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.