A new look at the moduli space of stable hyperelliptic curves

2008 ◽  
Vol 264 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Donghoon Hyeon ◽  
Yongnam Lee
Keyword(s):  
Moduli Space ◽  
Hyperelliptic Curves ◽  
New Look

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 $g\geqslant 2$.

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.

scholarly journals Hyperelliptic Curves with Extra Involutions

2005 ◽  
Vol 8 ◽  
pp. 102-115 ◽  
Author(s):  
J. Gutierrez ◽  
T. Shaska
Keyword(s):  
Automorphism Group ◽  
Moduli Space ◽  
Hyperelliptic Curves ◽  
Rational Model ◽  
Field Of Moduli ◽  
Field Of Definition

AbstractThe purpose of this paper is to study hyperelliptic curves with extra involutions. The locusLgof such genus-ghyperelliptic curves is ag-dimensional subvariety of the moduli space of hyperelliptic curvesHg. The authors present a birational parameterization ofLgvia dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈Hgwith |Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈Lgsuch that the Klein 4-group is embedded in the reduced automorphism group ofp. Further, forg= 3, they show that for every moduli point p ∈H3such that |Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

scholarly journals Pencils on Surfaces with Normal Crossings and the Kodaira Dimension of

2021 ◽  
Vol 9 ◽  
Author(s):  
Daniele Agostini ◽  
Ignacio Barros
Keyword(s):  
Recent Result ◽  
Moduli Space ◽  
Hyperelliptic Curves ◽  
Upper Bounds ◽  
Kodaira Dimension ◽  
Canonical Divisor ◽  
Curves On Surfaces ◽  
Negative Kodaira Dimension

Abstract We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline {\mathcal {M}}_{g,n}$ is not pseudoeffective in some range, implying that $\overline {\mathcal {M}}_{12,6}$ , $\overline {\mathcal {M}}_{12,7}$ , $\overline {\mathcal {M}}_{13,4}$ and $\overline {\mathcal {M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline {\mathcal {M}}_{12,8}$ and $\overline {\mathcal {M}}_{16}$ . We also show that the moduli space of $(4g+5)$ -pointed hyperelliptic curves $\overline {\mathcal {H}}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the classification of moduli of pointed hyperelliptic curves with negative Kodaira dimension.

scholarly journals The p-rank strata of the moduli space of hyperelliptic curves

2011 ◽  
Vol 227 (5) ◽  
pp. 1846-1872 ◽  
Author(s):  
Jeffrey D. Achter ◽  
Rachel Pries
Keyword(s):  
Moduli Space ◽  
Hyperelliptic Curves

scholarly journals WHITHAM-TODA HIERARCHY AND N=2 SUPERSYMMETRIC YANG-MILLS THEORY

1996 ◽  
Vol 11 (02) ◽  
pp. 157-168 ◽  
Author(s):  
TOSHIO NAKATSU ◽  
KANEHISA TAKASAKI
Keyword(s):  
Moduli Space ◽  
Toda Lattice ◽  
Homogeneous Solution ◽  
Hyperelliptic Curves ◽  
Low Energy ◽  
Integrable Hierarchy ◽  
Quasiperiodic Solution ◽  
Yang Mills ◽  
Toda Hierarchy ◽  
Toda Lattice Hierarchy

The exact solution of N=2 supersymmetric SU(N) Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy (Whitham-Toda hierarchy) describes modulation of a quasiperiodic solution of the (generalized) Toda lattice hierarchy associated with the hyperelliptic curves over the quantum moduli space. The relation between the holomorphic pre-potential of the low energy effective action and the τ-function of the (generalized) Toda lattice hierarchy is also clarified.

On the Moduli of a Quantized Elastica in ℙ and KdV Flows: Study of Hyperelliptic Curves as an Extension of Euler's Perspective of Elastica I

2003 ◽  
Vol 15 (06) ◽  
pp. 559-628 ◽  
Author(s):  
Shigeki Matsutani ◽  
Yoshihiro Ônishi
Keyword(s):  
Moduli Space ◽  
Loop Space ◽  
Riemann Sphere ◽  
Hyperelliptic Curves ◽  
Energy Functional ◽  
Topological Spaces ◽  
Natural Topology ◽  
Kdv Hierarchy ◽  
Schwarz Derivative ◽  
Induced Topology

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli [Formula: see text] of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere ℙ. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space [Formula: see text]. Using the topology, [Formula: see text] is classified. Studies on a loop space in the category of topological spaces Top are well-established and its cohomological properties are well-known. As the moduli space of a quantized elastica can be regarded as a loop space in the category of differential geometry DGeom, we also proved an existence of a functor between a triangle category related to a loop space in Top and that in DGeom using the induced topology. As Euler investigated the elliptic integrals and its moduli by observing a shape of classical elastica on [Formula: see text], this paper devotes relations between hyperelliptic curves and a quantized elastica on ℙ as an extension of Euler's perspective of elastica.

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