scholarly journals The uniformization of the moduli space of principally polarized abelian 6-folds

2020 ◽  
Vol 2020 (761) ◽  
pp. 163-217
Author(s):  
Valery Alexeev ◽  
Ron Donagi ◽  
Gavril Farkas ◽  
Elham Izadi ◽  
Angela Ortega
Keyword(s):  
Weyl Group ◽  
Abelian Variety ◽  
Moduli Space ◽  
Projective Line ◽  
Geometric Description ◽  
Canonical Class ◽  
Hurwitz Space ◽  
The Way

AbstractStarting from a beautiful idea of Kanev, we construct a uniformization of the moduli space \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_{6}-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

scholarly journals THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

2020 ◽  
pp. 1-23
Author(s):  
MICHELE BOLOGNESI ◽  
NÉSTOR FERNÁNDEZ VARGAS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Hyperelliptic Curve ◽  
Geometric Description ◽  
Birational Model ◽  
Rank 2 ◽  
Semistable Vector Bundles ◽  
Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

scholarly journals Principal co-Higgs bundles on ℙ1

2020 ◽  
Vol 63 (2) ◽  
pp. 512-530 ◽  
Author(s):  
Indranil Biswas ◽  
Oscar García-Prada ◽  
Jacques Hurtubise ◽  
Steven Rayan
Keyword(s):  
Moduli Space ◽  
Borel Subgroup ◽  
Algebraic Groups ◽  
Projective Line ◽  
Higgs Bundles ◽  
Complex Projective Line ◽  
Theoretic Characterization ◽  
Complex Projective

AbstractFor complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

scholarly journals Cross-ratio Dynamics on Ideal Polygons

2020 ◽  
Author(s):  
Maxim Arnold ◽  
Dmitry Fuchs ◽  
Ivan Izmestiev ◽  
Serge Tabachnikov
Keyword(s):  
Hyperbolic Space ◽  
Moduli Space ◽  
Hyperbolic Plane ◽  
Projective Line ◽  
Cross Ratio ◽  
Poisson Structures ◽  
Completely Integrable ◽  
The Cross

Abstract Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.

scholarly journals THE STRONG FRANCHETTA CONJECTURE IN ARBITRARY CHARACTERISTICS

2003 ◽  
Vol 14 (04) ◽  
pp. 371-396
Author(s):  
STEFAN SCHRÖER
Keyword(s):  
Moduli Space ◽  
Rational Points ◽  
Picard Group ◽  
Moduli Space Of Curves ◽  
Canonical Class ◽  
Generic Curve ◽  
Picard Scheme

Using Moriwaki's calculation of the ℚ-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g ≥ 3. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties.

An Aspect of Icosahedral Symmetry

2002 ◽  
Vol 45 (4) ◽  
pp. 686-696 ◽  
Author(s):  
Jan Rauschning ◽  
Peter Slodowy
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Irreducible Module ◽  
Projective Line ◽  
Icosahedral Symmetry ◽  
3 Dimensional ◽  
Icosahedral Group ◽  
Regular Icosahedron

AbstractWe embed the moduli space Q of 5 points on the projective line S5-equivariantly into (V), where V is the 6-dimensional irreducible module of the symmetric group S5. This module splits with respect to the icosahedral group A5 into the two standard 3-dimensional representations. The resulting linear projections of Q relate the action of A5 on Q to those on the regular icosahedron.

scholarly journals Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties

2019 ◽  
Vol 2019 (750) ◽  
pp. 123-156 ◽  
Author(s):  
Thomas H. Lenagan ◽  
Milen T. Yakimov
Keyword(s):  
Weyl Group ◽  
Maximal Torus ◽  
Explicit Description ◽  
Twist Map ◽  
Schubert Cell ◽  
Quantum Clusters ◽  
Prime Factors ◽  
Quantum Version ◽  
Large Families ◽  
The Way

Abstract The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac–Moody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construct large families of quantum clusters for all of these algebras and the quantum Richardson varieties associated to arbitrary symmetrizable Kac–Moody algebras and all pairs of Weyl group elements. Along the way we develop a quantum version of the Fomin–Zelevinsky twist map for all quantum Richardson varieties. Furthermore, we establish an explicit relationship between the Goodearl–Letzter and Cauchon approaches to the descriptions of the spectra of symmetric CGL extensions.

scholarly journals Finite Generation of the Algebra of Type A Conformal Blocks via Birational Geometry

2018 ◽  
Author(s):  
Han-Bom Moon ◽  
Sang-Bum Yoo
Keyword(s):  
Moduli Space ◽  
Type A ◽  
Projective Line ◽  
Birational Geometry ◽  
Finite Generation ◽  
Conformal Blocks ◽  
Parabolic Bundles ◽  
Mori Dream Space ◽  
Birational Model ◽  
Effective Cone

Abstract We study the birational geometry of the moduli space of parabolic bundles over a projective line, in the framework of Mori’s program. We show that the moduli space is a Mori dream space. As a consequence, we obtain the finite generation of the algebra of type A conformal blocks. Furthermore, we compute the H-representation of the effective cone that was previously obtained by Belkale. For each big divisor, the associated birational model is described in terms of moduli space of parabolic bundles.

scholarly journals DEFORMATIONS OF FOUR-DIMENSIONAL LIE ALGEBRAS

2007 ◽  
Vol 09 (01) ◽  
pp. 41-79 ◽  
Author(s):  
ALICE FIALOWSKI ◽  
MICHAEL PENKAVA
Keyword(s):  
Lie Algebras ◽  
Moduli Space ◽  
Geometric Description ◽  
Exceptional Points ◽  
Versal Deformations

We study the moduli space of four-dimensional ordinary Lie algebras, and their versal deformations. Their classification is well known; our focus in this paper is on the deformations, which yield a picture of how the moduli space is assembled. Surprisingly, we get a nice geometric description of this moduli space essentially as an orbifold, with just a few exceptional points.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

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