
Course unit
RIEMANN SURFACES
SC01111818, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2017/18
ECTS: details
Type 
ScientificDisciplinary 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 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
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 (RiemannHurwitz formula and RiemannRoch 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 1dimensional complex tori;
 Divisors on compact Riemann surfaces, linear systems;
 Differential forms and RiemannRoch theorem, applications;
 First notions of homology, Jacobians of Riemann surfaces, AbelJacobi 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)

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

