I am studying the blow-up construction using $\mathrm{Proj}$. 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 other bad pathologies could arise?
Bad behaviour of a blow-up
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algebraic-geometry
blowup
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2You should st$a$te your conditions more cle$a$rly. Wh$a$t 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. – 2011-07-05