|SCIENTIFIC DISCIPLINARY SECTOR
AIMS AND CONTENT
The course has two main goals. First of all, it offers an introduction to the theory of affine and projective algebraic varieties over an algebraically closed field, with a special focus on the case of plane algebraic curves. By specializing to the field of complex numbers, we then introduce Riemann surfaces (of which non-singular plane curves represent important examples) and prove some of the fundamental results in this area.
AIMS AND LEARNING OUTCOMES
The aim of the course is to provide an introduction to the theory of Riemann surfaces from a topological, analytic, geometric and algebraic perspective. One of the highlights of these ideas will be the Riemann-Roch theorem, whose main application shows that any compact Riemann surface is in reality an algebraic projective smooth curve. Moreover, this will lead us straight into the realm of algebraic geometry and our goal is to discuss some of the basic principles of this field with the main focus on the correspondence between the algebra of rings and the geometry of shapes defined by the solutions of polynomial equations. The most important and unifying message of the course is that it is concieved as an ideal meeting ground for topology, analysis, geometry and algebra and is displaying as a consequence the unity of mathematics.
Basic knowledge of topology, complex analysis and commutative algebra are welcome, but not strictly necessary.
In presence or Teams, depending on the pandemic and regulatory situation.
Riemann surfaces including many examples. Holomorphic maps between Riemann surfaces. Multiplicity, degree, Riemann-Hurwitz theorem and the genus of a smooth projective plane curve. Meromorphic functions and divisors on Riemann surfaces. Linear systems and their connection to holomorphic maps to projective spaces. Differential forms and Riemann-Hurwitz theorem for them. Riemann-Roch theorem and its many applications with the main focus on showing that any compact Riemann surface is a smooth projective algebraic curve. Algebraic varieties and their connections to noetherian rings. Zariski topology and the dimension of an algebraic variety. Projective varieties and the associated graded rings. Bézout's theorem and its many consequences on the geometry of curves over the complex numbers and also over the real numbers.
- 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.
TEACHERS AND EXAM BOARD
Ricevimento: See Aulaweb
Ricevimento: The teacher will be available for explanations by appointment, that will be agreed by email (at the email address firstname.lastname@example.org)
The class will start according to the academic calendar.
L'orario di tutti gli insegnamenti è consultabile all'indirizzo EasyAcademy.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the professor at the beginning of the semester to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.
Oral exam (including seminar chosen by the student among recommended topics) and evaluation of written exercises that will be proposed during the course.
|Esame su appuntamento
Attendance: advisable, but still essential as for all the course at the university.