Admission requirements
Linear Algebra 1,2 (required)
Analysis 1 (required)
Algebra 1, 2 (recommended)
Complex Analysis (recommended)
Topology (recommended)
Description
This lecture provides a broad introduction to algebraic curves, beginning with an overview of their algebraic, number theoretic, and complex analytic properties. We will state and sketch the proof of Bézout's theorem, which counts the number of intersection points of two projective plane curves. We will also define and briefly study the Riemann surfaces associated with these plane curves. Furthermore, we will discuss methods to extract the topology of the underlying Riemann surface from the equations. Time permitting, we will cover two additional topics: the explicit desingularization of plane curves and the integrals of differential forms on Riemann surfaces.
Course Objectives
Conceptualize the passage from two-variable polynomials to Riemann surfaces
Understand the various ways in which two plane curves can intersect
Be able to use Bézout's theorem to count the number of intersection points of two curves
Understand the topological classification of Riemann surfaces
Be able to relate the degree and genus of a smooth plane curve
(Optional) Be able to study the topology of a possibly singular plane curve from the defining equations
(Optional) Understand how integrals on Riemann surfaces appear naturally and learn their basic properties
Timetable
You will find the timetables for all courses and degree programmes of Leiden University in the tool MyTimetable (login). Any teaching activities that you have sucessfully registered for in MyStudyMap will automatically be displayed in MyTimeTable. Any timetables that you add manually, will be saved and automatically displayed the next time you sign in.
MyTimetable allows you to integrate your timetable with your calendar apps such as Outlook, Google Calendar, Apple Calendar and other calendar apps on your smartphone. Any timetable changes will be automatically synced with your calendar. If you wish, you can also receive an email notification of the change. You can turn notifications on in ‘Settings’ (after login).
For more information, watch the video or go the the 'help-page' in MyTimetable. Please note: Joint Degree students Leiden/Delft have to merge their two different timetables into one. This video explains how to do this.
Mode of Instruction
Lectures and exercise classes.
Assessment method
70% Final exam.
30% Quizzes during exercise classes. (About 10 in total. The best 5 will be counted towards your grade.)
0% No homework. (There will be suggested exercises to practice at home.)
Reading list
"Complex Algebraic Curves" by Frances Kirwan.
https://doi.org/10.1017/CBO9780511623929
Registration
Please register for the course in MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December.
Please note that it is compulsory to register your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Not being registered for an exam means your grade will not be processed.
Contact
Lecturers:
Dr. Emre Can Sertöz: e.c.sertoz@math.leidenuniv.nl
Dr. Haowen Zhang: h.zhang@math.leidenuniv.nl
Course assistant:
George Politopoulos: g.politopoulos@math.leidenuniv.nl
Remarks
This course is designed to celebrate the technical machinery you've acquired throughout your undergraduate studies in mathematics. Rather than introducing new techniques, we will apply the ones you already know to explore a beautiful and central subject in mathematics: algebraic curves.