OVERVIEW

The aim of this course is to develop the notions that have been introduced in the course Introduzione alla Geometria Algebrica and to generalize them to Scheme Theory: the notion of affine scheme will be introduced, as well as the general notion of scheme, and it will be shown how this generalizes the notion of algebraic variety; similarily, the notion of morphism between schemes will be introduced, and the main properties of morphisms of schemes will be studied. The core of the course will the the introduction and the study of some of the main objects of modern algebraic geometry, such as coherent and quasi-coherent sheaves. These notions will finally be used to introduce invertible sheaves, divisors, the Picard group of a scheme and how these objects allow to embed a projective scheme in a projective space..

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to introduce selected advanced topics considered fundamental for students who wish to pursue further study and
research in the field of modern Algebraic Geometry.

AIMS AND LEARNING OUTCOMES

During the coruse we will introduce the notion of scheme as a generalization of the notion of algebraic variety given in Introduzione alla Geometria Algebrica. In particular, we will introduce the notion of scheme as the set of prime ideals of a unitary commutative ring, and on this set we will define a topology and a structure sheaf by imitating the analogous notions for algebraic varieties. A scheme will be defined by glueing affine schemes. We will see how the category of schemes generalizes the category of algebraic varieties.

Using the language of schemes we will introduce some particular families of sheaves on a scheme, like coherent and quasi-coherent sheaves, of which we will define the categories and we will study the properties. In particular we will study locally free sheaves, and we will define fundamental notions like sheaf cohomology, with a particular attention to Cech cohomology, and we will introduce the notion of invertible sheaf and of divisor (and the relation between these two notions).

The aim of the course is then to introduce one of the most important languages of modern algebraic geometry, preparing the students to start research in in Algebraic Geometry.

At the end of the course the student will be able to describe some basic properties of a scheme or of an algebraic variety, as study its divisors and determine its fundamental properties (base points, ampleness). The student will be moreover able to algebrize a geometric problem in order to solve it in a rigorous way.

PREREQUISITES

The main prerequisite is the course "Introduzione alla Geometria Algebrica"

TEACHING METHODS

Traditional

SYLLABUS/CONTENT

1. Affine schemes

2. The category of schemes

3. Coherent and quasi-coherent sheaves

4. Sheaf cohomology

5. Invertible sheaves and divisors

RECOMMENDED READING/BIBLIOGRAPHY

R. Hartshorne, Algebraic Geometry, Springer

D. Eisenbud, J. Harris, The Geometry of Schemes, Springer

D. Mumford, The Red Book of Varieties and Schemes, Springer

I. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Springer

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The lessons will start in accordance with the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam will be oral, and it will considered as passed if the obtained evaluation is at least 18/30.

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

ASSESSMENT METHODS

The oral exam requires the knowledge and the ability to present the definitions, the statements and the proofs that have been treated alla long the course, the ability to give examples that illustrate the main notions of the course, and the ability to establish if a given statement is true or false by means of proofs or counterexamples.