Let $X$ be a smooth projective variety over $\mathbb{C}$.
And let $L$ be a big and nef line bundle on $X$.
I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$).
The only way I know is using Kawamata basepoint-free theorem:
Theorem. Let $(X, \Delta)$ be a proper klt pair with $\Delta$ effective.
Let $D$ be a nef Cartier divisor such that $aD-K_X-\Delta$ is nef and big for some
$a > 0$. Then $|bD|$ has no basepoints for all $b >> 0$.
Question. What other kinds of techniques to prove semi-ampleness or basepoint-freeness
of given line bundle are?
Maybe I miss some obvious method. Please don't hesitate adding answer although you think your idea on the top of your head is elementary.
Addition : In my situation, $X$ is a moduli space $\overline{M}_{0,n}$.
In this case, Kodaira dimension is $-\infty$.
More generally, I want to think genus 0 Kontsevich moduli space of stable maps to
projective space, too.
$L$ is given by a linear combination of boundary divisors.
It is well-known that boundary divisors are normal crossing,
and we know many curves on the space such that we can calculate intersection numbers with boundary divisors explicitely.
Best Answer
I don't think that your assertion is true; for example, Lazarsfeld gives an example (PAG, 2.3.3) of a big and nef divisor on a surface such that its graded algebra is not finitely generated, so that the divisor can't be semiample.
But there are some close results for nef and big divisors, or even for good divisors (when the Kodaira dimensions equals the numerical dimension) as Mourougane and Russo showed : for example, Wilson's theorem asserts that for any nef and big divisor on an irreducible projective variety, there exists $m_0\in \mathbb N$ together with an effective divisor $N$ such that for all $m\geq m_0$, the linear system $|mD-N|$ has no base-point. (PAG, 2.3.9)