Chapter 1. Affine algebraic varieties.
Contents:
algebraic sets, Noetherian rings, resultant, irreducible components,
polynomial maps, regular functions, localisation of rings, rational maps,
abstract affine varieties.
Theorems:
Hilbert Basis Theorem (1.6), Nullstellensatz (1.19), decomposition in
irreducible components (1.37), polynomial maps and ring homomorphisms (1.62),
field of rational functions (1.74).
Chapter 2. Projective varieties.
Contents:
projective space, homogeneous coordinates, irrelevant ideal, rational maps,
Segre embedding, maps defined by linear systems, blow-up of a point.
Theorems:
projective Nullstellensatz (2.19), projective closure (2.23).
Chapter 3. Projective plane curves.
Contents:
Bezout, intersection multiplicity, inflection points, group law on cubic curve.
Theorems:
Bezout's theorem (3.3), group law (3.31).
Chapter 4. Dimension.
Contents:
dimension, integral extension, integral closure, going up and down,
Noether normalisation.
Theorems: Characterisation of intergral extensions (4.8),
Noether normalisation (4.33).
Chapter 5. Tangent space and nonsingularity.
Contents:
tangent space, Jacobian criterion, Zariski tangent space, singular point,
closedness of image of projective varieties.
Theorems: Intrinsic characterisation of tangent space (5.11),
dimension of fibres (5.19).
Chapter 6. Lines on hypersurfaces.
Contents: Grassmann variety, Plücker coordinates, incidence correspondence,
27 lines on a
cubic surface and their configuration.
Theorems:
every cubic surface contains a line (6.10), a nonsingular cubic has 27 lines
(6.15).