Università di Roma “Tor Vergata”
ALGEBRAIC GEOMETRY
Tentative program
1) Algebraic preliminaries: Noetherian rings, Hilbert basis theorem, K-algebras and finiteness conditions, Zariski Lemma (if necessary, see Commutative Algebra course)
2) Affine space, affine closed subsets and the Zariski topology. Radical ideals. Hilbert Nullstellensatz (weak and strong form). Irreducibility and irreducible components. Affine and quasi-affine varieties: examples. Coordinate ring and field of rational functions of an affine variety. Affine rational normal curves, in particular affine twisted cubic and its radical ideal (determinantal variety).
3) Homogeneous polynomials. Vector spaces of homogeneous polynomials of given degree. Graded rings and homogeneous ideals.
4) Projective space and projective closed subsets. Affine and projective cones. Homogeneous Hilbert Nullstellensatz. Projective varieties: homogeneous coordinate ring. Projective rational normal curves: in particular projective twisted cubic and its homogeneous radica ideal. Quasi-projective varieties.
5) Other algebraic preliminaries: modules, localizations(if necessary, see Commutative Algebra course)
6) Presheaves and sheaves on a topological space. Regular and rational functions over an algebraic variety. Structural sheaf of an algebraic variety. Local and global sections. Affine case, projective case, some consequences.
7) Morphisms between algebraic varieties. Examples: Veronese embedding. Dominant morphisms. Rational maps, birational maps. Examples: linear systems of hypersurfaces in a projective space, projections, blow-up of a projective space at a point, resolution of singularities of some singular plane curves, stereographical projection of the smooth quadric on a projective plane.
8) Products of algebraic varieties. Segre embedding and Segre variety.
9) Diagonals and graph of a morphism.
10) Main theorem of elimination theory and applications: completeness of projective varieties.
11) Embedded tangent spaces of an affine (projective) variety and non-singularity. Dimension of an algebraic variety
12) Zarisky tangent space and derivations (if time permits)
Further topics (either if time permits or for seminars/thesis)
· Hilbert function and Hilbert polynomial of a projective variety. Degree and arithmetic genus of a projective variety. Examples.
· 27 lines in a smooth cubic surface in projective 3-dimensional space
· Cartier divisors and line bundles on a smooth projective curve; global sections; base-point-free linear systems.
· The canonical divisor and the canonical curves. Riemann-Roch theorem and its geometric interpretation on the canonical curve. Enriques-Babbage Theorem.
· Dual projective space. Other examples of projective varieties: Grassmannians and Pluecker embedding.
· Projective curves in the plane and their families. Parameter spaces. Chow variety of curves in 3-dimensional projective space.
· Finite morphisms. Semi-continuity of the fibre-dimension of a dominant morphism.