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 (1.5), Nullstellensatz (1.11), decomposition in
irreducible components (1.21), polynomial maps and ring homomorphisms (1.39),
field of rational functions (1.49).
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 (2.14), projective closure (2.17).
Chapter 3. Projective plane curves.
Contents:
Bezout, intersection multiplicity, inflection points, group law on cubic curve.
Theorems:
Bezout's theorem (3.2), dimension of linear system of cubics (3.23),
group law (3.29).
Chapter 4. Dimension.
Contents:
dimension, integral extension, integral closure, going up and down,
Noether normalisation.
Theorems: Characterisation of intergral extensions (4.6),
Noether normalisation (4.26).
Chapter 5. Tangent space and nonsingularity.
Contents:
tangent space, Jacobian criterion, Zariski tangent space, singular point.
Theorems: Intrinsic characterisation of tangent space (5.7),
dimension of fibres (5.15).
Chapter 6. Lines on hypersurfaces.
Contents: Grassmann variety, Plücker coordinates, incidence correspondence,
closewdness of image of projective varieties, 27 lines on a
cubic surface and their configuration.
Theorems:
every cubic surface contains a line (6.6),
main theorem of elimination theory (6.7).
The numbers refer to the newest version of the notes.