Università di Roma “Tor Vergata”
1) Algebraic preliminaries: Noetherian rings, Hilbert basis theorem, K-algebras and finiteness conditions, Zariski Lemma, Specm(R) for a commutative ring R.
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 ideal).
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: ring of homogeneous coordinates. Projective rational normal curves: in particular projective twisted cubic and its homogeneous ideal. Quasi-projective varieties.
5) Other algebraic preliminaries: modules, localizations. Presheaves and sheaves
6) Regular and rational functions over an algebraic variety. Ringed space and structural sheaf. Affine case, projective case and other consequences.
7) Morphisms between algebraic varieties. Constructible sets. 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, monoids, stereographical projection of the smooth quadric.
8) Products of algebraic varieties. Segre embedding and Segre variety. Diagonals and graph of a morphism. Main theorem of elimination theory and applications: completeness of projective varieties. (Tensor product of integral K-algebras of finite type).
9) Trascendence degree of an integral K-algebra of finite type. Dimension of an algebraic variety. (Other definition of dimension)
10) Embedded tangent spaces and non-singularity. Zarisky tangent space.
Further topics (either if time permits or for seminars/thesis)
Finite morphisms. Semi-continuity of the fibre-dimension of a dominant morphism.
Hilbert function and Hilbert polynomial of a projective variety. Degree and arithmetic genus of a projective variety. Examples.
Dual projective space. Other examples of projective varieties: Grassmannians and Pluecker embedding.
Projective curves in the plane and their families. Resolution of singularities of plane curves.
Parameter spaces. Chow variety of curves in 3-dimensional projective space.
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.