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It is rather simple to study elliptic curve themselves. But since your objective is stacks, you are going to have to be much more abstract than is usual. If you want to avoid algebraic geometry as muc …

I note that it is possible to move over from the manifold point of view, to use the more refined approach of algebraic geometry. This is because the Plucker embedding is actually "algebraic". In tha …

This is a question posed to me in private communication by this user. Given a scheme $T$, let $\Gamma (T) = Mor (T, \mathbb{A}^1)$ be the ring of global sections. Note that there is a canonical map $ …

This is a very minor point, but one which had been grating me for a while. I apologize for asking a relatively trivial question, but nevertheless hope that it is suitable for MO since it should have a …

In number theory, base change of a scheme or a variety is with respect to the underlying ring or field, is viewing the same scheme/variety over an extended ring or field, but with the "same" set of eq …

I think the books of Shafarevich meet your criteria. He gives analytic intuitions when he starts explaining about schemes. I had found it to be very helpful.

Given a moduli problem, it appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space(is this strictly true?). In any case, assuming the above, what add …

$H^1$ is the first derived functor of the functor $H^0$ of global sections. In the Cech cohomology construction, note that we look whether the local sections glue together to form global sections. On …

This question arose after I thought about Ben Webster's comments to this question. There he asked me what was my definition of a moduli problem. When I came to think of it, I never saw a precise defi …

Look at Darmon's article on "Arithmetic of Curves".

See C. Chevalley, Introduction to the theory of Algebraic functions of one variable. That book develops the whole theory of algebraic curves using just its function field.

Try Serre, Topics in Galois Theory.

This perspective is given in C. Chevalley, "Introduction to the theory of algebraic functions in one variable". There instead of considering smooth algebraic curves over the complex numbers(or Riemann …

I am surprised that no one mentioned this so far; I am only imagining that everyone thought it so natural that it escaped their mind. Most "standard courses" would be following Hartshorne's book, I …

Read the excellent proof of Yoneda's lemma in wikipedia. Read the short section on "functor of points" in Mumford's "Lectures on curves on an algebraic surface. For reviewing the basics of schemes et …