CODE 66453 2024/2025 6 cfu anno 3 MATEMATICA 8760 (L-35) - GENOVA 7 cfu anno 3 MATEMATICA 8760 (L-35) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA MAT/03 Italian GENOVA 2° Semester AULAWEB

## OVERVIEW

The course "Institutions of Higher Geometry" aims to introduce students to the fundamental concepts of algebraic varieties and Riemann surfaces. Through a theoretical approach and explicit examples, students will acquire a solid foundation for further advanced studies in geometry and topology.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The course aims to provide students with an in-depth understanding of algebraic varieties and Riemann surfaces, developing their analytical skills and ability to solve complex problems in geometry

### AIMS AND LEARNING OUTCOMES

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

• Understand and apply the basic concepts of algebraic varieties.
• Analyze the properties of Riemann surfaces.
• Solve theoretical and practical geometric problems using algebraic and analytical tools.
• Prove fundamental theorems related to these topics.

### PREREQUISITES

A full mastery of the content of the first two years' courses in algebra, analysis, and geometry is required.

### TEACHING METHODS

In-person lectures. Remote lectures via Teams only if necessary.

### SYLLABUS/CONTENT

• Introduction to algebraic varieties:
• Definition and initial properties.
• Fundamental examples.
• Morphisms and projective varieties:
• Definition of morphism.
• Affine and projective varieties.
• Riemann surfaces:
• Definition and fundamental properties.
• Topological and geometric invariants:
• Genus and Euler characteristic.

•    R. Cavalieri and E. Miles - "Riemann surfaces and algebraic curves", Cambridge University Press, 2016.
•    A. Gathmann -  "Algebraic geometry" (see the lecture notes at https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2019/alggeom-2019.pdf)
•    A. Gathmann - "Plane algebraic curves" (see the lecture notes at https://www.mathematik.uni-kl.de/~gathmann/class/curves-2018/curves-2018.pdf)
•    F. Kirwan - "Complex algebraic curves", Cambridge University Press, 1992.
•    R. Miranda - "Algebraic curves and Riemann surfaces", American Mathematical Society, 1995.
•    I. R. Shafarevich - "Basic algebraic geometry I", Springer-Verlag, 1994, 2013.

## LESSONS

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

The timetable for this course is available here: Portale EasyAcademy

## EXAMS

### EXAM DESCRIPTION

The exam consists of a written test and an oral interview.

### ASSESSMENT METHODS

The written test that will involve solving problems and developing new proofs. Those who achieve a passing grade in the written test can take an oral exam about theoretical topics covered during the course or further exercises.

### FURTHER INFORMATION

Attendance is recommended.

Communications and additional teaching materials will be available through Aulaweb.

Please be reminded that students with disabilities or specific learning disorders (SLD) must follow the detailed instructions on Aulaweb https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3 to request exam accommodations.

In particular, accommodations must be requested well in advance (at least 10 days) before the exam date by emailing the instructor and copying the School Referent Instructor and the relevant Office (see instructions).