Course Identification

Good knowledge of linear algebra.

Basic familiarity with the notions of a ring, an ideal, a topological space.

[1] Affine varieties

[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.

The students will learn the powerful machinery and language of commutative algebra and algebraic geometry. This will give them preliminary access to the vast literature on algebraic geometry, and allow to start understanding and using the tremendous progress in the field that was achieved in the 20th centuary.

[1] Atiyah-Macdonalds "Introduction to commutative algebra"
[2] Eisenbud "Commutative Algebra With a View Toward Algebraic Geometry"
[4] Kempf "Algebraic varieties"
[5] A course by A. Gathmann:
http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf