Algebraic geometry 1000-M2AlgGeometry

(1) Affine algebraic sets and their zero ideals.
(2) Noetherian rings. Hilbert's basis theorem, Hilbert's nullstellensatz.
(3) Zariski topology. Noetherian and irreducible topological spaces.
(4) Regular functions on affine sets. Spectrum and maximal spectrum of rings.
(5) Regular maps between affine sets.
(6) Ring localizations. Rational functions on affine sets. Birational equivalences.
(7) Constructible sets, Chevalley's theorem.
(8) Products of affine sets.
(9) Sheaves of regular functions and affine schemes.
(10) Projective spaces, projective algebraic sets, graded algebras.
(11) Regular and rational functions on projective sets.
(12) Morphisms and products of projective sets.
(13) Abstract algebraic sets and their morphisms. Schemes: separeted, of finite type, reduced. Morphisms: affine, finite, proper, flat.
(14) Algebraic groups and their actions on algebraic sets.
(15) Grassmannians. Homogeneous spaces.
(16) Dimensions of algebraic sets.
(17) Local rings of algebraic sets, tangent spaces.
(18) Smooth and normal algebraic sets. Normalizations and desingularizations.
(19) Geometric genus of smooth sets.
(20) Vector bundles and divisors of algebraic sets.