Algebraic Geometry

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/03 Geometry.

Examination method

Oral exam.

Learning
objectives

The aim of the course is to provide an introduction to algebraic geometry; then to provide the fundamental tools for the study of algebraic varieties and affine schemes, to discuss the most important results, and to illustrate the main techniques of proof and problem solving.

Syllabus

Affine space and algebraic sets. The Zariski topology and its properties. Reducible and irreducible sets. Noetherian rings and Hilbert’s basis theorem. Noetherian topological spaces and the correspondence between closed sets and ideals. Radical and Hilbert’s Nullstellensatz. Affine algebraic varieties. Coordinate ring and dimension of an algebraic variety. Height, Krull dimension and transcendence degree. Gauss’s lemma and factorial rings. Graded rings and projective varieties. Local rings, localisations and regular functions. Morphisms, dominant morphisms and properties of the fibres of a morphism. Constructible sets. Finite morphisms and projections. Regular points and tangent space. Regular local rings and the Zariski cotangent space. Derivations on a module. Presheaves, sheaves and morphisms between them. Sheafification of a presheaf. Monomorphisms and epimorphisms of sheaves. Kernel and image of a morphism of sheaves. Locally ringed spaces. Spectrum of a ring. Affine schemes. Examples of affine schemes.

Expected learning
outcomes

By the end of the course, students must demonstrate that they

know and understand the language of algebraic geometry and have acquired a basic knowledge of the topics presented in the course;
are able to apply the knowledge acquired to the study and solution of concrete examples, making correct use of proof techniques;
are able to communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
are able to identify the most appropriate methods to analyse and solve a problem relating to the course topics and to interpret the results correctly.

Learning outcomes
to be assessed

Knowledge and skills acquired on the course topics, the student’s presentation skills and command of language, the ability to apply the knowledge acquired to the solution of simple problems, the ability to support a discussion with examples and counterexamples, and command of the mathematical tools used in the course.

Algebraic Geometry