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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?)



Blowing up of a singular subvariety
Possibly the simplest example is to consider the blowup of a reduced and irreducible curve $C$ in a smooth 3-fold $X$ with a point $P\in C\subset X$ which is locally analytically isomorphic to
$$ 0\in \Gamma=\mathbb{V}(xy,yz,zx)\subset \mathbb{A}^3\_{x,y,z} $$
(i.e. in a small neighbourhood of $P\in X$, the curve $C$ looks like the union of the three coordinate lines $\Gamma\subset \mathbb{A}^3$). Thus we can understand what happens in a neighbourhood above $P\in X$ by considering the blowup of $\Gamma\subset\mathbb{A}^3$, even though $\Gamma$ is not irreducible.


The ideal $I=(xy,yz,zx)$ defining $\Gamma$ is generated by three equations $f=xy$, $g=yz$, $h=zx$ and these three equations satisfy two syzygies $zf=xg$, $zf=yh$. Thus the blowup is isomorphic to the complete intersection of codimension 2
$$ \sigma : \operatorname{Bl}\_\Gamma \mathbb{A}^3 = \mathbb{V}(zf-xg, \; zf-yh) \subset \mathbb{A}^3\_{x,y,z}\times \mathbb{P}^2\_{f,g,h} \to \mathbb{A}^3\_{x,y,z}. $$
There are four exceptional divisors. Three of them are isomorphic to $\mathbb{A}^1\times\mathbb{P}^1$ and they each dominate one of the three irreducible components of $\Gamma$. The last one isomorphic to $\mathbb{P}^2$ and it dominates the origin.


As this example shows, once you start blowing up subvarieties $V\subset X$ with non-lci singularities I believe this phenomenon (of having extra exceptional divisors which dominate the singularities of $V$) is essentially pretty typical.


