Complex Geometry(FIRST PART)
A.Y. 2024/2025
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.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
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.
·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
material on the ariel page of the course
[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.
[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.
- 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
Professors:
Colombo Elisabetta, Lombardi Luigi
Shifts:
Professor(s)
Reception:
friday.8.45-11.45
Office2101, second floor, via C. Saldini 50
Reception:
Wednesday 15:30-16:30 or by appointment
Math Department in via C. Saldini 50. Office: 1.109