# Blowing up a subvariety - what can happen to the singular locus?

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

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 given 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).
• 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