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Has somebody translated J.-P. Serre's "Faisceaux algébriques cohérents" into English? At least part of it? In a fit of enthusiasm, I started translating it and started TeXing. But after section 8, I …

This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil co …

For the moduli problem of a curve of genus $g$ with $n$ marked points, how large an $n$ is needed to ensure the existence of a fine moduli space? For this question, terminology is that of Mumford's G …

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 …

The other book suggestions are all so far excellent; the only caveat with them is that they all get into the number theoretic aspects very soon. I am taking the guess that you are more geometrically m …

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.

First case: Complex numbers. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Rob …

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 …

I suggest that you read Deligne's wonderful paper “Le Groupe Fondamental de la Droite Projective Moins Trois Points”. I read a little bit of it and was astonished. Please do take it up and read it wit …

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 …

$H^1$ 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 …

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 …

Can someone provide examples of Kähler manifolds which are not algebraic? This question came to my mind seeing the post of Andrea Ferretti.

This question was prompted by the post here, and I asked this earlier, deleted it, and due to pressure exerted by Ilya Nikokoshev, I am asking it again. Apologies to Pavel Etingof. Q1. Let $\Lambda$ …

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 …