Syllabus

Objectives

KNOWLEDGE AND UNDERSTANDING
By the end of the course the student is expected to be familiar with the fundamental objects of classical algebraic geometry, both affine and projective, and of basic concepts of commutative algebra.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING
By the end of the course the student is expected to be able to apply the notions of basic algebraic geometry acquired to solve problems and exercises of medium difficulty. The exercises can also be proposed as easy theoretical results.
JUDGMENT AUTONOMY
By the end of the course the student is expected to be able to recognize and apply the most basic techniques of algebraic geometry and also to recognize the situations and problems in which these techniques can be used advantageously.
COMMUNICATIVE SKILLS
By the end of the course the student is expected to be able to express himself with proficient command of language and exposure security on the topics of the course.
LEARNING CAPACITY
By the end of the course the student is expected to be able to consult the standard texts of algebraic geometry and commutative algebra.

Prerequisites

Linear algebra, affine and projective space and their subspaces; a basic knowledge of plane algebraic curves is useful but not essential.

Contents

Algebraic geometry of affine and projective varieties.
Zariski topology on affine and projective varieties. Hilbert Nullstellensatz. Regular and rational maps. Tangent spaces and singular points. Blow-up and outline of the resolution of singularities. Selected topics in commutative algebra (commutative rings and modules).

Didactical methods

Lectures and problem sessions. During the course some exercises will be assigned as homework, to be delivered in written form. The solutions will be discussed in class.

Exams

The exam program coincides with the arguments of the lectures. The exam will be held in oral form only, but the students who will not deliver the assigned exercises will have to take a written test, consisting in solving exercises modeled on those assigned during the course. The oral exam aims to carry out an assessment of the student’s familiarity with the program, comprehension of the contents (definitions and proofs) and command of language.

Texts

I. R. Shafarevich: Basic Algebraic Geometry 1: Varieties in Projective Space,
Third edition. Springer, Heidelberg, 2013

J. Harris: Algebraic geometry. A first course, Graduate Texts in Mathematics, 133, Springer-Verlag, New York, 1995.

R. Hartshorne: Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. (First chapter)

S.D. Cutkosky, Introduction to Algebraic Geometry, Graduate Studies in Mathematics 188, AMS 2018

E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhauser

More informations

Information about the progress of the program and teaching materials will be posted on this site http://moodle2.units.it