Alphabeta MathStable Curves, Algebraic Virtual Classes and DT/PT Geometry
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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
- 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
- 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
- Abelian Varieties, Base Change, and Arithmetic Models
- Affine Algebraic Sets and Coordinate Rings
- Algebraic Differentials Separability and Smooth Local Presentations
- Analytic Constructible Sheaf Foundations
- Birational Morphisms, Contractions, and Surface Singularities
- Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch
- Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps
- Coherent Duality on Projective Cohen-Macaulay Schemes
- Complex Algebraic Stratified Topology
- Constructible Incidence And Cochain Models
- Dimension Constructible Images and Dimensions of Fibres
- Etale Covers and the Etale Fundamental Group
- Intersection Products on Smooth Projective Surfaces
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Point Blowup Resolution on Arbitrary Regular Surfaces
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Projective GIT from Linearized Line Bundles
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
- Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini
Algebraic Topology
- 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
- Simple Homotopy, Whitehead Groups, and Torsion
- 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
- Normalization Finiteness for Affine Domains
- 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
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Divisors, Riemann--Roch, and Duality
- Elliptic Functions and Complex Tori
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Hodge Theory on Compact Riemann Surfaces
- Holomorphic Functions of Several Complex Variables
- Isolated Singularities and Laurent Series
- Mittag-Leffler and Runge's Theorem
- Periods, Jacobians, and Abel--Jacobi Theory
- Riemann Surfaces, Branched Maps, and Differentials
- Simply Connected Plane Domains: the Grand Equivalence
- The Argument Principle and Rouché's Theorem
- 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 Residue Theorem and the Evaluation of Real Integrals
- The Riemann Sphere and Möbius Transformations
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
- Chern–Weil Theory and Characteristic Forms
- Connections Levi Civita and Parallel Transport
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Measurable Densities and Radon Volume on Manifolds
- 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
- The Gauss Bonnet Theorem for Riemannian Surfaces
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
- Gradient Like Vector Fields and Morse Trajectories
- Handle Decompositions Duality and Rearrangement
- Intersection Pairings Self Intersection and Euler Classes
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Morse Inequalities and the Handle Chain Complex
- Oriented and Mod Two Intersection Numbers
- Relative smooth triangulation and bundle fitting
- Sublevel Deformation and the Handle Attachment Theorem
- Vector Field Index Euler Characteristic and Poincare Hopf
Fourier Analysis
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Valued Integration and the Radon Nikodym Property
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Compact Self Adjoint Hilbert Schmidt and Trace Class Operators
- Complex Norms and Realification
- Complex Subspace Realification
- Distributions Test Functions and Differentiation
- Dual Spaces Adjoint Operators and Annihilators
- Finite Dimensional Normed Spaces and Riesz Lemma
- Hilbert Space Geometry and Riesz Representation
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Orthonormal Bases, Parseval and Fourier Series
- Reflexivity and Eberlein Smulian
- Schwartz Space and the Plancherel Theorem
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Unbounded Self Adjoint Operators and Stones Theorem
- Weak and Weak Star Topologies
Group Theory
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
- Graded Bimodules and Tensor Functors
- Grothendieck Groups and Graded Cartan Pairings
- Grothendieck Spectral Sequences and Computations
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Mapping Cones Cylinders and Chain Triangles
- Perfect Complexes and Triangulated Grothendieck Groups
- 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
- Complexification, Realification and Real Structures
- 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
Measure Theory
- Absolute Continuity and the Sharp Fundamental Theorem of Calculus
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Differentiation of Monotone Functions and the Vitali Covering Theorem
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Radon Measures and the Riesz Markov Kakutani Theorem
