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Suppose I have a smooth algebraic surface $X$ and a subscheme $Z$ such that the blowup $$Y = Bl(X,Z)$$ is smooth.

Now $Z$ does not have to be smooth, say if it is given by some power of the maximal ideal at a point, but is it nevertheless the case that $Y$ is isomorphic to the iterated blowup of $X$ along distinctsmooth points?

Suppose I have a smooth algebraic surface $X$ and a subscheme $Z$ such that the blowup $$Y = Bl(X,Z)$$ is smooth.

Now $Z$ does not have to be smooth, say if it is given by some power of the maximal ideal at a point, but is it nevertheless the case that $Y$ is isomorphic to the blowup of $X$ along distinct points?

Suppose I have a smooth algebraic surface $X$ and a subscheme $Z$ such that the blowup $$Y = Bl(X,Z)$$ is smooth.

Now $Z$ does not have to be smooth, say if it is given by some power of the maximal ideal at a point, but is it nevertheless the case that $Y$ is isomorphic to the iterated blowup of $X$ along smooth points?

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anon
anon
  • 1
  • 1

Smooth blowup of a surface

Suppose I have a smooth algebraic surface $X$ and a subscheme $Z$ such that the blowup $$Y = Bl(X,Z)$$ is smooth.

Now $Z$ does not have to be smooth, say if it is given by some power of the maximal ideal at a point, but is it nevertheless the case that $Y$ is isomorphic to the blowup of $X$ along distinct points?