Question

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 http://math.stackexchange.com/questions/53676/blowing-up-a-subvariety-what-can-happen-to-the-singular-locus. Apologies for wasting time if so.

Answer 1

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).