# Algebraic Curves (BM)

Vak
2024-2025

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

In MyTimetable, you can find all course and programme schedules, allowing you to create your personal timetable. Activities for which you have enrolled via MyStudyMap will automatically appear in your timetable.

Questions? Watch the video, read the instructions, or contact the ISSC helpdesk.

Note: Joint Degree students from Leiden/Delft need to combine information from both the Leiden and Delft MyTimetables to see a complete schedule. This video explains how to do it.

## 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.)

"Complex Algebraic Curves" by Frances Kirwan.

https://doi.org/10.1017/CBO9780511623929

## Registration

As a student, you are responsible for enrolling on time through MyStudyMap.

In this short video, you can see step-by-step how to enrol for courses in MyStudyMap.
Extensive information about the operation of MyStudyMap can be found here.

There are two enrolment periods per year:

• Enrolment for the fall opens in July

• Enrolment for the spring opens in December

Note:

• It is mandatory to enrol for all activities of a course that you are going to follow.

• Your enrolment is only complete when you submit your course planning in the ‘Ready for enrolment’ tab by clicking ‘Send’.

• Not being enrolled for an exam/resit means that you are not allowed to participate in the exam/resit.

## 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.

Software
Starting from the 2024/2025 academic year, the Faculty of Science will use the software distribution platform Academic Software. Through this platform, you can access the software needed for specific courses in your studies. For some software, your laptop must meet certain system requirements, which will be specified with the software. It is important to install the software before the start of the course. More information about the laptop requirements can be found on the student website.