Riemann Surfaces & Algebraic Curves - MAT00111M

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  • Department: Mathematics
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2025-26

Module summary

Riemann surfaces lie at the interface of complex analysis, complex geometry and algebraic geometry. This course begins by developing some more advanced complex analysis before providing an introduction to Riemann surfaces, from the initial concepts through to more advanced applications. The course will emphasise concrete examples of algebraic curves to help visualise the important theorems.

Related modules


Additional information

A course in complex analysis that covers holomorphic and meromorphic functions, Laurent series and complex integration will give a good background for this course. The course will begin by reviewing these concepts, however it is expected that students have knowledge of all of these topics before the course begins.

The M level version of this module is not able to be taken if the H level version has already been taken.

Module will run

Occurrence Teaching period
A Semester 2 2025-26

Module aims

Riemann surfaces lie at the interface of complex analysis, complex geometry and algebraic geometry. This course begins by developing some more advanced complex analysis before providing an introduction to Riemann surfaces, from the initial concepts through to more advanced applications. The course will emphasise concrete examples of algebraic curves to help visualise the important theorems.

Module learning outcomes

By the end of the course, students will be able to:
1. Use the implicit function theorem to construct complex charts for a given algebraic curve.
2. Construct examples of Riemann surfaces as algebraic curves and compute their genus.
3. Integrate a differential form on a Riemann surface.
4. Describe the meromorphic functions on a Riemann surface that are bounded below by a given divisor.

Module content

H/M level content.
1. Analytic continuation
2. Definition of a Riemann surface and examples given by algebraic curves
3. Functions on Riemann surfaces and holomorphic maps between Riemann surfaces
4. Integration on Riemann surfaces
5. Divisors associated to meromorphic functions and one forms
M Level only.
6. Divisors and maps to projective space

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

Main Text:

  • Miranda, Algebraic Curves and Riemann Surfaces

Supplementary Texts:

  • Donaldson, Riemann Surfaces
  • Kirwan, Algebraic Curves