Alphabeta MathBridgeland Stability and DT Frontiers
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- 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
- Sylow's Theorems, p-Groups and Nilpotent Groups
- 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
- Geometric Fixed Parts And Gauss Manin Semisimplicity
- Higher-Dimensional Resolution of Singularities
- Intersection Products on Smooth Projective Surfaces
- Logarithmic Hodge Complexes And Smooth Open Weights
- Mixed Hodge Structures And Strictness
- Morphisms Local Rings and Rational Maps of Affine Varieties
- Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties
- Normal Varieties, Normalization, and Zariski's Main Theorem
- Point Blowup Resolution on Arbitrary Regular Surfaces
- Products Segre and Veronese Embeddings and Grassmannians
- Projective Algebraic Sets Projective Morphisms and Cones
- Smooth-Projective Serre Duality and Flag-Variety Line Bundles
- Surface Riemann-Roch and the Hodge Index Theorem
- 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
- 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
- Analytic Continuation, Monodromy, and Riemann Surfaces
- Analyticity of Holomorphic Functions; Liouville and Morera
- Compact Kahler Hodge Foundations
- 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
- Domains of Holomorphy, Plurisubharmonicity and Pseudoconvexity
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Harmonic Functions and the Poisson Integral
- Holomorphic Functions of Several Complex Variables
- Hörmander Estimates and the Levi Problem
- Isolated Singularities and Laurent Series
- Projective Hodge Polarization Foundations
- Riemann Surfaces, Branched Maps, and Differentials
- Subharmonic Functions and the Dirichlet Problem
- The Argument Principle and Rouché's Theorem
- The Dolbeault Complex and Integral Solutions
- The Hartogs Phenomena
- 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
- 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
- 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
- Vector Fields Flows and Lie Derivatives
- Whitney Embedding Tubular Neighbourhoods and Approximation
Differential Topology
- Exotic Smooth Structures and Milnor Spheres
- Gradient Like Vector Fields and Morse Trajectories
- Handle Decompositions Duality and Rearrangement
- Morse Critical Points Hessians and Indices
- Morse Functions Critical Values and Genericity
- Reeb Stability and Global Foliation Constructions
- Relative smooth triangulation and bundle fitting
- Sublevel Deformation and the Handle Attachment Theorem
Fourier Analysis
Functional Analysis
- Banach Alaoglu Goldstine and Krein Milman
- Banach Valued Integration and the Radon Nikodym Property
- Banach-Space Differential Calculus and Banach Manifolds
- Bounded Linear Operators and Quotient Spaces
- Compact Operators and Riesz Schauder Theory
- Complex Norms and Realification
- Complex Subspace Realification
- Distributions Test Functions and Differentiation
- Dual Spaces Adjoint Operators and Annihilators
- Fourier Transform Convolution and Approximate Identities
- 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
- Tempered Distributions and the Fourier Transform
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- Unbounded Self Adjoint Operators and Stones Theorem
- Weak and Weak Star Topologies
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Delta Functors and Universality
- Derived Categories
- Derived Functors
- Differential Graded Algebras and Homological Foundations
- Double Complexes Exact Couples and Convergence
- Ext and Balanced Resolutions
- Grothendieck Groups and Graded Cartan Pairings
- Grothendieck Spectral Sequences and Computations
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Lefschetz Splitting In Bounded Triangulated Categories
- 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
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- 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
- Euclidean Surface Measure, Divergence, and Green Identities
- Fredholm Elliptic Problems and the Elliptic Spectrum
- Fundamental Solutions Newtonian Potentials and Green Functions
- Harmonic Functions and Mean Values in Rn
- Interior and Boundary Sobolev Elliptic Regularity
- Lax--Milgram and Weak Elliptic Solutions
- Maximum Principles Harnack and Liouville in Rn
- Poisson Problems and Interior Harmonic Estimates
- Rellich Kondrachov and Sobolev Compactness
- Schauder and Lᵖ Elliptic Estimates
- Smooth Approximation and Sobolev Extension
