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you might be looking for Kollar's book Lectures on Resolution of Singularities

As for the first question, the class of X has to be the product of the Chern roots of the bundle, so in the Chow ring, it is the class of a complete intersection. As for the second question, you woul …

I think you are looking for this http://arxiv.org/pdf/math/0506120 which is the same as: Rizov, Jordan Moduli stacks of polarized $K3$ surfaces in mixed characteristic. Serdica Math. J. 32 (2006), …

The classical (pre Deligne-Mumford) approach is to map $\mathcal{M}_g$ into $\mathcal{A}_g$ using the Torreli map. Whereas the classical way to see that $\mathcal{A}_g$ is quasi-projective is to defin …

The funniest example I know of is the number of conics tangent to five conics. There are now a number of different proofs, all based on modern intersection theory. Over the years, and until Fulton and …

In Alain Roberts "Elliptic curves: notes from postgraduate lectures given in Lausanne 1971/72" page 11 (available on google books unless you already tried to read another chapter), there is a hand dra …

The trick I know (learned it from Ron Livne) is to project it to some space with known homotopy / homology, throw away the ramification and branch loci to get a covering map (and you better pray it's …

One of the annoying aspects with sheaf cohomology in algebraic geometry is that - even for curves over the complex numbers - the cohomological dimensions are not what you want them to be, if you are u …

The "standard" definition of a K3 surface is field independent (unless you are a physicist): $p_g=1, q=0$, and trivial canonical class. Some results: Mumford and Bombieri showed that you get (just …

you can blow up a point on a general type surface, and get a general type surface containing a copy of C.

Regarding EGA, I think the most appropriate answer is: "wy bother ?". Unless you have a really special interest, you shouldn't. Edit Expanding on this (it seems a lot of people seems it's just flame …

Let $C$ be a non hyperelliptic complex algebraic curve of genus $g$, then the vector space $I_2(C)$ of quadrics containing the canonical image of $C$ is $\binom{g+1}{2}-h^0(2\omega_C) = (g-2)(g-3)/2$ …

Look at Dan Abramovich's Birational geometry for number theorists

AFAIK, these are know only up to genus 3. Genus 2: Igusa (classical). Hyperelliptic genus 3: Shioda (classical). Non hyperelliptic genus 3: a decade ago by Dixmier & Ohno - see https://www.win.tue.n …

There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and Doyle-McMullen approaches (and then some more).