SUGGESTED SYLLABUS - MATH 233 A-B-C

ALGEBRAIC GEOMETRY

Quarter 1.

Basics of Commutative algebra.

Classical Algebraic Geometry -

Algebraic varieties

Morphisms - finite maps , projections

Noether Normalization theorem

Birational maps - Blow ups, Cremona

The main idea of this quarter is to do as many examples as possible.

Textbook- first 2 Chapters of Shafarevich Basic Algebraic geometry,

Commutative algerbra - Athiyah , McDonald.

Quarter 2

Theory of Schemes , Algebraic Spaces, Sheaves,

Cartie and Weyl divisors, Dualising sheaf, Cohomology

Serre's theorems about finiteness and Serre duality,

Higher direct images, Standard exact sequences computation of cohomology

Chern classes and characters, Projections formulae, interesection theory.

Textbook- Hartshorne Chapter 2,3

Quarter 3

Algebraic curves and surfaces.

Rieman Roch and Canonical embedings,

Rieman Roch for Vector Bundles

Moduli spaces (Jacobian, High Ranks) , Torelli theorem.

Algebraic surfaces

Zarislci factorization, Stein Factorization

Riemann Roch for surfaces

Luroth Theorem and Enriquess criteria for rationality

Very light treatment of classification of surfaces

Textbook - Hartshorn Ch 4,5 and GH chapters 2,3 .

If time permits one can try to bit of Hodge theory, weak and hard Lefschetz or Moduli spaces of abelain varieties ( Rieman Frobenius relations) or Moduli spaces of bundles on algebraic surfaces.

This course should be done with parallel problem sessions, which will provide examples. In Quarter 3 proofs should be just outlined.