Related papers: Hurwitz Spaces and Moduli Spaces as Ball Quotients…
We describe hypergeometric functions of Deligne-Mostow type for open subsets of the configuration space of six points on P^2, induced from those for seven points on P^1. The seven point ball quotient example DM(2^5,1^2) does not appear on…
We extend results by Mirzakhani in [Mir07] to moduli spaces of Hurwitz covers. In particular we obtain equations relating Weil-Petersson volumes of moduli spaces of Hurwitz covers, Hurwitz numbers and certain Hurwitz cycles on…
We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne-Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of…
We consider the moduli space $\mathfrak{M}_{g,n}$ of Riemann surfaces of genus $g\ge0$ with $n\ge1$ ordered and directed marked points. For $d\ge 2g+n-1$ we show that $\mathfrak{M}_{g,n}$ is homotopy equivalent to a component of the…
In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and…
In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational…
We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points…
In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological…
Using Atiyah-Bott localization on the space of stable maps to the stack quotient $[\mathbb{P}^1/\mathbb{Z}_2]$, we find recursions that determine all Hodge integrals with descendent insertions at one marked point on the hyperelliptic locus…
All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In a…
In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces $\overline{\cal M}_{g,n}$. This allows us to prove via algebraic geometry a recursion between the…
In this paper we describe explicit generating functions for a large class of Hurwitz-Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz…
The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $\Delta^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$.…
Hurwitz spaces are moduli of isotopy classes of covers. A specific space is formed from a finite group G and C, r of its conjugacy classes and an equivalence relation \dagger. Components, interpret as a braid orbits on Nielsen classes.…
Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of…
These are lecture notes based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to a recent work on the…
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves.…
Moduli spaces of algebraic curves and closely related to them Hurwitz spaces, that is, spaces of meromorphic functions on the curves, arise naturally in numerous problems of algebraic geometry and mathematical physics, especially in…
Volumes of moduli spaces of hyperbolic cone surfaces were previously defined and computed when the angles of the cone singularities are at most 2pi. We propose a general definition of these volumes without restriction on the angles. This…
We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livn\'e…