Alphabeta MathProj Projective Schemes Twisting Sheaves and Ampleness
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1 · Prerequisites
Abstract Algebra
- 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
- 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, 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
Category Theory
Commutative Algebra
Linear Algebra
Real Analysis
Scheme Theory
- Affine Schemes and the Structure Sheaf
- Diagonals Separated Morphisms and Valuative Uniqueness
- Fibre Products Base Change and Scheme Theoretic Fibres
- Finite Proper and Projective Morphisms
- Presheaves Sheaves Stalks and Sheafification
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Schemes Subschemes and Morphisms Locally of Finite Type
- Sheaf Operations Exactness Ringed Spaces and Module Pullback
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- 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
2 · Logical flowchart
3 · Items
Definition
Points of Proj of a graded ring
Definition
Graded shift convention for Proj
Definition
Standard opens of Proj
Lemma
Prime correspondence on a Proj chart
Theorem
Proj carries a scheme structure
Lemma
Standard opens are affine
Definition
Associated sheaf of a graded module on Proj
Definition
Relative very ampleness in the finite projective-space convention
Definition
Absolute ampleness by affine section opens
Definition
Global generation by the evaluation map
Lemma
A line-bundle section cuts an affine open inside an affine scheme
Lemma
Sections of a graded-module sheaf on a standard open
Definition
Twisting sheaf on Proj
Theorem
Projective space is Proj of a polynomial ring
Lemma
Affine-local graded algebras glue their Proj charts
Definition
Relative ampleness over an arbitrary base
Lemma
Extend a quasi-coherent section after multiplying by a power
Lemma
Ampleness is invariant under positive powers
Lemma
Finite pullback preserves absolute ampleness
Lemma
Empty Proj and irrelevant torsion
Theorem
Invertible twists for degree-one generated rings
Lemma
Proj is invariant under Veronese regrading
Lemma
Saturation detected on projective charts
Definition
Relative Proj of a graded quasi-coherent algebra
Definition
Zero scheme of a line-bundle section
Lemma
Relative very ampleness implies relative ampleness
Theorem
Closed subschemes of projective space and saturated ideals
Theorem
Relative Proj commutes with arbitrary base change
Theorem
Generating line-bundle sections define a morphism to projective space
Remark
Proj forgets irrelevant torsion and grading scale
Theorem
Maps to projective space equal generating line-bundle data
Lemma
A projective morphism has a relative Proj presentation
Theorem
Serre global-generation criterion for ampleness
Theorem
Segre embedding and its line bundle
Theorem
Veronese embedding pulls O(1) back to O(d)
Definition
Projective bundle in the quotient convention
Theorem
High powers of an ample line bundle embed a proper scheme
Theorem
Projective bundle represents line quotients
Sources
- The Stacks Project, Constructions of Schemes, Section 27.8 (Tag 01M3)
- Ravi Vakil, The Rising Sea, August 2022 draft, Section 4.5
- The Stacks Project, Constructions of Schemes, Section 27.10 (Tag 01MM)
- The Stacks Project, Constructions of Schemes, Section 27.9 (Tag 01MJ) and Section 27.10 (Tag 01MM)
- The Stacks Project, Morphisms of Schemes, Definition 29.38.1 (Tag 01VG) and Section 29.40 (Tag 01VU)
- Ravi Vakil, The Rising Sea, August 2022 draft, Section 17.6
- The Stacks Project, Properties of Schemes, Definition 28.27.1 (Tag 01PS)
- The Stacks Project, Properties of Schemes, Section 28.27 (Tag 01PS)
- Ravi Vakil, The Rising Sea, August 2022 draft, Section 17.4
- The Stacks Project, Properties of Schemes, Lemma 28.27.4 (Tag 01PV)
- The Stacks Project, Constructions of Schemes, Sections 27.8-27.21
- Ravi Vakil, The Rising Sea, 29 August 2022, Sections 4.5, 7.4, 9.3, 10.6, 17.4, 17.6, 18.2
- Gao-Zhang, Lectures on Algebraic Geometry, Chapter 5
- The Stacks Project, Morphisms of Schemes, Definition 29.38.1 (Tag 01VG)
- The Stacks Project, Properties of Schemes, Lemma 28.18.2 (Tag 01PW)
- The Stacks Project, Properties of Schemes, Sections 28.18, 28.27
- The Stacks Project, Morphisms of Schemes, Sections 29.38, 29.40
- The Stacks Project, Lemma 31.32.1 (Tag 0801)
- The Stacks Project, Properties of Schemes, Proposition 28.27.13 (Tag 01Q3)
- The Stacks Project, Modules, Lemma 17.9.7 (Tag 01BB)
- Ravi Vakil, The Rising Sea, August 2022 draft, Sections 17.4, 17.6
- The Stacks Project, Morphisms of Schemes, Sections 29.38 and 29.40, Lemma 29.40.3 (Tag 01VS)