Related papers: Factorization semigroups and irreducible component…
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,…
This article is a continuation of the article with the same title (see arXiv:1003.2953v1). Let {\rm $\text{HUR}_{d,t}^{G}(\mathbb P^1)$} be the Hurwitz space of degree $d$ coverings of the projective line $\mathbb P^1$ with Galois group…
We study the irreducible components of special loci of curves whose group of symmetries is given as certain group extension. We introduce some relative Hurwitz data, which we show by using mixed \'etale cohomology theory, identifies some…
For a fixed finite group $G$, we study the fields of definition of geometrically irreducible components of Hurwitz moduli schemes of marked branched $G$-covers of the projective line. The main focus is on determining whether components…
A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each…
Going beyond the studies of single and double Hurwitz numbers, we report some progress towards studying Hurwitz numbers which correspond to ramified coverings of the Riemann sphere involving three nonsimple branch points. We first prove a…
We study the components of the Hurwitz scheme of ramified coverings of $\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the…
In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of $\Delta_n ^2$ where the elements of the factorization are semi-frame are Hurwitz equivalent. The results…
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…
In 1891, Hurwitz introduced the enumeration of genus $g$, degree $d$, branched covers of the Riemann sphere with simple ramification over prescribed points and no branching elsewhere. He showed that for fixed degree $d$, the enumeration…
Hurwitz numbers count covers of curves satisfying fixed ramification data. Via monodromy representation, this counting problem can be transformed to a problem of counting factorizations in the symmetric group. This and other beautiful…
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…
Let $\mathcal{H}$ be a Hurwitz space that parametrises holomorphic maps to $\mathbb{P}^1$. Abramovich, Corti and Vistoli, building on work of Harris and Mumford, describe a compactification $\overline{\mathcal{H}}$ with a natural boundary…
We introduce and develop a language of semigroups over the braid groups for a study of braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application we give a new proof of Orevkov's theorem on…
The universal centralizer of a semisimple algebraic group is the family of centralizers of regular elements, parametrized by their conjugacy classes. When the group is of adjoint type, we construct a smooth, log-symplectic fiberwise…
The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and…
We compute the Picard groups with integral coefficients of the Hurwitz stacks parametrizing degree $4$ and $5$ covers of $\mathbb{P}^1$. As a consequence, we also determine the integral Picard groups of the Hurwitz stacks parametrizing…
The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to…
We introduce a ramified covering of small categories, and we show three properties of the notion: the Riemann-Hurwitz formula holds for a ramified covering of finite categories, the zeta function of $C$ divides that of $\widetilde{C}$ for a…
Neretin and Segal independently defined a semigroup of annuli with boundary parametrizations, which is viewed as a complexification of the group of diffeomorphisms of the circle. By extending the parametrizations to quasisymmetries, we show…