Related papers: Almost-Regular Dessins on a Sphere and Torus
Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called…
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with…
Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only…
Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary…
To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but…
Enumerating ramified coverings of the sphere with fixed ramification types is a well-known problem first considered by A. Hurwitz. Up to now, explicit solutions have been obtained only for some families of ramified coverings, for instant,…
Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and…
We introduce and study a semigroup structure on the set of irreducible components of the Hurwitz space of marked coverings of a complex projective curve with given Galois group of the coverings and fixed ramification type. As application,…
Branched covers between Riemann surfaces are associated with certain combinatorial data, and Hurwitz existence problem asks whether given data satisfying those combinatorial constraints can be realized by some branched cover. We connect…
In this note we provide a new partial solution to the Hurwitz existence problem for surface branched covers. Namely, we consider candidate branch data with base surface the sphere and one partition of the degree having length two, and we…
We give an alternative proof of the Hurwitz existence problem for branched covers of $\mathbb{P}^1$ in the case where the number of ramification points equals the number of branch points, that is, where all the ramification profiles are of…
We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group…
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic…
We introduce a natural structure of a semigroup (isomorphic to a factorization semigroup of the unity in the symmetric group) on the set of irreducible components of Hurwitz space of marked degree $d$ coverings of $\mathbb P^1$ of fixed…
We present the multi-matrix models that are the generating functions for branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus,…
In this paper, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target -- $\mathbb P^1$ with a list of $n$ points $q_1,\dots,q_n\in \mathbb P^1$ -- and partitions…
This is the first of two papers on the uniform asymptotics for real double Hurwitz numbers with triple ramification. Real double Hurwitz numbers with triple ramification count the number of real ramified coverings of the complex projective…
In this paper, we study a certain type of Hurwitz numbers which count branched covers over the Riemann sphere admitting several branch points with fixed ramification types, one branch point with a fixed number of preimages, and one branch…
Dessins d'enfants are combinatorial structures on compact Riemann surfaces defined over algebraic number fields, and regular dessins are the most symmetric of them. If G is a finite group, there are only finitely many regular dessins with…
We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain…