# Algebraic Curves

Topics:

• basics of algebraic varieties over the complex numbers (with focus on dimension 1),
• singularities of curves (what are they and when is a curve nonsingular),
• desingularization of curves by normalization,
• the relationship between nonsingular algebraic curves and complex manifolds of dimension 1,
• nonsingular projective algebraic curves and function fields,
• Riemann-Roch, and hopefully also some
• applications of Riemann-Roch.

Prerequisites:

• Basic set theory (cardinality of sets, etc).
• Fields, and vector spaces over fields. You should know what a field is and that every vector space has a basis, and moreover that the cardinality of the basis is independent of the choice of the basis.
• You should know what it means for a field to be algebraically closed.
• Complex numbers: You should be aware that the complex number field is an algebraically closed field (fundamental theorem of algebra).
• Basic commutative algebra: Rings, modules, ideals, prime ideals, maximal ideals. Isomorphism theorems, euclidean domains, UFD, PID, etc. We will review this as we go along, but you should have had a basic algebra course.
• Complex functions of one variable. It would be great if you knew what it means to say that a function is holomorphic, and some basic results on these. We will have to use this at some point in the course. At this point we will simply record what is true and continue.

I will be using the book by William Fulton, Algebraic Curves, allthough some of the material will be from outside of this book.

It is strongly encouraged to go to the lectures, which are on Tuesday and Thursday, 1:10-2:25 in Mathematics 307.

Problem sets will appear here. Please find below the current set.

The TA for the course is Alex Perry.

Grades are determined by an average of homework scores and final exam.

The final will be a written exam, intended mainly to see how much you actually got out of the course.

The weekly problem sets will be sections of exercises.pdf which you can download below.

• First problem set due Thursday, January 28 in class.
• Second problem set due Thursday, February 4 in class.
• Third problem set due Thursday, February 11 in class.
• Fourth problem set due Thursday, February 18 in class.
• Fifth problem set due Thursday, February 25 in class.
• Sixth problem set due Thursday, March 4 in class.
• Seventh problem set due Thursday, March 11 in class.
• Eight problem set due Thursday, March 26 in class.
• Ninth problem set due Thursday, April 1 in class.
• Tenth problem set due Thursday, April 8 in class.
• Eleventh problem set: Catch up with the homeworks you are behind on.
• Twelth problem set due Thursday, April 22 in class.
• Thirteenth problem set: Catch up with the homeworks you are behind on.