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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 …

Polya's "How to Solve It" is a good one. When prepping for the Putnam, I used "Problem Solving Through Problems"

I'm teaching a course over the summer (it's a sort of make-your-own course for non-majors) and I'm planning on organizing it as a math history course, hitting on major threads through about 1900, and …

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 …

"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 …

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 …

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to …

I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard g …

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 …

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 …

Is there a good reference for the structure of the Chow ring of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection th …

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

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.