Stable base loci

Let $\mathbb{B}(L):=\bigcap_{m \in \mathbb{N}} Bs(|mL|)$ be the stable locus of Cartier divisor. I have read the paper "Restricted volumes and base loci of linear sistems" in which it's proved that the base locus doesn't contain isolated points, but at the end there's a remark that states that it's possible to prove it using the machinery of multiplier ideals. Someone can tell me how it works? Or suggest me a refererence? thank you schumacher

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1 Answer

I don't know if the following is the elementary theory of multiplier ideals mentioned in the paper.

In the case of $B_+(D)$, I think the key point is that we can assume that $M$ is a nef and big line bundle. Since $M$ is big, the multiplier ideal $\mathcal{J}(|M|)$ is not trivial at $Bs(|M|)=\{\{f^{-1}(x)\}\}$ (Theorem 11.2.21, Lazarsfeld). But for a nef and big line bundle $M$, the multiplier ideal $\mathcal{J}(|M|)$ is trivial (Proposition 11.2.18, Lazarsfeld).

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I'm sorry but I don't understand the notation...what are $M$ and $f$? – Gianni Bello Aug 28 '11 at 20:45
$f$ is the normalized blowing up of $Bs(|mD|)\setminus \{x\}$. $M=f^∗(mD)−F$, where $F$ is the fixed part. – Fei YE Sep 4 '11 at 12:08