Positive square gives effective powers

For $L$ ample on $S$ and a line bundle $M$ with $M^2>0$ and $L\cdot M\ge0$, there is $N>0$ such that for all $n\ge N$ the bundle $M^n$ has a nonzero section.

Notation. $L^2$ is selfIntersection, namely $\chi(L^2)-2\chi(L)+\chi(\mathcal O_S)$. $L\cdot M$ is mixedEuler, namely $\chi(L\otimes M)-\chi(L)-\chi(M)+\chi(\mathcal O_S)$. IsAmple means that for each point $x$ and open $V\ni x$, some positive power has a global section with nonvanishing open containing $x$, inside $V$, and affine. A Surface is an integral scheme smooth of relative dimension 2 over $\operatorname{Spec}\mathbb C$ with a closed immersion into some complex projective space over $\mathbb C$.

Formalization Note The statement is verbatim from OpenAI's Lean development, with Definitions files of this development imported in place of the upstream modules; these are supporting code for the statement.