# Blowing up of a singular subvariety

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 $$a(E,X,\Delta) = k-1-\sum_i a_i\cdot \mathrm{mult}_Z(D_i)$$

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?)

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.
• If $X=\mathbb{A}^3$, then by definition the blowup is the relative proj construction Proj$_XA\to X$, where the graded ring $A=\bigoplus _{n\geq0}A_n$ has $n$th term $A_n=I^n$ and is generated by $I=(xy,yz,zx)$ (in degree 1) as an algebra over the ring $k[x,y,z]$ (in degree 0). I introduce $f=xy$ etc. to distinguish between $xy\in k[x,y,z]$ (an element in degree 0) and $xy\in I$ (a generator in degree 1). The syzygies are the precisely the relations that hold between the generators $xy,yz,zx$ of $I$, and hence give the relations between the generators $f,g,h$ of $A$. Sep 26 at 8:53
Just blow up the singular point in a variety $$X_1$$ which is obtained from a smooth, irreducible manifold $$X$$ by identifying points $$x$$ and $$y$$. The blow-up divisor is $${\Bbb P} T_xX\coprod {\Bbb P}T_yX$$, disconnected.