I have recently started a phd in the subject of algebraic curves and arithmetic geometry. In general I have a good background in algebra and analysis but only an introductory course in algebraic geometry (varieties and schemes). I also lack a thorough knowledge of geometry with only a course in manifolds and alg. topology. My teacher suggested learning Riemann surfaces first, up to the Riemann-Roch, and then building on that. My questions are the following;
- What are the relevant subjects that I must learn in my first year? (Besides Riemann surfaces, I believe function fields are related).
- What are some good books for self learning in the subject? I found Rick Miranda's book astonishing but a bit slow.
- Reading Fulton's section on Riemann surfaces (from the alg. topology book), I noticed that I should read cohomology from scratch. What is a good (and fast) introduction for that?
Thanks a lot!