Riemann sphere Let CP = C ∪ {∞} ∼= S be the Riemann sphere, or equivalently, the complex projective line. Our main topic of discussion will be branched covers of CP; these are pairs (C, f) with C a connected, compact Riemann surface, and f : C → CP a surjective holomorphic map. Of course, this may be stated equivalently in terms of smooth projective complex curves. The key properties of such a branched cover are: • for every point P ∈ C, there is some eP ≥ 1 such that in a neighborhood of P , the map f looks like z 7→ zP (with P corresponding to the origin); • there is some d ≥ 1 such that for any Q ∈ CP, we have ∑ P∈f−1(Q) eP = d. Definition 1.1. We say that the above d is the degree of f . We say that P ∈ C is a ramification point of f (of index eP ) if eP > 1. We then say that f(P ) is a branch point of f . If Q ∈ CP is a branch point, we say that the branch type of f at Q is the non-trivial partition of d obtained from the multiset of eP for P ∈ f−1(P ). We observe the following consequences: • since C is compact, there are only finitely many ramification points of f , and hence only finitely many branch points; • if Q ∈ CP is not a branch point, #f−1(Q) = d. Up to isomorphism, there are only finitely many covers of given type with given branch points (we will see later why this is true). This leads to a basic question in the field: Question 1.2. If we fix d ≥ 1, branch points Q1, . . . , Qn ∈ CP, and branch types T1, . . . , Tn, how many branched covers of CP of degree d, branched only over the Qi, with branch type Ti over each Qi?