scholarly journals Moduli spaces of abstract and embedded Kummer varieties

2021 ◽  
pp. 2150054
Author(s):  
Mattia Galeotti ◽  
Sara Perna
Keyword(s):  
Equivalence Class ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
The Other ◽  
Dimension Formula ◽  
Moduli Stacks ◽  
Other Hand ◽  
Kummer Surfaces ◽  
General Fact

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].

scholarly journals A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

2021 ◽  
Author(s):  
Anna Gori ◽  
Alberto Verjovsky ◽  
Fabio Vlacci
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Higher Dimension ◽  
Geometric Constructions ◽  
Flat Torus ◽  
Flat Tori

AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

scholarly journals On some dimension formula for automorphic forms of weight one I

1982 ◽  
Vol 85 ◽  
pp. 213-221 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Fuchsian Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
The Other ◽  
Dimension Formula ◽  
Linearly Independent ◽  
Other Hand ◽  
Algebraic Properties ◽  
Lecture Notes ◽  
Weight One

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

scholarly journals Congruence classes and extensions of rings with an application to braces

2020 ◽  
pp. 2050010 ◽  
Author(s):  
Tomasz Brzeziński ◽  
Bernard Rybołowicz
Keyword(s):  
Equivalence Class ◽  
Congruence Relation ◽  
Ring Theory ◽  
The Other ◽  
Ideal Formula ◽  
Other Hand ◽  
Congruence Classes

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring [Formula: see text] by an ideal [Formula: see text] is a paragon in the truss [Formula: see text] associated to [Formula: see text]. Second, an extension of a truss by a one-sided module is described. Even if the extended truss is associated to a ring, the resulting object is a truss, never a ring, unless the module is trivial. On the other hand, if the extended truss is associated to a brace, the resulting truss is also associated to a brace, irrespective of the module used.

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 PERTURBATION OF THE GROUND VARIETIES OF c=1 STRING THEORY

1993 ◽  
Vol 08 (33) ◽  
pp. 3187-3199 ◽  
Author(s):  
DEBASHIS GHOSHAL ◽  
PORUS LAKDAWALA ◽  
SUNIL MUKHI
Keyword(s):  
String Theory ◽  
Moduli Space ◽  
The Self ◽  
The Other ◽  
Cosmological Perturbation ◽  
First Order ◽  
Singular Varieties ◽  
Universal Deformations ◽  
Other Hand

We discuss the effect of perturbations on the ground rings of c=1 string theory at the various compactification radii defining the AN points of the moduli space. We argue that perturbations by plus-type moduli define ground varieties which are equivalent to the unperturbed ones under redefinitions of the coordinates and hence cannot smoothen the singularity. Perturbations by the minus-type moduli, on the other hand, lead to semi-universal deformations of the singular varieties that can smoothen the singularity under certain conditions. To first order, the cosmological perturbation by itself can remove the singularity only at the self-dual (A1) point.

TWO REMARKS ON SUBVARIETIES OF MODULI SPACES

2008 ◽  
Vol 19 (02) ◽  
pp. 237-243 ◽  
Author(s):  
KIRTI JOSHI
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Abelian Varieties ◽  
Moduli Space Of Curves ◽  
Rank Zero ◽  
Zero Locus

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.

scholarly journals On some Dualities Concerning Abelian Varieties

1966 ◽  
Vol 27 (2) ◽  
pp. 447-462
Author(s):  
Hideyuki Matsumura ◽  
Masayoshi Miyanishi
Keyword(s):  
Abelian Variety ◽  
Multiplicative Group ◽  
Abelian Varieties ◽  
The Other ◽  
Other Hand ◽  
The Relationship

The group of extensions Ext(A, Ga) and Ext(A, Gm) of an abelian variety A by the additive or multiplicative group Ga, Gm have been investigated in detail ([9], [10], [GACC]). On the other hand, F. Oort [8] and M. Miyanishi [6] recently studied the groups Ext(Ga, A) and Ext(Gm, A). The purpose of the present paper is to clarify the relationship between these groups. Our results show that the latter groups can be derived from the former.

scholarly journals Characterization of products of theta divisors

2014 ◽  
Vol 150 (8) ◽  
pp. 1384-1412 ◽  
Author(s):  
Zhi Jiang ◽  
Martí Lahoz ◽  
Sofia Tirabassi
Keyword(s):  
Euler Characteristic ◽  
Abelian Varieties ◽  
The Other ◽  
Other Hand ◽  
Birational Equivalence ◽  
Points Of View ◽  
The One

AbstractWe study products of irreducible theta divisors from two points of view. On the one hand, we characterize them as normal subvarieties of abelian varieties such that a desingularization has holomorphic Euler characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$. On the other hand, we identify them up to birational equivalence among all varieties of maximal Albanese dimension. We also describe the structure of varieties $X$ of maximal Albanese dimension, with holomorphic Euler characteristic $1$ and irregularity $2\dim X-1$.

Second flip in the Hassett–Keel program: existence of good moduli spaces

2017 ◽  
Vol 153 (8) ◽  
pp. 1584-1609 ◽  
Author(s):  
Jarod Alper ◽  
Maksym Fedorchuk ◽  
David Ishii Smyth
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
General Criterion ◽  
Stable Curves ◽  
Algebraic Stack ◽  
Moduli Stacks ◽  
Local Description ◽  
Compositio Math

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a generalization of the Keel–Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne–Mumford stack. We apply this result to prove that the moduli stacks $\overline{{\mathcal{M}}}_{g,n}(\unicode[STIX]{x1D6FC})$ parameterizing $\unicode[STIX]{x1D6FC}$-stable curves introduced in [J. Alper et al., Second flip in the Hassett–Keel program: a local description, Compositio Math. 153 (2017), 1547–1583] admit good moduli spaces.

scholarly journals $$\pi _1$$ of Miranda moduli spaces of elliptic surfaces

2021 ◽  
Author(s):  
Michael Lönne
Keyword(s):  
Fundamental Group ◽  
Moduli Spaces ◽  
Classical Result ◽  
Projective Line ◽  
The Other ◽  
Fundamental Groups ◽  
Elliptic Surfaces ◽  
Generators And Relations ◽  
Moduli Stacks ◽  
Finite Presentations

AbstractWe give finite presentations for the fundamental group of moduli spaces due to Miranda of smooth Weierstrass curves over $${\mathbf {P}}^1$$ P 1 which extend the classical result for elliptic curves to the relative situation over the projective line. We thus get natural generalisations of $$SL_2{{\mathbb {Z}}}$$ S L 2 Z presented in terms of $$\Bigg (\begin{array}{ll} 1&{}1\\ 0&{}1\end{array} \Bigg )$$ ( 1 1 0 1 ) , $$\Bigg (\begin{array}{ll} 1&{}0\\ {-1}&{}1\end{array} \Bigg )$$ ( 1 0 - 1 1 ) on one hand and the first examples of fundamental groups of moduli stacks of elliptic surfaces on the other.Our approach exploits the natural $${\mathbb {Z}}_2$$ Z 2 -action on Weierstrass curves and the identification of $${\mathbb {Z}}_2$$ Z 2 -fixed loci with smooth hypersurfaces in an appropriate linear system on a projective line bundle over $${{\mathbf {P}}}^1$$ P 1 . The fundamental group of the corresponding discriminant complement can be presented in terms of finitely many generators and relations using methods in the Zariski tradition.

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