Learn · Introduction To Algebraic Geometry
Lecture 22 of 75Week 4
Projective & Quasi-Projective Varieties
Prof. Arijit Dey · IIT Madras
Credits and licence
- From
- Introduction To Algebraic Geometry, IIT Madras (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- Tweens profiles and older. How courses are placed
All the lectures
- Topics in Commutative Algebra - Part 1
- Topics in Commutative Algebra - Part 2
- Topics in Commutative Algebra - Part 3
- Topics in Commutative Algebra - Part 4
- Topics in Commutative Algebra - Part 5
- Tutorial 1: Cayley-Hamilton Theorem, Nakayama's Lemma
- Topics in Commutative Algebra: Part 6
- Topics in Commutative Algebra: Part 7
- Topics in Commutative Algebra: Part 8
- Affine Algebraic Sets - Part 1
- Affine Algebraic Sets - Part 2
- Tutorial 2: Noether Normalization Lemma, Some Important Results in Dimension Theory
- Regular Morphisms
- Abstract Algebraic Sets
- Zariski Topology on Affine Space
- Irreducible Affine Algebraic Sets
- Ring of Regular Functions
- Projective Space
- Tutorial 3: Some Applications of Dimension Theory
- Zariski Topology on Projective Space
- Affine Open Cover of Projective Space
- Projective & Quasi-Projective Varieties
- Regular Functions on Quasi-Projective Varieties
- Presheaves & Sheaves
- Morphism of Presheaves/Sheaves
- Tutorial 4: More Applications of Dimension Theory
- A Brief Overview of Sheaf Theory - Part 1
- A Brief Overview of Sheaf Theory - Part 2
- A Brief Overview of Sheaf Theory - Part 3
- Prevarieties
- Sheaf of Regular Functions
- Ring of Germs of Regular Functions at a point, Field of Rational Functions
- Tutorial 5: Sheafification
- Ring of Regular Functions, Local Ring at a Point, and Field of Rational Functions of an Affine Variety
- Equivalence of Categories of the Category of Affine Varieties over a Field k and the Category of Finitely Generated Integral Domains over k
- Equivalence of Categories of the Category of Affine Varieties over a Field k and the Category of Finitely Generated Integral Domains over k (Contd.)
- Some Examples, Open Immersions & Closed Immersions
- Product of Quasi-affine Varieties
- Diagonal Morphisms, Abstract Varieties
- Tutorial 6: Normal Varieties & Normalization of a Variety
- Projective Varieties Revisited: Part 1
- Projective Varieties Revisited: Part 2
- Global Regular Functions on Projective Varieties are Constants : Part 1
- Global Regular Functions on Projective Varieties are Constants : Part 2
- Product of Prevarieties : Part 1
- Product of Prevarieties : Part 2
- Tutorial 7 : A Result on Tensor Products of k-algebras
- Morphisms of Prevarieties : Part 1
- Morphisms of Prevarieties : Part 2
- Finite Morphisms : Part 1
- Finite Morphisms : Part 2
- Fiber Products
- Tutorial 8 : Finite Morphisms
- Immersions
- Fiber Products, Separatedness
- Criterion of Separatedness
- Proper Morphisms & Complete Varieties
- Tutorial 9 : Closed Immersions & Graph of a Morphism
- Projective Varieties are Complete
- Zariski Tangent Space, Singular & Nonsingular Points
- Smooth Points Form a Non-empty Open Subset
- Blow-Ups, Rational Maps & Birational Maps
- Tutorial 10 : Zariski Tangent Space at a Point of an Affine Variety
- Blow-Ups ( Contd.)
- Smooth Morphisms
- Bertini's Theorem
- Sard's Theorem
- Tutorial 11 : Dimension of fiber of a morphism
- Introduction to Affine Schemes : Spectrum of a Ring
- Introduction to Affine Schemes : Topology on Spec A
- Introduction to Affine Schemes : Topology on Spec A ( Contd. )
- Introduction to Affine Schemes : Sheaf Structure on Spec A
- Abstract Non-singular Curves : Part 1
- Abstract Non-singular Curves : Part 2
- Tutorial 12 : Extension of Regular Functions