Hurwitz spaces
نویسندگان
چکیده
1.1 The classical Hurwitz space and the moduli of curves The classical Hurwitz space first appeared in the work of Clebsch [5] and Hurwitz [17] as an auxiliary object to study the moduli space of curves. Let X be a smooth projective curve of genus g over C. A rational function f : X → P of degree n is called simple if there are at least n − 1 points on X over every point of P. Such a cover has exactly r := 2g + 2n − 2 branch points. Let Hn,r denote the set of isomorphism classes of simple branched covers of P of degree n with r branch points. Hurwitz [17] showed that the set Hn,r has a natural structure of a complex manifold. In fact, one can realize Hn,r as a finite unramified covering
منابع مشابه
Riemann-Hilbert problem associated to Frobenius manifold structures on Hurwitz spaces: irregular singularity
Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius ma...
متن کاملHurwitz Spaces and Braid Group Representations Partially Supported by the National Science Foundation
In this paper we investigate certain moduli spaces (\Hurwitz spaces") of branched covers of the Riemann sphere S 2 , and representations of nite index subgroups of the spherical braid group which arise from these Hurwitz spaces. (By spherical braid group, we mean the group of braids in the 2-sphere; we will refer to the more classical group of braids in the plane as the planar braid group.) Hur...
متن کاملGeometry of Meromorphic Functions and Intersections on Moduli Spaces of Curves
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. Then we show, how intersection numbers can be expressed via Hurwitz numbers. And then we obtain an algorithm expressing intersection numbers 〈τn,m ∏r−1 i=1 ...
متن کاملNew Frobenius Structures on Hurwitz Spaces in Terms of Schiffer and Bergmann Kernels
New family of flat potential (Darboux–Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius structure on Hurwitz spaces found by B. Dubrovin more than 10 years ago. The branch points of a ramifie...
متن کاملThe Irreducibility of Certain Pure-cycle Hurwitz Spaces
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 theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a...
متن کاملCounting Ramified Coverings and Intersection Theory on Hurwitz Spaces Ii (local Structure of Hurwitz Spaces and Combinatorial Results)
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 obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these ...
متن کامل