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.
Best Answer
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.)