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I'm reading Kollár Mori Chapter 2.3. And they state the following lemma:

Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum a_iD_i$ a sum of distinct prime divisors. Let $Z\subseteq X$ be a closed subvariety of codimension $k$. Let $p:B_Z\rightarrow X$ be the blow up of $Z$ and $E\subseteq B_ZX$ the irreducible component of the exceptional divisor which dominates $Z$.(If $Z$ is smooth, then this is the only component.) Then $$a(E,X,\Delta) = k-1-\sum_i a_i\cdot \mathrm{mult}_Z(D_i)$$

I have a question about this sentence:

Let $p:B_Z\rightarrow X$ be the blow up of $Z$ and $E\subseteq B_ZX$ the irreducible component of the exceptional divisor which dominates $Z$.(If $Z$ is smooth, then this is the only component.)

Does that mean the exceptional divisor of the blowing up of a singular irreducible subvariety can be reducible? Are there any examples of this phenomenon? In general, what does a blowing up of singular subvariety look like?(what contributes to nondominate component?)