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Let $X$ be a variety defined over a number field $k$. If I blow-up along some arbitrary subvariety of $X$, what are the possible outcomes for the dimension of the singular locus of the variety? If the subvariety lies outside the singular locus of $X$, then it stays the same, if it is carefully chosen, it might go down. Can it go up?

To be more specific, my variety is a high dimensional hypersurface, and the subvariety I am blowing up is a linear space of much smaller dimension than the singular locus. I don't know if this changes the situation.

I have a feeling this question might be more suited to stackexchange, but it didn't spark much interest over there Apologies for wasting time if so.

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up vote 4 down vote accepted

Any birational map $\pi:X'\to X$ is the blow-up of some ideal sheaf on $X$, so in general one must expect singularities on $X'$, even if the ideal is reduced (as you assume).

As a concrete example, let $X=\mathbb{A}^n$ and blow-up the complete intersection subvariety gven by the ideal $I=(f,g)\subset k[x_1,\ldots,x_n]$. Then the blow-up of $X$ is the Proj of the Rees algebra $R[It]$ which is given by $k[x_1,\ldots,x_n,S,T]/(fS-gT)$. By choosing $f$ and $g$ appropriately one can produce varieties with singular locus of high dimension.

For your specific example, when $Y$ is a linear space of small dimension, I don't know if the above can happen, but there are certainly cases where the dimension of the singular locus will be unchanged after the blow-up, (e.g when $Y$ a point on a singular surface).

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Thanks, that's exactly what I was looking for. – samian86 Aug 25 '11 at 13:01
No problem. Actually, I think it might to say more about your specific case. In that case the Rees algebra is given by $k[x_0,\ldots,x_n,y_1,\ldots,y_k](f,x_iy_j-x_jy_k,\ldots)$ and it seems doable to investigate the singularities in each chart $y_i=1$ by hand. Perhaps you can show that the dimension of the singular locus does indeed stay the same for some choices of $f$. – J.C. Ottem Aug 25 '11 at 15:07

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