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

I know that blow ups can be used for the resolution of singular points on a variety $X$.

What I need to know is - 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 a carefully chosen singular point, it might go down. $\textbf{Can it go up?}$

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

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