Mt822: Introduction to Algebraic Geometry

1. Algebraic varieties 2 1.1. Affine varieties 2 1.2. Projective varieties 2 1.3. Zariski topology 3 1.4. Algebraic geometry and analytic geometry 3 1.5. Singular varieties 3 1.6. Ideals 4 1.7. Regular functions and maps 5 2. Sheaves and cohomology 6 2.1. The Mittag-Leffler problem 7 2.2. Sheaves 7 2.3. Maps of sheaves 8 2.4. Stalks and germs 10 2.5. Cohomology of sheaves 11 3. Vector bundles, line bundles and divisors 16 3.1. Holomorphic vector bundles 16 3.2. Vector bundles on a variety and locally free sheaves 18 3.3. Divisors 19 3.4. Line bundles 20 3.5. Sections of a line bundle 22 4. Algebraic curves 24 4.1. The Riemann-Roch formula 24 4.2. The Riemann-Hurwitz formula 25 4.3. Genus formula of plane curves 27 4.4. Base point free and very ample line bundles 28 4.5. Canonical maps 31 4.6. An informal moduli counting 33 4.7. Special linear systems 34 4.8. Weierstrass points 36 4.9. Plane curves and their duals 41 5. Chern classes 45 5.1. Chow ring and rational equivalence 45 5.2. Formal calculation of Chern classes 47 5.3. The splitting principle 48 5.4. Determinantal varieties 50