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

I ask the same question on MathStackExchange but receive no answer.

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?)

I ask the same question on mathstackexchange but receive no answer.

I'm reading Koll'ar 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 $$a(E,X,\Delta) = k-1-\sum_i a_i\cdot mult_Z(D_i)$$

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

I ask the same question on MathStackExchange but receive no answer.

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