B3.3 Algebraic Curves (2025-26)
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- Lecturer: Profile: Damian Rossler
Algebraic curves in the complex projective plane. Irreducibility, singular and nonsingular points, tangent lines.
Bezout's Theorem (the proof will not be examined). Points of inflection, and normal form of a nonsingular cubic.
Nonsingular algebraic curves as Riemann surfaces. Meromorphic functions, divisors, linear equivalence. Differentials and canonical divisors. The group law on a nonsingular cubic.
The Riemann-Roch Theorem (the proof will not be examined). The geometric genus. Applications.
Section outline
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Lecture notes by Nigel Hitchin. Based on the textbook "Complex Algebraic Curves" by Frances Kirwan, which is an excellent reference for this course.
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This is an initial optional problem sheet to get you started before the classes. Do not hand in solutions. Sample solutions are on the course web page. Reading: sections 1 and 2 of the Hitchin lecture notes.
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Opened: Tuesday, 20 January 2026, 11:13 AM
This is the problem sheet for the first class in weeks 2 or 3. It is based on the first three lectures on projective geometry, Hitchin notes chapter 1.
Only Section B will be marked. The file 25B33Sheet1.pdf just contains the questions, whilst the file 25B33Soln1A+C.pdf contains the questions and the solutions to Sections A and C.
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Opens: Tuesday, 3 February 2026, 11:14 AMThis is the problem sheet for the second class in weeks 4 or 5. It is based on lectures 4-8, Hitchin notes chapters 2 and 3.
Only Section B will be marked. The file 25B33Sheet2.pdf just contains the questions, whilst the file 25B33Soln2A+C.pdf contains the questions and the solutions to Sections A and C. -
Opens: Tuesday, 17 February 2026, 11:15 AMThis is the problem sheet for the third class in weeks 6 or 7. It is based on lectures 7 - 11, Hitchin notes sections 3-4.2.
Only Section B will be marked. The file 25B33Sheet3.pdf just contains the questions, whilst the file 25B33Soln3A+C.pdf contains the questions and the solutions to Sections A and C. -
Opens: Tuesday, 3 March 2026, 11:16 AMThis is the problem sheet for the fourth class in week 8 or week 1 of TT23. It is based on lectures 11 - 16, Hitchin notes sections 4-5.
Only Section B will be marked. The file 25B33Sheet4.pdf just contains the questions, whilst the file 25B33Soln4A+C.pdf contains the questions and the solutions to Sections A and C.
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Registration start: Friday, 16 January 2026, 12:00 PMRegistration end: Friday, 13 February 2026, 12:00 PM
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Class Tutor's Comments Assignment
Class tutors will use this activity to provide overall feedback to students at the end of the course.
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