Alphabeta MathProj Projective Schemes Twisting Sheaves and Ampleness — Examples
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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
- Proj Projective Schemes Twisting Sheaves and Ampleness
- 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
Example
Polynomial Proj charts
Example
A nilpotent irrelevant ideal gives empty Proj
Example
Twist transitions on the projective line
Example
A nowhere-vanishing section has empty zero divisor
Example
A projective hypersurface as a homogeneous quotient
Counterexample
O(-1) has no global generator
Counterexample
Two graded rings with the same Proj
Example
The conic map from O(2)
Counterexample
Global generation does not imply very ampleness
Example
Projective bundle of a trivial module
Sources
- 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, Properties of Schemes, Section 28.27 (Tag 01PS) and Morphisms of Schemes, Definition 29.38.1 (Tag 01VG)
- Ravi Vakil, The Rising Sea, 29 August 2022, Sections 17.4, 17.6 and 18.2