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In this article the author considers the limit, in some appropriate sense, of the expression $$\lim_{j\to \infty}\frac{[\Gamma(\mathcal{L}^{\otimes j},X)^s]}{[\Gamma(\mathcal{L}^{\otimes j},X)]}$$ Where $\Gamma(\mathcal{L}^{\otimes j},X)^s$ are the divisors in the linear system associated to $\mathcal{L}^{\otimes j}$ with $s$ singular points. The author interprets this expression with $s=0$ as being in some the probability that a divisor on $X$ is smooth. To me this suggests that the authors has in mind something along the lines of $$\lim_{j\to\infty} \Gamma(\mathcal{L}^{\otimes j},X)=\text{"All divisors on }X\text{"}$$ Now this might be obvious, but I don't know why this interpretation is correct. So maybe to formulate a precise question:

Let $\mathcal{L}$ be any ample line bundle on $X$, and $D$ any effective Cartier divisor on $X$. Is $D$ contained in the linear system associated to $\mathcal{L}^{\otimes j}$ for large enough $j$?

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    For all large $j$, $\mathcal{L}^{\otimes j}=D+E$, where $E$ is another effective divisor. Is this what you are asking? – 2017-02-07
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    The answer to the question is no. If $D$ is in the linear system associated to $\mathcal{L}^{\otimes j}$, then $\mathcal{O}(D)\cong\mathcal{L}^{\otimes j}$. In particular, $\mathcal{O}(D)$ is ample. But not every effective divisor is ample. – 2017-02-08

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