Let f : Σ̃ d:1 −→ Π Σ denote a branched cover between closed, connected, and orientable surfaces, where d is the global degree, Π = {Π1, . . . ,Πn}, n is the number of branching points, and Πi is the partition of d given by the local degrees over the i-th branching point. If l(Π) is the total length of Π, then the Riemmann-Hurwitz formula asserts that χ(Σ̃) − l(Π) = d · (χ(Σ) − n). A candidate br...

We address the question of when a covering of the boundary of a surface F can be extended to a covering of the surface (equivalently: when is there a branched cover with a prescribedmonodromy ). If such an extension is possible, when can the total space be taken to be connected? When can the extension be taken to be regular? We give necessary and sufficient conditions for both finite and infini...

Consider genus g curves that admit degree d covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family Y that naturally maps into the moduli space of stable genus g curves Mg. We study the geometry of Y , and produce a combinatorial method by which to investigate its slope, irreducible components, genus and orbifold...

There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F ). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois g...

We give a description of asymptotic quadratic growth rates for geodesic segments on covers of Veech surfaces in terms of the modular fiber parameterizing coverings of a fixed Veech surface. To make the paper self contained we derive the necessary asymptotic formulas from the Gutkin-Judge formula. As an application of the method we define and analyze d-symmetric elliptic differentials and their ...

There is a natural action of SL(2,R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup U = {( 1 ∗ 0 1 )} . We classify the U -invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2,R)-orbit of the set of branched covers of a fixed Veech surface.) For the U -action on these submanifolds, this is...

The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. This research was inspired by the analysis of Calabi–Yau manifolds that arise as smooth models of double covers of P branched along singular octic surfaces ([4, 3]). It is of considerable interest to determine the Hodge numbers for these manifolds, but t...

In this paper we show the existence of a Chow–Künneth decomposition for the moduli stack of stable curves with marked points Mg,r, for low values of g, r. We also look at the moduli space R3,2 of double covers of genus three curves, branched along 4 distinct points. We first obtain a birational model of R3,2 as a group quotient of a product of two Grassmannian varieties. This provides a Chow–Kü...

Consider a finite groupG acting on a Riemann surface S, and the associated branched Galois cover πG : S → Y = S/G. We introduce the concept of geometric signature for the action of G, and we show that it captures much information: the geometric structure of the lattice of intermediate covers, the isotypical decomposition of the rational representation of the group G acting on the Jacobian varie...

We present a complex matrix gauge model defined on an arbitrary two-dimensional orientable lattice. We rewrite the model’s partition function in terms of a sum over representations of the group U(N). The model solves the general combinatorial problem of counting branched covers of orientable Riemann surfaces with any given, fixed branch point structure. We then define an appropriate continuum l...