Is the blowup of an integral normal Noetherian scheme along a coherent sheaf of ideals necessarily normal?

I can show that there is an open cover of the blowup by schemes of the form $\text{Spec } C$, where $B \subset C \subset B_g$ for some integrally closed domain $B$ and some $g \in B$, but I don't see why this would imply that $C$ is integrally closed. Intuitively, it seems reasonable that a blowup would be at least as "nice" as the original scheme, but that intuition may have more to do with how blowups are generally used than what they are capable of.

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    $\begingroup$ Nope; every birational map between projective varieties over a field is a blow-up along some closed subscheme of the target. General blow-ups are totally wild things. $\endgroup$ – BCnrd Sep 4 '10 at 19:27
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    $\begingroup$ If you want a criterion for when blow-up at a point preserves some nice properties, the singularity should be fairly mild. There is an affirmative result involving normality in the case of rational double point singularities, for example. (Look at the end of Artin's article on Lipman's resolution for excellence surfaces in the book "Arithmetic geometry", for example.) $\endgroup$ – BCnrd Sep 4 '10 at 19:30
  • $\begingroup$ I can't seem to come up with a birational map from a nonnormal projective variety to a normal projective variety. Can someone supply an example? $\endgroup$ – Charles Staats Sep 4 '10 at 21:55

For an explicit example, blow up any sufficiently complicated isolated singularity of a surface in affine 3-space, and the result will in general have singularities along curves so is not normal. I think x2+y4+z5 = 0 will do for example: blowing this up gives x2+y4z2+z3 = 0 on one of the coordinate charts, which is singular along the line x=z=0.

(Hypersurfaces in affine space are normal if and only if they are regular in codimension 1.)


Richard Borcherds sort of example will certainly work. There's another type of example where the ambient space is smooth.

For example, blowing up $(x^2, y^2)$ in $\mathbb{A}^2$ will yield a pinch-point (aka Whitney's Umbrella) singularity.

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    $\begingroup$ This example was extremely helpful for me. $\endgroup$ – Charles Staats Sep 5 '10 at 0:43
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    $\begingroup$ With regards to the assertion that the "blow up is at least as nice", you can make statements like this with regards to "normalized" blow-ups, but even then, smooth points can become singular points. For example, blowing up $(x^2, y)$ blown up gives a quadric cone singularity. This blow-up can also be obtained by blowing up $(x,y)$, blowing up a point on the exceptional $\mathbb{P}^1$, and then blowing down that first $\mathbb{P}^1$ $\endgroup$ – Karl Schwede Sep 5 '10 at 18:43

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