This book presents original peer-reviewed contributions from the London Mathematical Society (LMS) Midlands Regional Meeting and Workshop on 'Galois Covers, Grothendieck-Teichmüller Theory and Dessinsd'Enfants', which took place at the University of Leicester, UK, from 4 to 7 June, 2018. Within the theme of the workshop, the collected articles cover a broad range of topics and explore exciting new links between algebraic geometry, representation theory, group theory, number theory and algebraic topology. The book combines research and overview articles by prominent international researchers and provides a valuable resource for researchers and students alike.
Author(s): Frank Neumann, Sibylle Schroll
Series: Springer Proceedings in Mathematics & Statistics, 330
Publisher: Springer
Year: 2020
Language: English
Pages: 250
City: Cham
Preface
Contents
Galois Covers, Grothendieck-Teichmüller Theory and Dessins d'Enfants - An Introduction
1 Overview of Themes
2 Summaries of Individual Contributions
References
On the Number of Rational Points of Classifying Stacks for Chevalley Group Schemes
1 Classifying Stacks, ell-Adic Cohomology, Frobenius Morphisms and Zeta Functions
2 Cohomology of Classifying Stacks and the Theorems of Borel and Steinberg
3 Applications to Chevalley Group Schemes and Finite Groups of Lie Type
3.1 Classical and Exceptional Chevalley Groups
3.2 Twisted Chevalley or Steinberg Groups
References
Cyclic Symmetry on Complex Tori and Bagnera–De Franchis Manifolds
1 Introduction
2 An Exact Sequence in Factorial Rings
3 An Arithmetic Application
4 R(n) : = mathbbZ[Cn] = mathbbZ[x] / (xn -1) -Modules Which Are Torsion Free Abelian Groups
5 Fully Ramified Cyclic Coverings of the Projective Line and Associated Hodge Structures
6 Structure Theorem for Bagnera–De Franchis Manifolds
7 The Intersection Product for the Homology of Fully Ramified Cyclic Coverings of the Line
References
Second Fundamental Form of the Prym Map in the Ramified Case
1 Introduction
2 The 2nd Fundamental Form of the Prym Map
3 Totally Geodesic Submanifolds
References
Strongly Real Beauville Groups III
1 Introduction
2 Preliminaries
3 The Finite Simple Groups
4 Characteristically Simple Groups
5 Abelian and Nilpotent Groups
6 Doubly Hurwitz/Minimal Beauville Groups
7 Miscelenia
7.1 Purity
7.2 Higher Dimensional Analogues
7.3 Reflection Groups
7.4 Beauville Spectra
7.5 Beauville Graphs
References
The Pluricanonical Systems of a Product-Quotient Variety
1 Introduction
2 Minimal Models of Quotients of Product of Two Curves
3 Product Quotient Varieties Birational to Calabi-Yau Threefolds
4 Examples of Numerical Calabi-Yau Product-Quotient Threefolds
5 The Sheaves of Ideals Id on a Smooth Projective Variety with a Finite Group Action
6 The Sheaves of Ideals Id for Cyclic Quotient Singularities
7 A Calabi-Yau 3-Fold
8 A Fake Calabi-Yau 3-Fold
9 Some Minimal Surfaces of General Type
References
Joining Dessins Together
1 Introduction
2 Preliminaries
2.1 Hurwitz Dessins
2.2 Jordan's Theorem
2.3 From Coset Diagrams to Dessins
2.4 Macbeath–Hurwitz Curves
3 Some Small Dessins
4 Joins for Hurwitz Dessins
4.1 y-Handles and y-Joins
4.2 The Double Cover mathcalD(k)mathcalD
4.3 Multiple Joins
4.4 y-Handles from PSL2(q)
5 x-Joins
5.1 The Triple Cover mathcalD(x)mathcalD(x)mathcalD
5.2 A Non-trivial Example of an x-Join
5.3 x-Handles from PSL2(q)
6 Handles and Joins for the Modular Group
7 Handles and Joins for Dessins of All Types
References
Arithmetic Chern–Simons Theory I
1 The Arithmetic Chern–Simons Action: Basic Case
2 The Arithmetic Chern–Simons Action: Boundaries
3 The Arithmetic Chern–Simons Action: The p-adic Case
4 Remarks
5 Towards Computation
6 Motivation: L-Functions
7 Appendix: Conjugation on Group Cochains
References
Branch Stabilisation for the Components of Hurwitz Moduli Spaces of Galois Covers
1 Introduction
2 Algebraic Setting
3 The Hurwitz Monoid
4 The Genus Zero Case
5 Generalisations
6 The Tautological Lift
References
Dessins d'Enfants and Brauer Configuration Algebras
1 Introduction
2 Dessins d'Enfants
2.1 A Permutation Representation of Dessins
2.2 Belyĭ's Theorem
2.3 Galois Action on Dessins
3 From a Dessin d'Enfant to a Path Algebra
3.1 Admissible Ideals of KQ and Bound Quiver Algebras
4 From a Dessin d'Enfant to a Brauer Configuration Algebra
5 Examples of (Well-Known) Algebras and Their Dessins
5.1 Symmetric Nakayama Algebras
5.2 Koszul Brauer Configuration Algebras
6 Brauer Configuration Algebra of the Dual Dessin
6.1 Paths Formed by the Faces of a Dessin
6.2 Dual Dessins
7 Galois Action on Brauer Configuration Algebras
References
On the Elliptic Kashiwara–Vergne Lie Algebra
1 Introduction
2 The Elliptic Kashiwara–Vergne Lie Algebra from [AKKN]
3 A Reformulation of the div Condition
4 The Mould Theoretic mathfrakkrvell from ch11RS
References
