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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.

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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 too elementary.

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# Technique to prove basepoint-freeness

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?

Maybe I miss some obvious method. Please don't hesitate adding answer although you think your idea on the top of your head is too elementary.
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