OVERVIEW

This course is intended for students enrolled in the Master's Degree Program and aims to introduce the fundamental concepts of Algebraic Geometry.

AIMS AND CONTENT

LEARNING OUTCOMES

The course objective is to present an elemental introduction to the concepts and methods of Modern Algebraic Geometry.

AIMS AND LEARNING OUTCOMES

The objective of the course is to provide an elementary introduction to the concepts and methods of modern Algebraic Geometry. After reviewing affine and projective varieties, the following topics will be addressed: sheaves, dimension, tangent spaces and singular points, sheaf cohomology, arithmetic and geometric genus, and divisors on curves. The course will conclude with a proof of the Riemann–Roch Theorem for curves.

By the end of the course, students are expected to be able to:

• recall and connect fundamental notions such as affine and projective algebraic varieties;
• relate the geometric features of algebraic varieties with their algebraic properties;
• characterize examples of algebraic varieties in terms of dimension and singularities;
• distinguish between different sheaves on a variety, using the properties faced in class such as sheaf cohomology;
• compute the geometric and arithmetic genus of a given curve;
• apply the Riemann-Roch Theorem to algebraic curves;
• interrelate the different notions presented in the course, justifying statements and proving recalled results;
• propose algebraic formulations of geometric problems naturally arising from the study of varieties.

PREREQUISITES

Students are expected to be familiar with the topics covered in the IGS course, particularly modules, localization, Zariski topology, coordinate rings, and affine and projective varieties.

It is recommended to have completed all Algebra courses from the Bachelor's Degree Program.

TEACHING METHODS

The course consists of lectures delivered by the lecturer, in which theoretical concepts will be presented and applied to various examples and exercises. Attendance is not mandatory but strongly encouraged.

SYLLABUS/CONTENT

1. Review on affine and projective algebraic sets
2. Sheaves and varieties
3. Dimension
4. Tangent spaces and singular points
5. Sheaf cohomology
6. Arithmetic genus of a curve and the Riemann-Roch Theorem
7. Rational maps, geometric genus, and rational curves

If time permits, the following topics may also be introduced:

1. Bézout’s Theorem
2. Canonical sheaf and divisor

RECOMMENDED READING/BIBLIOGRAPHY

Primary reference:

• Perrin, Daniel. Algebraic Geometry. An Introduction. Translated from the 1995 French original by Catriona Maclean. Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2008. xii + 258 pp. ISBN: 978-1-84800-055-1; 978-2-7598-0048-3

Additional recommended texts (partial use):

• Shafarevich, Igor R. Basic Algebraic Geometry 1: Varieties in Projective Space. Third edition. Translated from the 2007 third Russian edition. Springer, Heidelberg, 2013. xviii + 310 pp. ISBN: 978-3-642-37955-0; 978-3-642-37956-7

• Ellingsrud, Geir and Ottem, John Christian. Introduction to Schemes. Preliminary version available online.

Additional references may be provided during the course.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

September 22nd, 2025

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral exam on the course content and on the application of the concepts covered during lessons.

Students with disabilities or specific learning disorders (DSA) are reminded that, in order to request exam accommodations, they must first upload their certification to the University’s online services portal at servizionline.unige.it under the “Students” section. The documentation will be verified by the University’s Inclusion Services Office for students with disabilities and DSA.

Afterwards, well in advance (at least 7 days) before the exam date, students must send an email to the professor responsible for the exam, copying both the School’s Inclusion Contact Professor for students with disabilities and DSA (sergio.didomizio@unige.it) and the aforementioned Inclusion Services Office. The email must include the following information:

• Course name
• Date of the exam session
• Student’s last name, first name, and student ID number
• The compensatory tools and dispensatory measures deemed useful and being requested

The Inclusion Contact Professor will confirm to the exam instructor that the student is entitled to request accommodations and that such accommodations must be agreed upon with the instructor. The instructor will then reply, indicating whether the requested accommodations can be granted.

Requests must be submitted at least 7 days before the exam date to allow the instructor sufficient time to review them. In particular, if the student intends to use concept maps during the exam (which must be much more concise than those used for studying), failure to meet the submission deadline will result in insufficient time to make any necessary revisions.

For more information on requesting services and accommodations, please refer to the document: Linee guida per la richiesta di servizi, di strumenti compensativi e/o di misure dispensative e di ausili specifici.

ASSESSMENT METHODS

The oral exam will focus on the course content and the application of the concepts covered during the lectures. Students will be evaluated on the following aspects:

• ability to recall definitions and the main results introduced in class, including through examples;
• ability to prove the propositions and theorems discussed, with proper reasoning;
• ability to use, both in theoretical contexts and in concrete examples, the tools introduced in the course, such as sheaf theory, dimension, sheaf cohomology, arithmetic and geometric genus, and the Riemann–Roch theorem;
• ability to formulate geometrical problems in algebraic terms.

FURTHER INFORMATION

For further information not included here, please contact the lecturer.