scholarly journals A Note on Planarity Stratification of Hurwitz Spaces

2015 ◽  
Vol 58 (3) ◽  
pp. 596-609 ◽  
Author(s):  
Jared Ongaro ◽  
Boris Shapiro
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Plane Curve ◽  
Minimal Degree ◽  
Hurwitz Spaces ◽  
Birational Map ◽  
Severi Varieties ◽  
Closed Riemann Surface

AbstractOne can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to ℂℙ2and a projection of the image curve froman appropriate pointp∊ ℂℙ2to the pencil of lines throughp. We introduce a natural stratiûcation of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.

scholarly journals Remarks on the Elliptic Case of the Mapping Theorem for Simply-Connected Riemann Surfaces

1955 ◽  
Vol 9 ◽  
pp. 17-20 ◽  
Author(s):  
Maurice Heins
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Riemann Surfaces ◽  
Simple Pole ◽  
Conformal Map ◽  
Elliptic Case ◽  
Extended Plane ◽  
Simply Connected ◽  
Conformal Equivalence

It is well-known that the conformal equivalence of a compact simply-connected Riemann surface to the extended plane is readily established once it is shown that given a local uniformizer t(p) which carries a given point p0 of the surface into 0, there exists a function u harmonic on the surface save at p0 which admits near p0 a representation of the form(α complex 0; h harmonic at p0). For the monodromy theorem then implies the existence of a meromorphic function on the surface whose real part is u. Such a meromorphic function has a simple pole at p0 and elsewhere is analytic. It defines a univalent conformal map of the surface onto the extended plane.

HARMONIC MAPS FROM A CLOSED SURFACE OF HIGHER GENUS TO THE UNIT 2-SPHERE

1993 ◽  
Vol 04 (02) ◽  
pp. 359-365 ◽  
Author(s):  
GUOFANG WANG
Keyword(s):  
Riemann Surface ◽  
Harmonic Maps ◽  
Closed Surface ◽  
Higher Genus ◽  
Closed Riemann Surface

We obtain the existence of harmonic maps of degree one from a closed Riemann surface of genus greater than 1 with a metric admitting a plane of symmetry to the unit 2-sphere.

Covering properties of open continuous mappings having two valences between Riemann surfaces

1995 ◽  
Vol 118 (2) ◽  
pp. 321-340 ◽  
Author(s):  
Abdallah Lyzzaik
Keyword(s):  
Riemann Surface ◽  
Finite Number ◽  
Riemann Surfaces ◽  
Embedding Theorem ◽  
Branch Points ◽  
Continuous Mappings ◽  
Counting Multiplicity ◽  
Covering Properties ◽  
Closed Riemann Surface ◽  
Finite Genus

AbstractLet be an open Riemann surface with finite genus and finite number of boundary components, and let be a closed Riemann surface. An open continuous function from to is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in either p or q times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all non-trivial functions having two valences between and .The object of this paper is to study the geometry of (p, q)-maps and establish a generalized embedding theorem which asserts that the image surfaces of (p, q)-maps embed in p-fold closed coverings possibly having branch points off the image surfaces.

scholarly journals Complete Classification of Degree 7 for Genus 1

2021 ◽  
pp. 594-603
Author(s):  
Peshawa M. Khudhur
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Symmetric Group ◽  
Compact Riemann Surface ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Branched Cover ◽  
Transitive Subgroup ◽  
Genus One

Assume that  is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples (  of permutations in the symmetric group , in which  and s generate a transitive subgroup G of  This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

Hyperelliptic d-Tangential Covers and d× d-Matrix KdV Elliptic Solitons

2018 ◽  
Vol 2020 (23) ◽  
pp. 9539-9558
Author(s):  
Armando Treibich
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Kdv Equation ◽  
Rational Surface ◽  
Geometric Condition ◽  
Kp Equation ◽  
Nodal Curves ◽  
Severi Varieties ◽  
The Kdv Equation ◽  
Elliptic Solitons

Abstract More than $40$ years ago I. Krichever developed the Theory of (vector) Baker–Akhiezer functions and devised a criterion for a $d$-marked compact Riemann surface to provide $d\times d$-matrix solutions to the KdV equation. Later on he also found a criterion for a $d$-marked curve to provide $d\times d$-matrix solutions to the Kadomtsev-Petviashvili (KP) equation, doubly periodic with respect to $x$, the 1st KP flow. In particular, when both criteria apply, one should obtain $d\times d$-matrix KdV elliptic solitons. It seems, however, that the latter issue has been completely neglected until very recently (cf. [10] where the $d=2$ case is treated). In this article we fix a complex elliptic curve $X=\mathbb{C}/\Lambda$, corresponding to a lattice $\Lambda \subset \mathbb{C}$, and define so-called hyperelliptic $d$-tangential covers as $d$-marked covers of $X$ satisfying a geometric condition inside their Jacobians. They satisfy Krichever’s criteria and give rise, therefore, to families of $d\times d$-matrix KdV elliptic solitons. We also construct an anticanonical rational surface ${\mathcal S}$ naturally attached to $X$, with a Picard group of rank $10$. It turns out that the former covers of $X$ correspond to rational irreducible curves in suitable divisor classes of ${\mathcal S}$. We thus reduce their construction to proving that the associated Severi Varieties (of rational irreducible nodal curves) are not empty. The final key to the problem consists in finding rational reducible nodal curves in the latter divisor classes that can be deformed to irreducible ones, according to A. Tannenbaum’s criterion (see [5]). At last we deduce, for any $d\geq 2$, infinite families (of arbitrary high genus and degree) of hyperelliptic $d$-tangential covers, giving rise to $d\times d$-matrix KdV elliptic solitons.

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