- Sobolev Poincare and Morrey Inequalities
- Sobolev Traces and Zero Boundary Values
- Weak Derivatives and Sobolev Spaces
Quantum Field Theory Mathematics
- ADHM Linear Data, Positive-FI Quotients and Framed Monads
- Faithful flatness of finite-type algebraic-to-analytic local rings
- Finite-dimensional coherent cohomology on compact projective space
- Local coherent acyclicity on sufficiently small polydiscs
- Oka coherence and local coherent free resolutions
- Projective Dolbeault comparison and scalar structure-sheaf cohomology
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
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Further Trigonometric Identities and Inverse Functions
- Improper and Parameter-Dependent Multiple Integrals
- Improper 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
- Regular Surfaces and Surface Integrals
- 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 Real Gamma and Beta Functions
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
- Trigonometric and Oscillatory Examples in One Variable
- Volumes of Elementary Solids and Solids of Revolution
Representation Theory of Groups
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations
- Analytic Derived Six Operations And Duality
- 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 Covering Spaces
- 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
Bridgeland stability condition with support property
Definition
Local Calabi–Yau threefold from a canonical line bundle
Definition
Quiver potential and King theta stability data
Definition
Gamma-BMT inequality and curve Brill–Noether number
Lemma
A three-dimensional perfect residue-field complex forces regularity
Lemma
Curve-nef ample perturbations by a projective Nakai construction
Lemma
Algebraic surface derived McKay in characteristic zero
Lemma
Homogeneous representations preserve slope semistability in characteristic zero
Lemma
Polygraph common ideal bases and freeness of every alternating-ideal power
Lemma
Regular-base acyclicity for finite-length perfect complexes
Lemma
Smooth surface Hilbert schemes, semismallness and the canonical norm line
Lemma
Classical and nonhyperelliptic vector-bundle Brill–Noether bounds
Lemma
Canonical codimension-two surface sections have a nonzero first jet
Lemma
Canonical curve section-ring generation by inner projection
Lemma
Ample divisor relative handles and primitive integral restriction
Lemma
Coherent duality and rational slope-tilt heart noetherianity
Definition
Motivic Hall scattering data for a quiver with potential
Definition
Source data for local P1 by P1 DT and analytic wall crossing
Theorem
Local structure of the stability-condition space
Lemma
Finite-group induction of invariant stability conditions
Lemma
Analytic sector factorization and component propagation
Lemma
Log adjoint and nef big vanishing on smooth projective varieties
Lemma
Projective derived McKay for the CY threefold orbifolding input
Lemma
Global CY threefold flop model from perverse-point Quot schemes
Lemma
Ordinary slope semistable restriction to sufficiently high general surfaces
Lemma
Cohen–Macaulay isospectral schemes give all-n surface Hilbert kernel equivalence
Lemma
Nested planar Hilbert schemes from universal ideal matrices
Lemma
Smooth fundamental divisors from canonical double cuts and first jets
Lemma
Canonical section-ring lifting on K3 surfaces and Fano ladders
Lemma
The weighted degree one del Pezzo and double plane K3 ladder
Lemma
Tilt derived-dual truncation and canonical finite saturation
Definition
Strict Calabi–Yau manifold convention
Lemma
Hall HN factorization, regular logarithms and potential Euler integration
Lemma
Semiampleness of nef big line bundles on projective K trivial threefolds
Theorem
Ordinary sheaf Bogomolov Gieseker discriminant inequality
Lemma
Finite étale stability lifting and Hilbert norm-line intertwining
Theorem
All-n isospectral Hilbert Cohen–Macaulayness and the n!-flat family
Lemma
Ordered elliptic product stability and actual diagonal symmetries
Lemma
Picard-one Fano index divisor is the integral generator
Lemma
Three-dimensional subadjoint nonvanishing from a cubic Euler polynomial
Lemma
Low-dimensional nonvanishing from exact subadjoint resolution data
Lemma
Isolated lc-centre SNC vanishing and polynomial nonvanishing
Lemma
Projective minus-one contraction and crepant canonical-surface resolution
Lemma
Weak del Pezzo generation and the smooth degree-one pencil
Lemma
Weak quadric and projective-plane polarization generation
Lemma
K3 ample linear systems from Serre extensions and instability
