Algebraic curves

About the course

This course will run in Weeks 11-20, with two lectures per week and one workshop per fortnight. See your timetable for lecture/workshop times and locations (in case of sudden changes!).

Lecture notes.

This course will use ideas from MATH220 (including generalised eigenspaces) and MATH225 (rings, ideals). It should complement MATH424 (Galois theory) and MATH322 (Commutative algebra). MATH322 is a prerequisite, but this year (2023-24) you should be able to get away without having done it. MATH424 is not necessary, but I highly recommend taking it if you're interested in this module.

Assessment

There are three main components to the assessment: fortnightly coursework (worth 20% of the overall grade), a written project (worth 30%) and an exam in the summer (worth 50%).

Projects

The projects are due after Easter; you can read more about what they involve here.

Coursework

There will be five courseworks, to be submitted electronically via Moodle (scanned handwriting is fine: I don't require LaTeXed solutions!). They are due on Tuesdays at 5pm.

Sheet 1 is a guided exercise leading to the classification of plane conics. It shouldn't require any specialist knowledge beyond manipulating expressions and completing the square.

Courseworks will be awarded letter grades (A-F) based on how successfully you have engaged with the learning objectives, and how many/which questions you have attempted.

The remaining 4 courseworks have 12 questions each; these are categorised into three groups: gamma, beta and alpha questions. You are supposed to do all the gamma questions (there are usually 3-4). You can do as many or as few of the beta and alpha questions as you like: anything you submit will be marked. The catch is that in order to access the higher letter grades, you will need to do more of the beta and alpha questions. More precisely:

To get a C (2:2), you'll need to try at least one beta question with moderate success.
To get a B (2:1), you'll need satisfactory attempts on at least one beta and one alpha question.
To get a B+/A- (2:1-1st borderline) you'll need substantial success on at least two beta and one alpha questions.
To get an A (solid 1st, 80% or more) you'll need substantial success on at least two beta and two alpha questions.

Recommended books

The following books have been constantly at my side during the summer whilst I was writing these notes. They all go further than I will manage to get, and offer a complementary perspective on all of the content from the course.

Fulton (1969) Algebraic curves
Reid (1988) Undergraduate algebraic geometry
Walker (1950) Algebraic curves

Another book which I recommend very highly is:

Cox, Little, O'Shea (2015 fourth edition) Ideals, varieties, and algorithms

It takes a much more computational approach (basing everything on the computationally effective notion of Groebner bases). Their other book "Using algebraic geometry" is one of the few I know which discusses the eigenspace approach to local rings and multiplicities which I used as the starting point for defining intersection multiplicity in my notes. Happily, the more traditional proofs from Fulton's book were easy to carry over to this setting.

Elliptic functions and elliptic integrals only made a fleeting appearance in Chapter 2 of the lecture notes. If you want to read more about this topic and how it relates to algebraic curves, you could read:

McKean and Moll (1997) Elliptic curves: function theory, geometry, arithmetic
Jones and Singerman (1987) Complex functions: an algebraic and geometry viewpoint

On topic which I would have loved to cover in more detail but didn't (for lack of space) is the theory of elliptic curves and the relationship to number theory and to cryptography. For that, I can highly recommend:

Silverman and Tate (1992) Rational points on elliptic curves

If you feel that you are lagging behind in background (rings, ideals, generalised eigenspaces), and feel like your MATH105/MATH220/225/322 notes are not enough, you could try:

Axler (1996) Linear algebra done right (for generalised eigenspaces)
Van der Waerden (1949 English translation) Modern Algebra, Volume I, Chapters 3 and 4 (for rings/ideals/polynomials)