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No general numerical criterion

Numerical criteria for base-pointfreeness is are known except only in specific cases such as the Kawamata basepoint-free theorem and Reider's theorem (for surfaces$\dim X=2$).

In the case you mention, $X=\mathcal{M}_{0,n}$, the problem of classifying semi-ample divisors is an important problem. It is slightly easier in positive characteristic, thanks to a theorem of Keel which says that a nef line bundle $L$ is semi-ample if and only if the restriction $L|_E$ is semiample, where $E$ is the exceptional locus of subvarieties $Z$ such that $L^{\dim Z}.Z=0$. If $f:X\to Y$ is a morphism with exceptional locus $E$, then $L$ is semi-ample if and only $L^r$ is the pullback of an ample line bundle on $Y$ for $r>0$. For more the precise statements, you might want to take a look at

According to G. Farkas' article, there are currently no known examples of nef divisors which are not semi-ample.

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No general numerical criterion for base-pointfreeness is known except in specific cases such as the Kawamata basepoint-free theorem and Reider's theorem for surfaces.

In the case you mention, $X=\mathcal{M}_{0,n}$, the problem of classifying semi-ample divisors is an important problem. It is slightly easier in positive characteristic, thanks to a theorem of Keel which says that a nef line bundle $L$ is semi-ample if and only if the restriction $L|_E$ is semiample, where $E$ is the exceptional locus of subvarieties $Z$ such that $L^{\dim Z}.Z=0$. If $f:X\to Y$ is a morphism with exceptional locus $E$, then $L$ is semi-ample if and only $L^r$ is the pullback of an ample line bundle on $Y$ for $r>0$. For more precise statements, you might want to take a look at

According to G. Farkas' article, there are currently no known examples of nef divisors which are not semi-ample.