I intend to put supervision schedules and comments to the example sheets here. I posted some comments to the first example sheets (a bit late, sorry about that), see bottom of this page. The schedule for the third supervision is also posted below.

Some useful reading

Here is some advice for those of you who like algebraic/geometric/abstract things. If you like analysis, you should probably ignore this and instead talk to someone who does analysis.

From my point of view, the main point of this course is that it gives you a tiny little bit of algebraic geometry. I think it is fair to say that the lack of algebraic geometry is the only significant weakness in the Cambridge undergraduate course, and for those of you who plan to continue with research in some algebraic or geometric direction, it is worthwhile to try to get at least some idea of what algebraic geometry is about.

Algebraic geometry is by many people considered to be among the most difficult subjects in pure maths, but it is also a field in which many of the most significant advances have been made (for example a large proportion of the Fields medals in the past 50 years have been related to algebraic geometry). In my (very personal) opinion, almost all of the most beautiful things in pure maths have, in some way or another, their roots in Grothendieck's revolution in algebraic geometry in the 60s, where he introduced schemes, sheaf cohomology, and many other things, creating a language that unified geometry and number theory.

So here are the books. In any case, I would strongly recommend a quick read-through of Fesenko's online algebra notes and a brief look on some book on scheme theory.

Books on Riemann surfaces:

Kirwan: Complex algebraic curves (basic)
Farkas and Kra: Riemann surfaces (more advanced)

Books on algebraic geometry:

Reid: Undergraduate algebraic geometry
Hartshorne: Algebraic geometry (the standard reference, exposes you to the full horror and beauty of abstract algebraic geometry, i.e. scheme theory)
Liu: Algebraic geometry and arithmetic curves (also scheme theory, but more accessible than Hartshorne)
Eisenbud and Harris: Geometry of schemes (also more accessible than Hartshorne)
Griffiths and Harris: Principles of algebraic geometry (no schemes, but much nice stuff on complex algebraic varieties, i.e. higher-dimensional analogues of Riemann surfaces)
Some online sources: Milne's course notes (Algebraic geometry) and Vakil's course notes (very long).

Books on algebra (cover prerequisites for scheme theory)

Some highly recommended and quite brief online notes are found at Fesenko's web page (check Commutative algebra and Homological algebra). For a more thorough reading, I would start with the following two books:
Atiyah and Macdonald: Introduction to commutative algebra (I have an electronic version of this if anyone's interested)
Vermani: An elementary approach to homological algebra.

Relations to number theory

The course on Riemann surfaces has many connections to Wiles' proof of Fermat's last theorem. An excellent introduction to these ideas is found in Diamond and Shurman: A first course in modular forms.

Fourth supervision

Friday 18 Jan:
2pm-3pm: Christian and Ailsa
3pm-4pm: Edward and Ashley
Monday 21 Jan:
5pm-6pm: Amy and James
Tuesday 22 Jan:
4pm-5pm: Chris and Chris
5pm-6pm: Spencer and Alex

Third supervision

Thursday 22 Nov:
4-5: Ailsa Keating, Christian Johansson
5-6: Chris Doris, Chris Fairless
Monday 26 Nov:
5-6: Amy Pang, James West
Tuesday 27 Nov:
4-5: Edward Sanders, Ashley Ballard
5-6: Alex Helfet, Spencer Hughes

Second supervision

Thursday 8 Nov:
2-3: Chris Doris, Chris Fairless
3-4: Ailsa Keating, Christian Johansson
Friday 9 Nov:
4-5: Amy Pang, James West
5-6: Edward Sanders, Ashley Ballard
Tuesday 13 Nov:
4-5 (or 5-6?): Alex Helfet, Spencer Hughes

No particular comments on the example sheet.

First supervision

Wednesday 24 Oct:
1.30pm 2.30pm: Amy Pang, James West
5-6: Edward Sanders, Ashley Ballard
Thursday 25 Oct:
3.30-4.30 Ailsa Keating, Christian Johansson
Friday 26 Oct:
2.30 3.30: Alex Helfet
4-5: Chris Doris, Chris Fairless
Tuesday 30 Oct:
4-5: Spencer Hughes

Notes and comments on the first example sheet: (These include answers to some questions I couldn't answer during the supervision.)

5 (iii) Someone asked how you show that f cannot have an essential singularity at 0. One way of doing it is to use Picard's theorem (thanks to Ashley and Edward for telling me about this), which I think were mentioned without proof in some lecture. Quoting from Wikipedia: "The second theorem, also called "Big Picard" or "Great Picard", states that if f(z) has an essential singularity at a point w then on any open set containing w, f(z) takes on all possible values, with at most a single exception, infinitely often. This is a substantial strengthening of the Weierstrass-Casorati theorem, which only guarantees that the range of f is dense in the complex plane."
If you don't want to use Picard's theorem, you can use Weierstrass Casorati in combination with the Baire category theorem (solution due to Christian Johansson).

8 On question 8, I was not completely clear with all of you: for d=1, the answer is 0 and 2. For d > 1, the answer is all the integers from (d-1) up to 2d (inclusive).
7 The natural way to solve question 7 is to use that F must be rational. However, someone asked if it is possible to use the local mapping theorem instead. The argument would go like this: by compactness, the set of points Q for which the equation does not have d distinct solutions, must have a limit point in S^2. Consider a disc centred at this limit point. Obtain a contradiction. I don't think this will work. However, you can use compactness together with rationality, and use that branching points of a rational map are isolated.