= Blowing up
{disambiguate=algebraic geometry}
{title2=$\operatorname{Bl}_Z X$}
{wiki=Blowing_up}
= Blowup of an algebraic variety
{synonym}
= Blowups of algebraic varieties
{synonym}
The <projective morphism> $\operatorname{Proj}_X\bigoplus_{j\ge0}\mathcal I^j\to X$ associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an <isomorphism>. Blowing up a smooth point of an $n$-dimensional smooth variety replaces the point by a <projective space> $\mathbb P^{n-1}$, the <algebraic exceptional divisor>.
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