´úÊý´Ø(Ó¢ÎÄ°æ)ALGEBRAIC VARIETIES

1 Affine varieties 1.1 The Zariski topology 1.2 Irreducibility and decomposition 1.3 Finiteness properties and the Hilbert theorems 1.4 The affine category 1.5 The sheaf of regular functions 1.6 The product 1.7 The notion of a variety 1.8 Function fields and rational maps 1.9 Finite morphisms 1.10 Normalization 2 Local properties of varieties 2.1 Dimension 2.2 Smooth and singular points 2.3 Differentials and derivations 2.4 Sheaves of rings and modules 3 Projective varieties 3.1 Projective spaces 3.2 The Zariski topology on a projective space 3.3 The Segre embeddings 3.4 The \"proj\" construction and blowing up 3.5 Elimination theory and projections 3.6 The Veronese embeddings 3.7 Grassmannians 3.8 Fano varieties and the Gau¦Â map 3.9 Multiplicities of modules 3.10 Hilbert functions and Hilbert polynomials 4 Projective curves 4.1 Valuations and points 4.2 Divisors and invertible modules on a curve 4.3 The Riemann-Roch theorem 4.4 Residues and Serre duality 4.5 Some applications of the Riemann-Roch theorem . 4.6 The theorem of Riemann-Hurwitz Index Bibliography