Projective Algebraic Geometry

A.Y. 2016/2017
6
Max ECTS
42
Overall hours
SSD
MAT/03
Language
Italian
Learning objectives
The aim of the course is to give an introduction to affine and projective algebraic varieties.
Expected learning outcomes
Knowledge of some elementary properties of affine and projective varieties.
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course syllabus
Affine algebraic varieties.
Affine algebraic sets. The Zariski topology. Hilbert's Nullstellensatz. Affine varieties.
Regular functions and maps: morphisms and isomorphisms; rational and birational maps.

Projective algebraic varieties.
Projective algebraic sets. Projective varieties.
Morphisms, rational maps and birational equivalence.

Attributes of Varieties.
Tangent spaces and smoothness.
Definitions of dimension.
Hilbert polynomial and degree of a projective variety.

Examples
Rational normal curves. Segre and Veronese embeddings. Variety of projective conics. Projections and Normalization Lemma. Blow ups. Rational and unirational varieties. Grassmannians G(k,n) and Plücker embeddings. Examples of enumerative geometry: lines on a surface in P3.

Vector bundles.
Definition and examples of vector bundles on projective varieties. Sections of a vector bundle. Line bundles and rational maps. Hyperplane and tautological line bundles. Examples.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor: Bertolini Marina
Professor(s)