I'm very new to algebraic geometry, but am trying to read some papers, and am confused by a few things.
Let $X$ be a projective surface. My first question is notational:
(0) If an author writes $$ch_2(E) < c_1^2(E)$$ should I understand this to really mean $$\int_X ch_2(E) < \int_X c_1(E) \wedge c_1(E)?$$ Because I think of chern classes and chern characters as living in $H^\bullet(X,\mathbb Q)$ and $H^\bullet(X,\mathbb Z)$ respectively, I don't know how else to interpret the inequality.
Now let's say $T$ is a torsion sheaf supported in dimension 1.
(1) Why is is that $c_1(T)$ is effective?
If $T$ locally comes from a quotient $\mathcal O_X/ I$ with no nilpotence, I understand that the locus where functions vanish is cut out by a positive divisor, so $c_1$ is Poincare dual to an effective divisor. But I am not as familiar with the case that $T$ might in general might look define a ring of functions for some "divisor with infinitesimal neighborhood of order k". For instance, $\mathbb C[x,y]/x^n$.
Second, let's say that F is an ample divisor.
(2) Why do I know that $c_1(T) . F > 0$?
I know that for curves, I think of ample line bundles as effective divisors, hence (2) follows from (1); but I am uncomfortable with surfaces. The line bundle $L$ corresponding to an ample divisor $F$ eventually has enough global sections to generate it (i.e., $L^{\otimes N}$ is generated by global sections for $N$ large enough) but I am not sure why this implies that $L$ itself must have an effective divisor. A related question might be: How does the divisor $F$ change as you tensor $L$?
Third, let's say that $T$ is a torsion sheaf supported in dimension 0.
(3) Why is $ch_2(T) > 0$?
I believe this is more or less the question (1) above, but in codimension 2.
