scholarly journals Generic Torelli theorem for Prym varieties of ramified coverings

2012 ◽  
Vol 148 (4) ◽  
pp. 1147-1170 ◽  
Author(s):  
Valeria Ornella Marcucci ◽  
Gian Pietro Pirola
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Branch Points ◽  
Prym Variety ◽  
Prym Varieties ◽  
Torelli Theorem ◽  
Image Position ◽  
Prym Map ◽  
Double Coverings ◽  
Ramified Coverings

AbstractWe consider the Prym map from the space of double coverings of a curve of genus gwithrbranch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2is generically injective ifWe also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

scholarly journals COMPACTIFICATION OF THE PRYM MAP FOR NON-CYCLIC TRIPLE COVERINGS

2013 ◽  
Vol 24 (03) ◽  
pp. 1350015 ◽  
Author(s):  
HERBERT LANGE ◽  
ANGELA ORTEGA
Keyword(s):  
Moduli Space ◽  
Smooth Curve ◽  
Jacobian Variety ◽  
Prym Variety ◽  
Prym Varieties ◽  
Jacobian Varieties ◽  
Genus 2 ◽  
Proper Map ◽  
Dimension 2 ◽  
Prym Map

According to [H. Lange and A. Ortega, Prym varieties of triple coverings, Int. Math. Res. Notices2011(22) (2011) 5045–5075], the Prym variety of any non-cyclic étale triple cover f : Y → X of a smooth curve X of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map Pr of a moduli space [Formula: see text] of admissible S3-covers of genus 7 to the moduli space [Formula: see text] of principally polarized abelian surfaces. The main result is that [Formula: see text] is finite surjective of degree 10.

scholarly journals Rational points on abelian varieties over function fields and Prym varieties

2020 ◽  
Author(s):  
Abolfazl Mohajer
Keyword(s):  
Structure Theorem ◽  
Function Fields ◽  
Abelian Varieties ◽  
High Rank ◽  
Prym Variety ◽  
Projective Varieties ◽  
Prym Varieties ◽  
Weil Group ◽  
Galois Covers ◽  
Smooth Projective Varieties

AbstractIn this paper, using a generalization of the notion of Prym variety for covers of projective varieties, we prove a structure theorem for the Mordell–Weil group of abelian varieties over function fields that are twists of abelian varieties by Galois covers of smooth projective varieties. In particular, the results we obtain contribute to the construction of Jacobians of high rank.

scholarly journals Shimura subvarieties in the Prym locus of ramified Galois coverings

2021 ◽  
Author(s):  
Gian Paolo Grosselli ◽  
Abolfazl Mohajer
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Fixed Group ◽  
Galois Coverings ◽  
Base Curve ◽  
Prym Map

AbstractWe study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ A p , D of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $${{\mathbb {P}}}^1$$ P 1 . We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

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.

The generic Torelli theorem for the Prym map

1982 ◽  
Vol 67 (3) ◽  
pp. 473-490 ◽  
Author(s):  
Robert Friedman ◽  
Roy Smith
Keyword(s):  
Torelli Theorem ◽  
Prym Map

Holonomies of Yang–Mills connections for bundles on a surface with disconnected structure group

1994 ◽  
Vol 116 (2) ◽  
pp. 375-384 ◽  
Author(s):  
Johannes Huebschmann
Keyword(s):  
Moduli Space ◽  
Scalar Product ◽  
Additive Group ◽  
Closed Surface ◽  
Riemannian Metric ◽  
Compact Lie Group ◽  
Universal Central Extension ◽  
Yang Mills ◽  
Disconnected Structure ◽  
Image Position

AbstractLet Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, letbe the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.

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