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, sheaf cohomology, genus of a curve, 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;
• distinguish between different sheaves on a variety, using the properties faced in class such as sheaf cohomology;
• apply the Riemann-Roch Theorem to algebraic curves;
• connect the different notions presented during the course, justifying statements and proving the results recalled;
• propose algebraic formulations of geometric problems naturally arising from the study of varieties.

PREREQUISITES

Students are expected to be familiar with some basic notions covered in the IGS course, in particular modules, the Zariski topology, coordinate rings, and affine and projective varieties.

Students are advised to have attended all Algebra courses in the Bachelor’s Degree programme.

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. Sheaf cohomology
4. Arithmetic genus of a curve and the Riemann-Roch Theorem

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

1. DImension and singularities
2. Canonical sheaf and divisor
3. Rational maps, geometric genus and rational curves

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

See https://corsi.unige.it/corsi/11907/studenti-orario

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.

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, genus of a curve, and the Riemann–Roch theorem;
• ability to formulate geometrical problems in algebraic terms.

FURTHER INFORMATION

Compensatory tools and dispensatory measures – Disability/Recognised Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to make the examination easier.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the course instructor, in agreement with the relevant Inclusion Officer. To benefit from adaptations during an examination, students must fill in the Adaptation Request Form. The request will be automatically sent by the system to the course instructor, to the Inclusion Officer of the relevant School/Area/Department, and in copy to the relevant administrative office. Students will also receive a copy of the request by e-mail.

The following adaptations may be granted:

• Additional time (+30% for students with a Specific Learning Disorder)

• Additional time (+50% for students with a disability/recognised invalidity)

• Additional time during oral examinations to organise the answer

• Calculator (programmable and graphing calculators are not allowed)

• Concept maps

• Tables and/or formula sheets

• Taking the examination in written form

• Taking the examination in oral form

• Reading tutor (for written examinations only)

• Writing tutor (for written examinations only)

Requests for adaptations must be submitted at least 7 working days before the scheduled examination date.

Further information for students with disabilities or Specific Learning Disorders is available on the webpage: Services for students with disabilities or SLD | UniGe | University of Genoa

Inclusion Officer: Sergio Di Domizio – sergio.didomizio@unige.it