Let $X$ be a non-singular projective variety defined over the complex numbers, and let $Y\subset X$ be a non-singular subvariety. Denote by $\phi: \tilde{X} \dashrightarrow X$ the blow-up of $X$ along $Y$. We know that the exceptional divisor $E := \phi^{-1}(Y)$ need not be irreducible.
Question: What can we say about the intersection of the irreducible components of $E$? e.g.~are they pairwise disjoint?
Thanks!