Alphabeta MathGTS Elliptic Genus One Fibrations
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Fields and Cyclotomic Extensions
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules over a Principal Ideal Domain and the Canonical Forms
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Solvability by Radicals and Kummer Theory
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Dimension Constructible Images and Dimensions of Fibres
- Etale Covers and the Etale Fundamental Group
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Algebraic Topology
- Bocksteins Steenrod Squares and Cohomology Operations
- Chern and Pontryagin Classes by Splitting and Complexification
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Higher Homotopy Groups and Cofiber Sequences
- Homology Axioms Degree and Classical Applications
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences
- Local Coefficients, Twisted Homology, and Duality
- Obstruction Theory, Postnikov Towers, and Classifying Spaces
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- Stiefel Whitney and Euler Classes by Universal Constructions
- The Serre Spectral Sequence and Applications
- Topological Vector Bundles and Grassmannian Classification
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
- Algebraic Zariski Main for Quasi-Finite Morphisms
- Artinian Rings and Length
- Associated Primes and Primary Decomposition
- Dedekind Domains and Ideal Classes
- Depth and Cohen Macaulay Modules
- Flatness and Faithful Flatness
- Henselian Rings and Equicharacteristic Cohen Structure
- Homogeneous Resultants and Projective Intersection Length
- Integral Extensions and Going Up
- Inverse Limits and Noetherian Completion
- Koszul Complexes and Regular Sequences
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Rees Modules Artin Rees and Hilbert Samuel Theory
- Regular Local Rings and Homological Dimension
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Complex Analysis
- Compact Kahler Hodge Foundations
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- Riemann Surfaces, Branched Maps, and Differentials
- The Dolbeault Complex and Integral Solutions
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Compact Laplace-type operators and harmonic representatives
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemann Curvature and Riemannian Submanifolds
- Riemannian Metrics Length Distance and Volume
- Sard Theorem and Transversality
- Singular Cochains Mayer Vietoris and Smooth Singular Comparison
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The De Rham Complex Homotopy and Mayer Vietoris
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Grothendieck Spectral Sequences and Computations
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Projective and Injective Resolutions
- Spectral Sequences
- Tor Flatness and Global Dimension
- Triangulated Categories
- Universal Coefficients and Kunneth Theorems
- Yoneda Extensions and Homological Dimension
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Improper and Parameter-Dependent Multiple Integrals
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Divergence Theorem and Classical Stokes
- The Exponential Function
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
- Volumes of Elementary Solids and Solids of Revolution
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Cartier and Weil Divisors Line Bundles and Picard Groups
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Kahler Differentials Conormal Sequences and Infinitesimal Lifting
- Presheaves Sheaves Stalks and Sheafification
- Proj Projective Schemes Twisting Sheaves and Ampleness
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Residues Serre Duality for Curves and the Full Riemann Roch Theorem
- Riemann Roch for Curves via Euler Characteristics
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Cohomology Cech Cohomology and Comparison
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
- Smooth Proper Curves Divisors Genus and Ramification
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
String Theory
Topology
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Function Space Topologies and the Exponential Law
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
The small étale site of a scheme
Definition
Flat genus-one fibration
Lemma
A pointed genus-one curve has a regular group law over its ground field
Lemma
Jointly surjective étale families form a Grothendieck pretopology
Theorem
Proper flat genus-one fibrations have one-dimensional fibres
Definition
Elliptic fibration with a section
Definition
Presheaves and sheaves of sets on a site
Definition
Weierstrass model over a line bundle
Theorem
Every presheaf of sets on the small étale site has an associated sheaf
Lemma
Representable presheaves are sheaves on the small étale site
Theorem
Canonical bundle of a Weierstrass hypersurface
Definition
Étale torsors and contracted products
Theorem
Étale torsor classes form an abelian group
Definition
Torsors under a specified relative group scheme on the small étale site
Sources
- The Stacks Project, Étale Cohomology, Section 59.27
- J. S. Milne, Lectures on Étale Cohomology, Chapter I
- Timo Weigand, TASI lectures on F-theory, arXiv:1806.01854v1
- Volker Braun and David R. Morrison, F-theory on Genus-One Fibrations, arXiv:1401.7844v2
- J. S. Milne, Elliptic Curves, second edition
- The Stacks Project, More on Morphisms, Section 37.30
- Timo Weigand, TASI Lectures on F-theory, arXiv:1806.01854v1
- David R. Morrison and Daniel S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, arXiv:1208.2695v2
- The Stacks Project, Sites and Sheaves, Section 7.10