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?



Log resolutions of linear series
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 $|L|$.


