Lecture Notes

SES # TOPICS
1 Course Introduction, Zariski Topology (PDF)
2 Affine Varieties (PDF)
3 Projective Varieties, Noether Normalization (PDF)
4 Grassmannians, Finite and Affine Morphisms (PDF)
5 More on Finite Morphisms and Irreducible Varieties (PDF)
6 Function Field, Dominant Maps (PDF)
7 Product of Varieties, Separatedness (PDF)
8 Product Topology, Complete Varieties (PDF)
9 Chow's Lemma, Blowups (PDF)
10 Sheaves, Invertible Sheaves on P1 (PDF)
11 Sheaf Functors and Quasi-coherent Sheaves (PDF)
12 Quasi-coherent and Coherent Sheaves (PDF)
13 Invertible Sheaves (PDF)
14 (Quasi)coherent Sheaves on Projective Spaces (PDF)
15 Divisors and the Picard Group (PDF)
16 Bezout's Theorem (PDF)
17 Abel-Jacobi Map, Elliptic Curves (PDF)
18 Kähler Differentials (PDF)
19 Smoothness, Canonical Bundles, the Adjunction Formula (PDF)
20 (Co)tangent Bundles of Grassmannians (PDF)
21 Riemann-Hurwitz Formula, Chevalley's Theorem (PDF)
22 Bertini's Theorem, Coherent Sheves on Curves (PDF)
23 Derived Functors, Existence of Sheaf Cohomology (PDF)
24 Birkhoff–Grothendieck, Riemann-Roch, Serre Duality (PDF)
25 Proof of Serre Duality (PDF)

All lecture notes in one file: 18.725 Lecture Notes (PDF)