Let $X$ be a proper, finite type scheme over a field $k$. What useful properties of line bundles (e.g. amplitude, nefness) can be detected cohomologially?

For example, in our setting we have the Cartan-Grothendieck-Serre theorem characterizing ample line bundles on $X$ as those line bundles $L$ such that for any coherent sheaf $\mathscr{F}$ on $X$, we have $H^i(X, \mathscr{F} \otimes L^{\otimes n}) = 0$ for all $i > 0$ and $n \gg 0$. Restricting further to projective varieties, we have that a line bundle $L$ is big iff $h^0(X, L^{\otimes n})$ grows like $n^{\dim X}$.

Do other such characterizations hold for nef line bundles? semiample? other useful classes of line bundles which I've forgotten?