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


  1. Cohomology of Sheaves
  2. Serre Duality


  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

14 Responses to AG from the Beginning

  1. Thanks a lot, this is exactly what I asked for :-)

    Now I’m happy, learning your stuff.

  2. Pingback: » Math 2.0 (Konrad Voelkel's Blog)

  3. Pingback: 2 good resources « Discovering Math

  4. Pingback: Algebraic Geometry Blogging « The Unapologetic Mathematician

  5. Pingback: MaBloWriMo | Gracious Living and the Two Meat Meal

  6. Pingback: Remember, Remember the Fifth of November « Big Numbers

  7. Pingback: I still want to learn Algebraic Geometry | Honglang Wang's Blog

  8. lucasbraune says:

    This is rather nice and useful. Well done!

  9. Pingback: Algebraic geometry from the beginning | Mathesis universalis

  10. Pingback: Charles Siegel and Matt DeLand’s AG from the Beginning | Theories and Theorems

  11. Amritasu Sinha says:

    1. Very useful and informative.
    2. I am very much helped.
    3. Sir can you please give a hint as how do I think of a rank 2 bundle on an elliptic curve?

    • Thanks, I’m glad people are still finding this blog useful, though I haven’t touched it in years. I’m no longer in mathematics so I don’t have any deep insights on rank 2 bundles on elliptic curves, but the main thing is that it’s a family of planes for each point on the curve. If there’s some vector space they all fit into (there should be for some finite N, though no clue how to bound it) then that means its a map from the elliptic curve to the Grassmannian. I vaguely recall something about vector bundles over abelian group schemes decomposing into direct sums of line bundles, but I can’t verify it. Good luck!

  12. Anonymous says:

    Thank you for compiling such an organised list of articles!! This is legit!!

  13. Gomathy says:

    This is really helpful, thanks a lot!

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s