First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SC01111818, A.A. 2019/20

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course First cycle degree in
SC1159, Degree course structure A.Y. 2008/09, A.Y. 2019/20
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Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination RIEMANN SURFACES
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit


ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/03 Geometry 6.0

Course unit organization
Period Second semester
Year 3rd Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Practice 3.0 24 51.0 No turn
Lecture 3.0 24 51.0 No turn

Start of activities 02/03/2020
End of activities 12/06/2020
Show course schedule 2019/20 Reg.2008 course timetable

Prerequisites: Algebra, geometry and analysis of the first two years. Basic knowledge on holomorphic functions of one variable.
Target skills and knowledge: The course aims to develop the fundamental concepts regarding compact Riemann surfaces (in particular, on spheres and tori), introducing the notion of genus and its interpretations (Riemann-Hurwitz formula and Riemann-Roch theorem).
Examination methods: Written exam.
Assessment criteria: The exam test the acquired knowledge during the course and the capacity to apply this knowledge in particular cases. In particular the written exam will involve theory and exercises.
Course unit contents: Introduction to the geometry of compact Riemann surfaces. Topics:

- Definition of a Riemann surface;
- Elementary properties of holomorphic and meromorphic functions on a Riemann surface;
- Detailed study of the Riemann sphere and 1-dimensional complex tori;
- Divisors on compact Riemann surfaces, linear systems;
- Differential forms and Riemann-Roch theorem, applications;
- First notions of homology, Jacobians of Riemann surfaces, Abel-Jacobi theorem.
Planned learning activities and teaching methods: Lectures and exercise classes.
Additional notes about suggested reading: Other than the textbook, the professor's personal notes and other material will be avilable online.
Textbooks (and optional supplementary readings)
  • Miranda Rick, Algebraic curves and Riemann Surfaces. --: AMS - GSM 5, 1995. (per consultazione) Cerca nel catalogo

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)
  • One Note (digital ink)