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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?)

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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.

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  • $\begingroup$ Could you explain more about how to use syzygies to compute blowing up? $\endgroup$
    – Hydrogen
    Sep 24 at 17:09
  • $\begingroup$ 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 $A$. $\endgroup$
    – Tom Ducat
    Sep 26 at 8:53
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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.

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  • $\begingroup$ Sorry, my question maybe misleading and I will edit it. I actually mean blowup a singular subvariety in a smooth variety. (Which is the assumption in the lemma) $\endgroup$
    – Hydrogen
    Sep 17 at 22:27

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