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I have a smooth proejective fourfold $X$ with an effective divisor $D$ on it. The base locus ${\rm Bs}|D|$ of the linear system $|D|$ is a smooth rational curve $C$ and a generic member of $|D|$ has multiplicity two along $C$.

Let $\pi: Y \rightarrow X$ be the blow-up along $C$ and $D' = \pi^*(D) - 2 E$, where $E$ is the exceptional divisor over $C$. I guess that the base locus ${\rm Bs}|D'|$ lies on $E$.

Is it true that $\dim ({\rm Bs}|D'|) < 1$? In particular, I would like know when $|D'|$ is base-point free.

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    $\begingroup$ Intuitively (without thinking too much) I don't see what prevents the following scenario: that all members of $|D|$ are highly singular at one point of $C$, so that all their strict transforms contain some positive dimensional subvariety of the pre-image of that point. $\endgroup$
    – pinaki
    Commented Jun 4, 2021 at 17:37

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