A proof of the assertion in the last sentence: a nef and big line bundle gives a birational contraction $X \rightarrow Y$; if the bundle is not ample this contracts a (nonempty) finite set of curves, each of which has negative selfintersection by the Hodge Index theorem, hence is a $(-2)$-curve.

Fair enough. I don't think the extra condition will change things, though: even if $L$ is nef and big, it may well be trivial on some subvariety (e.g. the exceptional divisor of a birational morphism). Then $L$ still cannot "see" what is happening on that subvariety. I'll let someone else give you a precise answer, though.

My previous comment was written before the words "and big" appeared in the last line...

I am far from an expert, and Jason Starr will be along at some point to give a full answer, but here's a basic point that occurs to me: the point of (semi-)stability is to construct reasonable moduli spaces. If you weaken ampleness to basepoint-freeness then you could take $L=O_X$; then every sheaf is semistable of slope zero. The moduli space of all sheaves on $X$ is not a reasonable object, for several reasons.

The top Chern class of a vector bundle $V$ is the rational equivalence class of the zero-locus of a not-identically-zero section of $V$.

There have been several similar questions recently, e.g. mathoverflow.net/questions/215847/… and mathoverflow.net/questions/215665/… They appear to be very popular. I must say that I think "[Theorem statement] What is the intuition behind this theorem?" is not a good question template.

Nice question. For the benefit of "outsiders", it might be clearer to avoid using surface to mean two different things

Dear David, a smooth projective surface minus finitely many points cannot be affine, because functions which are regular outside a subset of codimension 2 are regular everywhere. (Sorry to keep being annoying.)

Nice question. Sorry to pick a nit, but what does "affine K3" mean?

I can't resist mentioning that a small band of rebels has been trying to replace $NE(X)$ with the notation $\operatorname{Curv}(X)$, which has the advantage that actually gives some clue to what it denotes. See for example: math.ucla.edu/~totaro/papers/public_html/cone.pdf As far as I can tell, it's not making much difference.

Dear Donu: to make your interesting answer more complete, could you say a word or two about what $W_\bullet(\mathcal O)$ is?

@MikhailBondarko: A quadric cone is not smooth.

Does Hartshorne really use that notation for the stalk of the tangent shead at a point? It is mightily confusing.

I am a little confused about this question. Already the simplest nontrivial example, namely line bundles of degree 0 on an elliptic curve, shows that $h^0$ is not locally constant (only semicontinuous). Can you clarify what I am misreading?

It looks like you just expand out the integrand as a power series in $\mathbf e$, set $\mathbf e^3=5$ and kill everything else. Am I missing something?