Complex geometry(first part)

A.Y. 2020/2021
Overall hours
Learning objectives
Learn the basic tools and methods in the theory of Riemann surfaces.
Expected learning outcomes
Students will learn some basic tools and results in the theory of Riemann surfaces including maps between Riemann surfaces, the Riemann Existence theorem, the theory of divisors, the Riemann Roch theorem and the canonical model of a Riemann surface.
Course syllabus and organization

Single session

Lesson period
Second semester
Teaching methods
Teaching activity will be held following current regulations.
All the lectures and exercise sessions will take place in presence at university or on Zoom and the students will be able both to attend in real time following the schedule of the first semester, and/or to watch the video later, since all the sessions will be recorded and made available for the students on Ariel. Moreover, tutoring activities will be organized for groups of students, also, if allowed, in place at the university.

Program and references
The program and references for the course will not change.

The examination will be at the university or on Zoom or Skype platform depending on current regulations.
Course syllabus
·Riemann surfaces: examples and properties.
·Maps between Riemann surfaces, the Riemann-Hurwitz theorem.
·The Riemann Existence theorem
·Algebraic curves and their holomorphic 1-forms.
·The Riemann Roch theorem, embeddings and projective models.
Prerequisites for admission
Knowledge of topics in geometry covered in the courses of Geometria 3, 4 and 5 is recommended
Teaching methods
Frontal lessons, with a selection of exercises
Teaching Resources
webpage of the course with related exercises and information (link from the teachers' webpages)
[D] S. Donaldson, Riemann Surfaces, Oxford Graduate Texts in Math. 22, Oxford, 2011.
[Fo] O. Forster, Lectures on Riemann Surfaces, GTM 81, Springer, New York, 1981.
[Fu] W. Fulton, Algebraic Topology. A First Course, GTM 153, Springer, New York, 1995.
[GH] Ph. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley and Sons 1978.
[Mi] R. Miranda, Algebraic Curves and Riemann Surfaces. AMS 1995.
[Na] M. Namba, Geometry of Projective algebraic Curves, Marcel Dekker, Inc. 1984.
[S] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151. Springer-Verlag 1994.
Assessment methods and Criteria
The exam consists of an oral interview.

- During the oral exam the student will be asked to illustrate some results of the course program, as well as to discuss the resolution of some of the problems assigned during the course, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to apply them .

The mark is expressed in thirtieths and will be communicated immediately at the end of the oral exam.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
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