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- The Duality of Lᵖ and L^q
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Radon Nikodym Theorem and Lebesgue Decomposition
Number Theory
Partial Differential Equations
- Analytic Majorants and the Cauchy–Kovalevskaya Theorem
- Euclidean Surface Measure, Divergence, and Green Identities
- Fredholm Elliptic Problems and the Elliptic Spectrum
- Harmonic Functions and Mean Values in Rn
- Interior and Boundary Sobolev Elliptic Regularity
- Lax--Milgram and Weak Elliptic Solutions
- Partial Differential Equations and Characteristics
- Poisson Problems and Interior Harmonic Estimates
- Rellich Kondrachov and Sobolev Compactness
- Smooth Approximation and Sobolev Extension
- Sobolev Poincare and Morrey Inequalities
- Sobolev Traces and Zero Boundary Values
- Weak Derivatives and Sobolev Spaces
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Arc Length and Rectifiable Curves
- Areas of Elementary Plane Figures
- 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
- Convex and Semicontinuous Functions on Rⁿ
- Convexity
- 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
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- Blowups, Exceptional Divisors, and Strict Transforms
- Cartier and Weil Divisors Line Bundles and Picard Groups
- Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes
- Compact Support Sheaf Calculus
- Deformation Theory of Schemes and Obstruction Spaces
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Flat Smooth and Etale Morphisms
- Group Schemes of Finite Type over a Field
- Hilbert Functors and Projective Hilbert Schemes
- 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 Compact Connected Surfaces
- Compactness
- Compactness in Metric Spaces
- Complete Metrizability, Čech-Completeness, and Baire Category
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Convergence: Nets and Filters
- 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 Ascoli–Arzelà Theorem
- 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
Classical Deligne–Mumford stack and cotangent complex
Lemma
Finite scheme covers of separated finite-type Deligne–Mumford stacks
Lemma
Rational DM Chow groups and the deformation construction of refined Gysin
Definition
Rational Chow groups and zero-section Gysin maps
Definition
Stable n-pointed algebraic curves and the moduli functor
Definition
Stable pairs and DT ideal-sheaf moduli on projective threefolds
Definition
BF perfect obstruction theory and intrinsic-normal-cone class
Theorem
Proper Deligne–Mumford stack of stable pointed curves
Theorem
Proper algebraic stack of stable maps with finite stabilizers
Lemma
Trace-free Euler characteristic of a stable-pair complex
Lemma
Stable-pair deformations and fixed-determinant complex deformations
Definition
Equivariant K-theoretic virtual structure sheaf and determinant twist
Lemma
Symmetric-obstruction virtual class is the degree-zero Aluffi class
Theorem
Behrend–Fantechi algebraic virtual class from a perfect obstruction theory
Theorem
Pandharipande–Thomas stable-pair virtual class
Theorem
Algebraic stable-map descendants and deformation/splitting identities
Definition
Equivariant stable-pair moduli on quasi-projective threefolds
Theorem
PT relative-family virtual-class restriction
Theorem
Behrend weighted Euler identity for proper symmetric-obstruction schemes
Theorem
Associativity from genus-zero WDVV under a splitting axiom
Sources
- Behrend and Fantechi, The Intrinsic Normal Cone
- The Stacks Project, The Cotangent Complex, Chapter 92
- Edidin, Hassett, Kresch and Vistoli, Brauer groups and quotient stacks
- Behrend and Fantechi, The intrinsic normal cone
- The Stacks Project, Moduli of Curves, Sections 20–25
- Finn F. Knudsen, The Projectivity of the Moduli Space of Stable Curves, II: The Stacks M_{g,n}
- William Fulton and Rahul Pandharipande, Notes on Stable Maps and Quantum Cohomology
- Pandharipande and Thomas, Curve Counting via Stable Pairs in the Derived Category
- Martin Kool, Four Lectures on Stable Pairs
- The Stacks Project, Moduli of Curves
- Valery Alexeev and G. Michael Guy, Moduli of Weighted Stable Maps and Their Gravitational Descendants
- The Stacks Project, Lemma 36.32.6
- R. Pandharipande and R. P. Thomas, Curve Counting via Stable Pairs in the Derived Category
- Nikita Nekrasov and Andrei Okounkov, Membranes and Sheaves
- Feng Qu, Virtual pullbacks in K-theory
- Young-Hoon Kiem and Michail Savvas, Localizing Virtual Structure Sheaves for Almost Perfect Obstruction Theories
- The Stacks Project, Morphisms of Algebraic Stacks
- Kai Behrend, Donaldson–Thomas type invariants via microlocal geometry
- D. Huybrechts and R. P. Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes
- Kai Behrend and Yuri Manin, Stacks of Stable Maps and Gromov–Witten Invariants
- D. Holmes and Y. Schuler, Experiments with membranes, maps and sheaves