Algebraic Geometry

Algebraic geometry is a part of mathematics
dealing with algebraic sets (zero sets of 
polynomials) and polynomial maps between them,
The methods of algebraic geometry are so
powerful that this definition is often expanded 
to include algebraic number theory and geometry of 
complex manifolds, which are treated by means of 
algebraic geometry along with more classical 
subjects.

I will give an introduction to basic methods of
algebraic geometry and commutative algebra, treating
commutative algebra geometrically.

Syllabus

1. Hilbert Nulstellensatz. Algebraic sets and algebraic
varieties. Categories and functors.

2. Localization. Ideals. Maximal ideals, prime ideals,
nilradical. Strong Nulstellensatz. Equivalence of the 
categories of affine varieties and of finitely generated rings. 

3. Irreducible components. Noetherian rings.
Hilbert basis theorem. Modules over a ring.
Noether theorem on rings of invariants.

4. Tensor product of rings and tensor products
of modules over a ring. 

5. Zariski topology, dominant morphisms, integrally
closed domains. Integral closure and integral dependency.
Quotient of a variety by a group action.

6. Noether normalization lemma and its applications.

Basic knowledge of algebra (rings, fields, groups, 
vector spaces) is necessary. I would explain more advanced
algebraic notions, such as ideals and modules over a ring, 
but very briefly. Some knowledge of geometry (implicit function
theorem, definition of a manifold) would also be helpful.
The last two sections will be given only if time permits. 

Useful books:

Atiyah, MacDonald, "Introduction To Commutative Algebra"
Mumford, "Algebraic Geometry I: Complex Projective Varieties"
Kollar, "The structure of algebraic threefolds", chapters 1-6
(Bull. Amer. Math. Soc. 17 (1987), 211-273 
http://www.ams.org/journals/bull/1987-17-02/S0273-0979-1987-15548-0/ ).