OVERVIEW

The course introduces students to scheme theory, a fundamental language and tool of modern algebraic geometry. The main ideas and objects of the theory will be presented, along with selected applications and complementary topics.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to introduce selected advanced topics considered fundamental for students who wish to pursue further study and
research in the field of modern Algebraic Geometry.

AIMS AND LEARNING OUTCOMES

The course is designed to provide students with a rigorous yet accessible introduction to scheme theory, with attention to the geometric and arithmetic motivations behind definitions and results. The goal is to lay a solid foundation for advanced study and research in algebraic geometry.

At the end of the course, students will be able to:

1. Describe the fundamental concepts of scheme theory;

2. Apply the main constructions in standard and novel contexts;

3. Interpret arithmetic and geometric examples through the lens of scheme theory;

4. Prove propositions and theorems using basic techniques from commutative algebra and algebraic geometry;

5. Clearly communicate mathematical ideas and arguments using proper terminology;

6. Connect the course material with other areas of mathematics, particularly commutative algebra and classical geometry, in preparation for advanced study.

PREREQUISITES

A solid understanding of the content of the Introduction to Algebraic Geometry course is essential. In addition, students are expected to be fully proficient in the core material from the standard undergraduate curriculum in algebra and geometry.

TEACHING METHODS

Traditional lecture-based teaching.

Students with certified Specific Learning Disorders (DSA), disabilities, or other special educational needs are encouraged to contact the instructor and the departmental disability coordinator at the beginning of the course to agree on possible accommodations that respect the learning objectives while addressing individual needs.

SYLLABUS/CONTENT

1. Affine schemes

2. The category of schemes

3. Coherent and quasi-coherent sheaves

4. Sheaf cohomology

5. Invertible sheaves and divisors

6. Complements

RECOMMENDED READING/BIBLIOGRAPHY

R. Hartshorne, Algebraic Geometry, Springer

G. Ellingsrud, J. C. Ottem, Introduction to Schemes

D. Eisenbud, J. Harris, The Geometry of Schemes, Springer

D. Mumford, The Red Book of Varieties and Schemes, Springer

I. Shafarevich, Basic Algebraic Geometry 2: Schemes and Complex Manifolds, Springer

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The lessons will start in accordance with the academic calendar.

https://corsi.unige.it/corsi/9011/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of an oral test. The exam is considered passed if the grade is equal to or greater than 18/30.

Students with certified DSA, disabilities, or other special educational needs are encouraged to contact the instructor at the beginning of the course to arrange examination procedures that, while maintaining the objectives of the course, take into account individual learning needs and provide suitable compensatory tools.

ASSESSMENT METHODS

During the oral examination, students will be asked to state and explain the definitions introduced in the course, as well as the statements and proofs of theorems, using appropriate terminology. They will be expected to determine whether given statements are true or false and justify their answers using proofs, examples, and counterexamples. They will also be expected to apply the concepts of the course to prove new statements.

FURTHER INFORMATION

Ask the

professor for other information not included in the teaching schedule