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You might want to look up some things about index theorems (particularly Atiyah-Singer). They tend to relate topological and geometric data, so you can put geometric data in and topological data out. …

You would probably do well with any of the resources under "D-modules" on Gaitsgory's page. Ginzburg's lectures are good, as is the book by Hotta, Takeuchi and Tanisaki.

Here's a few that I found back when I was considering doing enumerative geometry: Gromov-Witten Theory and Donaldson-Thomas Theory I and II, referred to as MNOP Maps, Sheaves and K3 Surfaces by Pand …

You can start with these notes by Vistoli, which talk about that stuff in the direction of doing stacks and descent theory. The other articles in FGA explained might be useful, as they do a lot of mo …

I have to agree strongly with Ame's answer, in part. Weibel is a great place to go for the formalism. Once you have a little bit of the formalism, though, where to go depends on interests. To reall …

I have to follow Alberto's answer with Deligne and Mumford's paper on irreducibility of the moduli of curves.

You need the field extension to have transcendence degree 1 over $\mathbb{C}$ to get a Riemann surface. More generally, you can get an algebraic curve at least for every transcendence degree 1 extens …

Ok, I might be missing something (I often am) but I believe that this does it. Scroll up to page 452, line 3.9. The book is Effective Methods in Algebraic Geometry by Rossi and Spangher.

And actually, as a partial answer to my own question, I just stumbled across Schubert's "Kalkul" on Google Books, and it looks complete, which makes me rather happy, though other portions of the quest …

For deformation theory and complex manifolds, I'm a fan of Manetti's lecture notes.

Every book on Riemann surfaces should. My personal favorite is Rick Miranda's "Algebraic Curves and Riemann Surfaces" but there's also Farkas and Kra, which gives a more analytic point of view.

You want concrete? Then you want curves and surfaces! Check out Chapter V of Miranda's "Algebraic Curves and Riemann Surfaces" to see lots of stuff about divisors, how they're made up of functions, ho …

I want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things. What I am looking for are references for classical enumerative geometry, back before …

I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the Problem …

"Algebraic Geometry and Arithmetic Curves" by Liu might be good, it covers a lot of the same material, but does it more arithmetically. There's also "An Invitation to Arithmetic Geometry" by Lorenzin …