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Summary: Introduction to projective varieties
by Enrique Arrondo(*)
Version of September 5, 2007
This is still probably far from being a final version, especially since I had no time yet to complete the
second part (which is so far not well connected with the first one). Anyway, any kind of comments are very
welcome, in particular those concerning the general structure, or suggestions for shorter and/or correct
proofs.
0. Algebraic background
1. Projective sets and their ideals; Weak Nullstellensatz
2. Irreducible components
3. Hilbert polynomial. Nullstellensatz
4. Graded modules; resolutions and primary decomposition
5. Dimension, degree and arithmetic genus
6. Product of varieties
7. Regular maps
8. Properties of morphisms
9. Resolutions and dimension
10. Ruled varieties
11. Tangent spaces and cones; smoothness
12. Transversality
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