This is a somewhat vague question, but given my general ignorance in algebraic geometry, I hoped that maybe someone could give me some hint...

Suppose that I have a variety $X$ (over an algebraically closed field of characteristic 0) and I want to prove that it's normal. Assume (for simplicity) that I know that away from one point $x\in X$.

Assume now that I have an explicit resolution $\pi:Y\to X$ and I know everything about its scheme-theoretic fibers (in my particular case they are irreducible but unfortunately not reduced; however, the corresponding reduced scheme is always smooth). Is there some statement about that resolution (maybe its fibers) that would imply normality of $X$ (once again: it is clear that if the fibers were irreducible and reduced, that would be it; in my case they are not reduced, but completely describe them. Does it help?)