Say $X$ is a smooth algebraic surface and $Z$ is some subscheme. When is the blowup of $X$ along $Z$ smooth?

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surface, it is always smooth if $Z$ is reduced... For non-reduced $Z$, the link Blup provided is useful. On the other hand, there is Zariski factorization of complete (integrally closed) ideals through which the surface question should be more or less tractable. – Karl Schwede Jan 30 '12 at 15:10