This is the same as asking for a classification of all rational surfaces. Every (smooth) rational surface is a blowup of $\mathbf P^2$ or a Hirzebruch surface. I don't think you can say anything more.

$PX$ is the image of the Segre embedding of $\mathbf P^{n-1} \times \mathbf P^{m-1}$ into $\mathbf P^{mn-1}$.

Does "resolution" here really mean "resolution" or just "partial resolution"? If the former the answer seems to be "yes" for a fairly trivial reason: let $X$ be a variety with one terminal point and one canonical point.

@René: but $\widetilde{A}$ has the same fundamental group as $A$.

Regarding your third point, maybe another nice example is intermediate Jacobians and irrationality of cubic threefolds.

Jason Starr will show up here soon to tell you why the commas are correct.

What is the problem with applying this definition directly to schemes?

Matsuki's book "Introduction to the Mori program" is intended, I think, as a more user-friendly alternative to Koll\'ar--Mori.

Can you give a reference for the statement about the Cox ring of a very general HK 4-fold? That is surprising to me.

Related: mathoverflow.net/questions/24503/…

@Y.Li: why not?

I am baffled by the closure of this question. The OP is asking how to do X. Andreas Blass has given clear examples from personal experience of how one can do X. The fact that our social conventions are strongly opposed to doing X is not the issue here.

@GunnarÞórMagnússon: I am only too familiar with this phenomenon.

@GunnarÞórMagnússon: how could there be a torsion class in $H^{1,1}(M,\mathbf R)$? That is a real vector space.

(I deleted a response to OP's follow question that said more or less: no, let $X$ be the Fermat $K3$ in characteristic 3.)

@LaurentMoret-Bailly: so it is. That feels strange, but I guess that is because I never thought carefully about it before. Thank you for educating me.

@LaurentMoret-Bailly: I don't think I follow. The OP wants to start with a purely insep. $\varphi$. In my comment $\varphi=i \circ \alpha$. Maybe I am being slow.

Let $X$ be the blowup of $\mathbf P^2$ in a point, $i: \mathbf P^2 \dashrightarrow X$ the obvious birational map, and $\alpha: \mathbf P^2 \dashrightarrow \mathbf P^2$ a purely inseparable endomorphism. Then $i \circ \alpha$ is purely inseparable, but there is no regular map $\mathbf P^2 \rightarrow X$.