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I am studying the Blow-up contruction using the $\mathrm{Proj}$. I am looking for examples in which this construction is singular. Does anyone know situations in which the transform of the blowed-up locus is not a divisor?Which are other bad pathologies that could arise?

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You should state your conditions more clearly. What are you blowing up in what, i.e. what parameters are assumed to be nonsingular? If you blow up something nonsingular in nothing, it will still be nonsingular, but that's probably not what you are looking for. – Jesko Hüttenhain Jul 5 '11 at 19:03

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