OVERVIEW

In the course of Geometria Superiore 1 we will introduce the theory of toric varieties, a peculiar class of algebraic varieties  constructed from euclidean polytopes. This construction provides a family of special algebraic varieties whose geometric properties (such as singularities, cohomology, intersection theory, line bundles) may be described explictely by means of the geometric properties of the polytopes they are built from. Toric varieties are hence very concrete examples of algebraic varieties on which one may test general theories and conjectures. Moreover it is possible to apply on them in an explicit way many general and abstract constructions from algebraic geometry. Finally, toric varieties have several interesting applications and relations with physics, code theory, commutative algebra and much more. Thanks to these deep relations, toric varieties forms a very modern subject that is widely present in a lot of current research in algebraic geometry (and further).

AIMS AND CONTENT

LEARNING OUTCOMES

The main educational outcome for the students will be to learn and work in a concrete way with many abstract tools from algebraic geometry, by applying them to the family of toric varities. This will allow students to have a clearer understanding of the general methods and notions in algebraic geometry.

PREREQUISITES

The course Introduzione alla Geometria Algebrica is strongly recommended.

TEACHING METHODS

The course will be held in a traditional way, with lectures and exercices.

SYLLABUS/CONTENT

The content of the course is the following:

1. Convex cones and affine toric varieties.
2. Polytopes, fans and projective toric varieties.
3. Singularities of toric varieties and their resolution.
4. Divisors, line bundles, tangent bundle and cohomology.
5. Intersection theory on toric varieties.
6. Applications of toric varieties.

RECOMMENDED READING/BIBLIOGRAPHY

The suggested books and notes are the following:

1. W. Fulton, "Introduction to toric varieties".
2. D. Cox, J. Little, H. Schenck, "Toric Varieties".
3. M. Mustață, "Lecture notes on toric varieties".

TEACHERS AND EXAM BOARD

Exam Board

VICTOR LOZOVANU (President)

ARVID PEREGO

LESSONS

LESSONS START

The lessons will start in accordance with the academic calendar as approved by the Consiglio del Corso di Studi.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam will consist in giving a talk on one of the topics that the teachers will propose.