Lemma
Ordinary zero-denominator BG positivity gives tilt HN, duality and the second heart
Theorem
Quiver stability scattering diagram
Lemma
Colored orbifold point vertex and central DT extraction
Lemma
Finite directed flops between smooth crepant threefold resolutions
Lemma
Cynk–Hulek local clusters and equivariant derived induction
Lemma
Isotrivial fibre stability restriction from torsion invariance
Theorem
All-n surface Hilbert symmetric-group kernel equivalence and norm-line compatibility
Lemma
Smooth surface tilt hearts, HN, discriminant, numerical-line transport and the line object
Lemma
Unique lc-place residue pushforward and subadjoint discriminant effectivity
Lemma
Fixed-resolution tie-breaking to one log canonical place
Lemma
Primitive Fano threefold generation from a canonical K3 section
Lemma
Rational tilt discriminant and numerical support from ordinary sheaf BG
Lemma
Exact semistable transport along numerical wall lines across slope hearts
Lemma
Rational Gamma-BMT gives geometric stability with finite-rank support
Lemma
Compact adjunction and induced stability with its actual image lattice
Lemma
Ramified finite-flat induction of stability and the elliptic double cover
Lemma
Ordered-root graph filtration for the symmetric quotient of a product of projective lines
Lemma
Rational elliptic product parameters and full intersection support
Lemma
Elliptic product isogeny rescaling and restriction order
Lemma
Projective-space stability with Bayer and restriction bounds
Lemma
Phase and cohomology bounds from semistable skyscrapers
Theorem
Stability conditions on smooth projective complex varieties
Theorem
Stability under specified smooth CY3 orbifolding
Theorem
Invariant stability and DT data for local P1 by P1
Lemma
Special product-charge uniqueness and actual block-permutation invariance
Lemma
K3 and del-Pezzo curve Brill–Noether bounds from actual boundary evaluation
Lemma
Curvette extraction and the effective cyclic different on a sole surface lc place
Lemma
Effective restriction to integral divisors from constructed tilt and duality theory
Lemma
The closed vertical Brill--Noether interval and its Gamma boundary
Lemma
Strict covers in the named Enriques and bielliptic Hilbert families
Lemma
Codimension-two subadjunction from explicit marked rational models
Lemma
Small-slope counterexample with uniform HN bounds on every integral hyperplane restriction
Lemma
Full Gamma BMT propagation from the closed vertical boundary
Theorem
Stability on listed higher-dimensional CY examples
Lemma
Canonical fundamental cuts and terminal first divisors in index-two fourfolds
Lemma
Very general K3 sections in the specified Fano regimes have the primitive ambient Picard group
Lemma
A smooth curve Brill–Noether gap gives strict threefold small-slope BG
Lemma
Smooth Fano fundamental ladders and the high-degree embedding bridge
Lemma
Primitive anticanonical K3 bridges for the nonprimitive Fano cases
Lemma
Smooth Brill–Noether gaps for the named weighted, Fano and cyclic families
Theorem
Gamma-BMT and stability on named polarized CY3 families
Sources
- Tom Bridgeland, Stability conditions on triangulated categories
- Emanuele Macrì and Benjamin Schmidt, Lectures on Bridgeland Stability
- Tom Bridgeland, Fabrizio Del Monte, and Luca Giovenzana, Invariant Stability Conditions on Certain Calabi-Yau Threefolds
- Tom Bridgeland, Scattering diagrams, Hall algebras and stability conditions
- Soheyla Feyzbakhsh, Naoki Koseki, Zhiyu Liu, and Nick Rekuski, Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves
- Bridgeland, King and Reid, The McKay correspondence as an equivalence of derived categories
- Bridgeland and Iyengar, A criterion for regularity of local rings
- Huneke and Lyubeznik, Absolute integral closure in positive characteristic
- Olivier Debarre, Introduction to Mori Theory (M2 course, 2010; revision 2011)
- The Stacks project, Theorem 37.53.4 (03H0), Stein factorization; Noetherian case
- Ramanan and Ramanathan, Some remarks on the instability flag
- Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, arXiv math/0010246, sections3.2--3.8 and4.1--4.11; local constructions supplied below
- Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, retrieved arXiv math/0010246, sections 3.8 and 5.1; explicit local constructions below
- Brambila-Paz, Grzegorczyk and Newstead, Geography of Brill–Noether loci for small slopes
- Feyzbakhsh, Koseki, Liu and Rekuski, Stability conditions on Calabi–Yau threefolds via Brill–Noether theory of curves
