Question

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

Answer 1

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).