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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;

  1. What are the relevant subjects that I must learn in my first year? (Besides Riemann surfaces, I believe function fields are related).
  2. What are some good books for self learning in the subject? I found Rick Miranda's book astonishing but a bit slow.
  3. 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!

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I would advice Forster's book, Lectures on Riemann surfaces. It is a incredible book, introducing Riemann surfaces from scratch in the first chapter and studying all basic properties of compact Riemann surfaces (Riemann-Roch, Abel-Jacobi and Serre Duality) using the langage of sheaf cohomology which is introduced in a very concrete way using Cech complex. It is definitely less slow that the book of Miranda.

Another book which I really liked is the book "Introduction to compact Riemann surfaces and dessin d'enfants". It is more oriented into the relation between algebraic curves, Riemann surfaces and field extension of degree 1. Lot of example are treated (especially hyperelliptic curves if I remember well). It is also a wonderful book, and the second part of the book is devoted to dessins d'enfants and the incredible Beyli theorem.

Finally, since you are doing arithmetic geometry I would suggest also more classical references, for example the book of Liu "Algebraic Geometry and Arithmetic Curves", and the book of Silverman about elliptic curves.

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