- Mella, Existence of good divisors on Mukai varieties
- Miles Reid, MA4L7 Algebraic Curves, Part 3 (2022)
- Claire Voisin, Hodge Structures and the Topology of Algebraic Varieties (Harvard revision, 2021)
- Bayer, Macrì and Toda, Bridgeland Stability Conditions on Threefolds I
- Maxim Kontsevich and Yan Soibelman, Analyticity and resurgence in wall-crossing formulas
- Maxim Kontsevich and Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
- Bridgeland, Del Monte and Giovenzana, Invariant Stability Conditions on Certain Calabi-Yau Threefolds
- Kontsevich and Soibelman, Analyticity and resurgence in wall-crossing formulas
- Demailly, Complex Analytic and Differential Geometry
- Fujino, Fundamental theorems for the log minimal model program
- Howard Nuer, Stability conditions on some families of Calabi–Yau threefolds via orbifolding
- Tom Bridgeland, Flops and derived categories
- The Stacks Project, Lemma 69.22.9 (Tag 0A4T)
- Hubert Flenner, Restrictions of semistable bundles on projective varieties, Commentarii Mathematici Helvetici 59 (1984), 635–650
- Mella, Existence of good divisors on Mukai varieties, Theorem 2.5
- Feyzbakhsh--Koseki--Liu--Rekuski, Stability conditions on Calabi--Yau threefolds via Brill--Noether theory of curves; direct section-ring lifting below
- FKLLR, Stability conditions on Calabi–Yau threefolds via Brill–Noether theory of curves
- Yiran Cheng, Bridgeland stability conditions on some higher-dimensional Calabi-Yau manifolds and generalized Kummer varieties
- Mark Gross and P. M. H. Wilson, Mirror symmetry via 3-tori for a class of Calabi-Yau threefolds
- The Stacks Project, schematic locus for reasonable algebraic spaces
- Bayer, Macri and Stellari, The space of stability conditions on abelian threefolds and on some Calabi–Yau threefolds
- Flenner, Restrictions of semistable bundles on projective varieties
- Yiran Cheng, Bridgeland stability conditions on some higher-dimensional Calabi–Yau manifolds and generalized Kummer varieties
- Yucheng Liu, Stability conditions on product varieties
- Dan Abramovich and Alexander Polishchuk, Sheaves of t-structures and valuative criteria for stable complexes
- Alexander Polishchuk, Constant families of t-structures on derived categories of coherent sheaves
- Mella, Existence of good divisors on Mukai varieties, fundamental-divisor convention
- Mella, Existence of good divisors on Mukai varieties, v1 nonvanishing context; cubic resolution argument below
- Kawamata, Subadjunction of Log Canonical Divisors for a Subvariety of Codimension 2, v1 Theorems 1 and 6; exact resolution data only
- Mella, Existence of good divisors on Mukai varieties, v1 Lemma 2.1 and Theorem 2.2
- Mella, Existence of good divisors on Mukai varieties, Lemma 5.1
- Stacks Project, Resolution of Surfaces, contraction context; direct projective construction using the supplied finite-cover purity theorem below
- Olivier Debarre, Surfaces K3
- János Kollár, Flops
- Bayer, Macrì and Stellari, The space of stability conditions on abelian threefolds, and on some Calabi–Yau threefolds
- Soheyla Feyzbakhsh, An effective restriction theorem via wall-crossing and Mercat’s conjecture
- Kawamata, Subadjunction of Log Canonical Divisors for a Subvariety of Codimension 2, v1 proof of Theorem 1 page 6; residue step constructed below
- Mella, Existence of good divisors on Mukai varieties, v1 Lemma 2.1
- Polishchuk, Constant families of t-structures on derived categories of coherent sheaves
- Chunyi Li, A remark on stability conditions on smooth projective varieties
- Howard Nuer, Stability conditions on some families of Calabi-Yau threefolds via orbifolding
- Bryan, Cadman, and Young, The orbifold topological vertex
- Cheng, Bridgeland stability conditions on some higher-dimensional Calabi–Yau manifolds and generalized Kummer varieties
- Soheyla Feyzbakhsh, Mukai’s program via wall-crossing
- Chunyi Li, On stability conditions for the quintic threefold
- Kawamata, Subadjunction of Log Canonical Divisors for a Subvariety of Codimension 2, v1 proof of Theorem 1; transverse positivity constructed below
- Keiji Oguiso and Stefan Schröer, Enriques manifolds
- Kawamata, Subadjunction of log canonical divisors for a subvariety of codimension 2
- Mella, Existence of good divisors on Mukai varieties, v1 Lemma 2.1 and Theorems 2.2--2.5
- Feyzbakhsh–Koseki–Liu–Rekuski, Stability conditions on Calabi–Yau threefolds via Brill–Noether theory of curves, named family geometry scope
- Chen, Chen and Chen, On quasismooth weighted complete intersections