How do you define blow-up? It should be straightforward to show that an explicit construction has relative dimension 0 over $X$.

How do you define blow-up? It should be straightforward to show that an explicit construction has relative dimension 0 over $X$.

Update: In the comments I suggest to take the formal two-dimensional ring $k[[x_1, x_2]]$ and work with it. This assumes that $X$ is smooth. If it's not, then Olivier gave an example of quadratic cone where blowup has relative dimension 1.