**Università
di Roma “Tor Vergata” **

**ALGEBRAIC
GEOMETRY**

**Tentative
program**

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.

**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)**