**Università
di Roma “Tor Vergata” **

**ALGEBRAIC
GEOMETRY**

**Tentative
program**

Miscellanea: Finite morphisms. Complete intersections, set-theoretical complete intersections. Semi-continuity of the fibre-dimension of a dominant morphism. 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. Composed morphisms and birational morphisms. 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, modules, localizations,
integral closure, K-algebras and finiteness conditions, Specm(R) for
a ring R. Presheaves and sheaves

**2)**
Affine space, affine closed subsets and the Zariski topology.
Radical ideals. Hilbert Nullstellensatz (weak and strong form).
Irreducibility and irreducible components. Affine twisted cubic.
Affine and quasi-affine varieties: examples. Coordinate ring and
field of rational functions of an affine variety.

**3)**
Graded rings and homogeneous ideals. Projective space and projective
closed subsets. Homogeneous ideals. Affine and projective cones.
Homogeneous Hilbert Nullstellensatz. Projective varieties: ring of
homogeneous coordinates. Projective twisted cubic and its
homogeneous ideal. Quasi-projective varieties.

**4)**
Regular and rational functions over an algebraic variety. Ringed
space and structural sheaf. Affine case, projective case and other
consequences.

**5****)
**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 projective space at a point, resolution of
singularities, Cremona group, the quadratic elementary
transformation of the projective plane

**6)**
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.

**7****)**
Trascendence degree of an integral K-algebra of finite type.
Dimension of an algebraic variety.

**8****)**
Embedded tangent spaces and non-singularity. Zarisky tangent space.

**9****)**
Hilbert function and Hilbert polynomial of a projective variety.
Degree and arithmetic genus of a projective variety. Examples.

**Further
topics (either if time permits or for seminars/thesis)**