MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

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