Let $X$ be a smooth projective variety over $\mathbb{C}$, and $Z \subset X$ be a reduced irreducible subvariety of $X$, in particular, I do not assume it is to be smooth. Let $f: Y \to X$ to be the blowup of $X$ along $Z$.
My question: is the exceptional locus of $f$ irreducible?
I can show that the exceptional locus is purely of codimensional $1$ by the smoothness of $X$, but I don't know if there is only one component.
Notice: there is one component which is the closure of the exceptional divisor of the blowup for $U$ along $Z\cap U$, where $U\subset X$ is an open set such that $Z\cap U$ is smooth. In $U$, everything is like smooth case, hence what I worry about is the part $Z\backslash U$ may contribute to some component.