AG from the Beginning

Here’s an index of all of the AG from the Beginning articles.  I reordered a little bit for coherent topics, and don’t THINK that I screwed up any dependencies (ie, anything should be understood in terms of things higher up on the page than it), but if I did, let me know and I’ll fix it.

Basic Set Up

1. Affine Varieties
2. Projective Varieties
3. Some Commutative Algebra and a bit of Geometry
4. Morphisms of Varieties
5. Algebraic Groups
6. Varieties over other Fields

Projective Geometry

1. Gradings on Rings and Modules
2. Hilbert Polynomials
3. Bezout’s Theorem
4. Elliptic Curves
5. Veronese Embedding
6. Segre Embedding
7. Grassmannians and Flag Varieties

Intrinsic Geometry

1. Tangent Spaces and Singular Points
2. Localization
3. Nakayama’s Lemma
4. Sheaves
5. Morphisms of Sheaves
6. Locally Ringed Spaces
7. Abstract Varieties
8. Complete Varieties and Chow’s Lemma

Computational Geometry

1. Groebner Bases and Buchberger’s Algorithm
2. Elimination and Extension Theorems
3. Resultants
4. Computing Hilbert Functions
5. Projective Elimination Theory

Relative Geometry

1. Sheaves of Modules
2. Locally Free Sheaves and Vector Bundles
3. Line Bundles and the Picard Group
4. Differential Forms and the Canonical Bundle

Linear Systems

1. Weil Divisors, Cartier Divisors and Line Bundles
2. Linear Systems
3. Bertini’s Theorem

Abstract Algebraic Geometry

1. Schemes
2. Proj of a Graded Ring and Basic Properties of Schemes

Cohomology

1. Cohomology of Sheaves
2. Serre Duality

Curves

1. Riemann-Roch Theorem for Curves
2. Hurwitz’s Theorem
3. Hurwitz’s Theorem on Automorphisms
4. Dual Curves
5. Plücker Formulas
6. Canonical Linear Systems
7. Geometric Form of Riemann-Roch
8. Constructing Nodal Curves

Schubert Calculus

1. Grassmannians Redux
2. Schubert Varieties
3. Schubert Classes and Cellular Cohomology
4. Pieri and Giambelli Formulas
5. Applications of the Schubert Calculus

Resolution of Singularities

1. Normalization and Normal Varieties
2. Blowing Things Up

Hilbert Schemes

1. Flat Modules and Morphisms
2. Representability of Functors
3. Moduli Spaces and Base Change
4. The Hilbert Scheme
5. Constructing the Hilbert Scheme II
6. Flattening Stratifications
7. Examples of Moduli Spaces

Matt’s Series on Group Schemes and Moduli

1. Group Schemes and Moduli I
2. Group Schemes and Moduli II
3. Group Schemes and Moduli III
4. Group Schemes and Moduli IV

Matt’s Post on Rational Varieties

Matt’s Series on Grothendieck-Riemann-Roch

1. Proper Maps and (Quasi) Projective Varieties
2. Direct Image Sheaves under Proper Maps
3. The Grothendieck Group of Coherent Sheaves on a Variety
4. The Grothendieck-Riemann-Roch Theorem, stated
5. The Grothendieck-Riemann-Roch Theorem, a proof-sketch

The Hilbert Polynomial Explained

https://rigtriv.wordpress.com/2008/01/11/elliptic-curves-2/