# Log resolutions of linear series

Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal. Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$.

Is $f$ a log resolution of the linear series $|L|$ (even if $X$ is not smooth)?

If it is do you have a proof or a reference for this?

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I don't think so. Suppose for example that $X$ is smooth, and the base locus of $|L|$ is set-theoretically a divisor with normal crossing, but it has an embedded component. In this case $X$ itself will be a log-resolution of the base locus, but not of the linear system $|L|$.
This seems pretty obvious to me. The linear system corresponds to a line bundle $L$ with a number of independent sections $s_i$, whose zero scheme is the base locus. When you pullback the $s_i$ the base scheme pulls back. – Angelo Jun 18 '10 at 5:25