Semester I.

1. Affine variaties

2. Rings, ideals, modules, Noetherianity, Hilbert basis theorem,
principle ideal domains, application: proof of primary
decomposition theorem and classification of finitely generated
commutative groups.

3. Algebraic sets, Zariski topology, Hilbert's Nulstellensatz

4. Morphisms

5. Sheaves of functions, Serre's lemma

6. Non-affine varieties, projective varieties

7. Dimension, Noether's normalization lemma, Chevalley theorem,
principal ideal theorem

8. Zariski tangent space, smooth varieties, blow up, 27 lines on a
smooth cubic surface

9. Product of varieties, separated and complete varieties, Chow's
lemma, valuation criteria.

Semester II.

1. Algebraic curves and their non-singular models

2. Riemann-Roch theorem - elementary approach

3. Sheaves, quasi-coherent sheaves, Serre's theorem, coherent
sheaves, Nakayama's lemma

4. Cohomologies

5. Higher cohomological operations with sheaves. Base change

5. Divisors, invertible sheaves, Picard group

6. Riemann-Roch theorem and applications.


Books:

1) Atiyah-Macdonalds "Introduction to commutative algebra"

2) Eisenbud "Commutative Algebra With a View Toward Algebraic
Geometry"

3) Kempf "Algebraic varieties"

4) A course by A. Gathmann
http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf

Learning Outcomes

The students will study the basic notions of commutative algebra
and algebraic geometry. They will learn to translate problems from
algebra to geometry and vice versa. They will acquire the ability
to use powerful algebraic techniques in geometric problems, and to
use their geometric intuition in abstract algebraic problems. the
language of algebraic geometry will gain the students access to
part the modern literature in the broad fields of algebra and
geometry.