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 $g\geqslant 2$.

scholarly journals Thomae-Weber Formula: Algebraic Computations of Theta Constants

2019 ◽  
Author(s):  
Turku Ozlum Celik
Keyword(s):  
Hyperelliptic Curve ◽  
Algebraic Method ◽  
Hyperelliptic Curves ◽  
Fourth Power ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Singular Quadric ◽  
Level 2 ◽  
Theta Characteristic

Abstract We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated with a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete level 2 structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.

scholarly journals On pliability of del Pezzo fibrations and Cox rings

2017 ◽  
Vol 2017 (723) ◽  
Author(s):  
Hamid Ahmadinezhad
Keyword(s):  
Moduli Space ◽  
Toric Varieties ◽  
Projective Spaces ◽  
Low Rank ◽  
Del Pezzo Surfaces ◽  
Coordinate Ring ◽  
Birational Transformations ◽  
Cox Rings ◽  
Del Pezzo

AbstractWe develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over

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.

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 Derived categories of quintic del Pezzo fibrations

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Fei Xie
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Derived Categories ◽  
Derived Category ◽  
The Other ◽  
Del Pezzo Surfaces ◽  
Linear Section ◽  
Projective Duality ◽  
Del Pezzo ◽  
Space Approach

AbstractWe provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

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.

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