On the structure of the Whittaker sublocus of the moduli space of algebraic curves

2006 ◽  
Vol 136 (2) ◽  
pp. 337-346
Author(s):  
Ernesto Girondo ◽  
Gabino González-Diez
Keyword(s):  
Differential Equations ◽  
Algebraic Equation ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Hyperelliptic Curves ◽  
Teichmuller Theory ◽  
Compactness Result ◽  
Teichmüller Theory ◽  
Fuchsian Differential Equations

We prove the compactness of the Whittaker sublocus of the moduli space of Riemann surfaces (complex algebraic curves). This is the subset of points representing hyperelliptic curves that satisfy Whittaker's conjecture on the uniformization of hyperelliptic curves via the monodromy of Fuchsian differential equations. In the last part of the paper we devote our attention to the statement made by R. A. Rankin more than 40 years ago, to the effect that the conjecture ‘has not been proved for any algebraic equation containing irremovable arbitrary constants’. We combine our compactness result with other facts about Teichmüller theory to show that, in the most natural interpretations of this statement we can think of, this result is, in fact, impossible.

ON THE CONNECTEDNESS OF THE LOCUS OF REAL ELLIPTIC-HYPERELLIPTIC RIEMANN SURFACES

2009 ◽  
Vol 20 (08) ◽  
pp. 1069-1080 ◽  
Author(s):  
JOSÉ A. BUJALANCE ◽  
ANTONIO F. COSTA ◽  
ANA M. PORTO
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Orbit Space ◽  
Previous Article ◽  
Connected Components ◽  
Hyperelliptic Involution ◽  
Hyperelliptic Surfaces ◽  
Genus One

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace [Formula: see text] of real elliptic-hyperelliptic algebraic curves in the moduli space [Formula: see text] of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space [Formula: see text] of real elliptic-hyperelliptic surfaces of genus > 5: we show that [Formula: see text] is connected if g is odd and has exactly two connected components if g is even; in both cases the closure [Formula: see text] of [Formula: see text] in the compactified moduli space [Formula: see text] is connected.

IV.—The Differential Equations Associated with the Uniformization of Certain Algebraic Curves

1958 ◽  
Vol 65 (1) ◽  
pp. 35-62 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Rational Function ◽  
Algebraic Equation ◽  
Algebraic Curves ◽  
Algebraic Equations ◽  
Branch Points ◽  
Unknown Parameters ◽  
Image Position ◽  
Automorphic Functions

SynopsisEvery algebraic equation can be uniformized by automorphic functions belonging to a certain group of bilinear transformations. In certain cases, such as for hyperelliptic equations, this group is a subgroup of the monodromic group of a differential equation of the formwhere R(z) is a rational function which, in general, contains unknown parameters as coefficients. A conjecture of E. T. Whittaker regarding the values of these parameters for the hyperelliptic case is proved for a wide variety of algebraic equations whose branch points possess certain symmetric properties, and is extended to equations of higher type. In several cases, the uniformizing functions belong to subgroups of the groups of the Riemann-Schwarz triangle functions.

Moduli Space

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Riemannian Metrics ◽  
Classifying Space ◽  
Compactness Criterion ◽  
Hyperbolic Structures ◽  
Isomorphism Classes ◽  
Aspherical Manifold ◽  
Finite Index Subgroup

This chapter focuses on the moduli space of Riemann surfaces. The moduli space parameterizes many different kinds of structures on Sɡ, such as isometry classes of hyperbolic structures on S, conformal classes of Riemannian metrics on S, biholomorphism classes of complex structures on S, and isomorphism classes of smooth algebraic curves homeomorphic to S. The chapter first considers the moduli space as the quotient of Teichmüller space before discussing the moduli space of the torus. It then examines the theorem (due to Fricke) that Mod(S) acts properly discontinuously on Teich(S), with a finite-index subgroup of Mod(S) acting freely such that M(S) is finitely covered by a smooth aspherical manifold. The chapter also looks at Mumford's compactness criterion, which describes what it means to go to infinity in M(S), and concludes by showing that M(Sɡ) is very close to being a classifying space for Sɡ-bundles.

ON THE PARTITION FUNCTION OF STRING THEORIES DEFINED ON ALGEBRAIC CURVES

1992 ◽  
Vol 07 (21) ◽  
pp. 5131-5154 ◽  
Author(s):  
FRANCO FERRARI
Keyword(s):  
Riemann Surface ◽  
String Theory ◽  
Partition Function ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Bosonic String ◽  
Branch Points ◽  
Bosonic String Theory ◽  
String Theories

In this paper the amplitudes of bosonic string theory on Riemann surfaces are studied taking the branch points as moduli. The case of a general Riemann surface of genus three is completely worked out, constructing the chiral determinants and the propagators. The chiral determinants and the partition function are given explicitly in terms of the moduli and of the parameters of the curve apart from a factor which is essentially a theta constant. The Beilinson-Manin formulae for Riemann surfaces whose moduli space is parametrized by branch points are discussed.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

scholarly journals The Volume of the Quiver Vortex Moduli Space

2021 ◽  
Author(s):  
Kazutoshi Ohta ◽  
Norisuke Sakai
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Gauge Theories ◽  
Target Space ◽  
Contour Integral ◽  
Quiver Gauge Theory ◽  
Volume Formula ◽  
Volume Operator ◽  
Space Volume

Abstract We study the moduli space volume of BPS vortices in quiver gauge theories on compact Riemann surfaces. The existence of BPS vortices imposes constraints on the quiver gauge theories. We show that the moduli space volume is given by a vev of a suitable cohomological operator (volume operator) in a supersymmetric quiver gauge theory, where BPS equations of the vortices are embedded. In the supersymmetric gauge theory, the moduli space volume is exactly evaluated as a contour integral by using the localization. Graph theory is useful to construct the supersymmetric quiver gauge theory and to derive the volume formula. The contour integral formula of the volume (generalization of the Jeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vorticity by the area of the Riemann surface divided by the intrinsic size of the vortex). We give some examples of various quiver gauge theories and discuss properties of the moduli space volume in these theories. Our formula are applied to the volume of the vortex moduli space in the gauged non-linear sigma model with CPN target space, which is obtained by a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abelian generalization of the quiver gauge theory and “Abelianization” of the volume formula.

scholarly journals THE RATIONALITY OF THE MODULI SPACE OF ONE-POINTED INEFFECTIVE SPIN HYPERELLIPTIC CURVES VIA AN ALMOST DEL PEZZO THREEFOLD

2017 ◽  
Vol 232 ◽  
pp. 121-150
Author(s):  
HIROMICHI TAKAGI ◽  
FRANCESCO ZUCCONI
Keyword(s):  
Moduli Space ◽  
Hyperelliptic Curves ◽  
Formula One ◽  
Del Pezzo

Using the geometry of an almost del Pezzo threefold, we show that the moduli space ${\mathcal{S}}_{g,1}^{0,\text{hyp}}$ of genus $g$ one-pointed ineffective spin hyperelliptic curves is rational for every $g\geqslant 2$.

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