**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.