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Reference and Question 7(a) of Problem Set 8

The main parts of Question 7(a) are:

Let $H$ be the line at infinity in $\mathbb{CP}^{2}$, and let $P$ and $Q$ be distinct points on $H$. Let $X$ be the blow up of $\mathbb{CP}^{2}$ at $P$ and $Q$; let $E_{1}$, $E_{2}$ be the exceptional divisors over $P$, $Q$ and let $L$ be the proper transform of $H$. What is the self-intersection number of $L$?

I know that the self-intersection number of $H$ and the exceptional divisors is $1$ and $-1$, respectively. And blowing up a point on $L$ reduces the intersection number by $1$. So, this would mean that $L$ now has the number $-1$? However, I do not know how one finds the self-intersection number of the proper transform of $L$? A non-expert explanation will be welcomed.

Alex Ravsky
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Radz
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1 Answers1

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You are right, $L^2 = -1$.

Giving a non-expert explanation depends on what prior results you have available. Here is a reasonable guess. You may know that for a divisor $D$ "downstairs" with total transform $\tilde{D}$, the self-intersection number is preserved: $\tilde{D}^2 = D^2$. Here one intersection number is computed "upstairs" (in this case, on $X$) and the other is computed "downstairs" (in this case, on $\mathbb{P}^2$). Granting this, with $D=H$, we have $\tilde{D} = L + E_1 + E_2$. We know $H^2 = 1$. So $(L+E_1+E_2)^2 = 1$. Distributing, we get $$ L^2 + 2L.(E_1+E_2) + (E_1^2 + 2 E_1.E_2 + E_2^2) = 1. $$ We know $L.E_1 = L.E_2 = 1$ since the curves intersect transversely (normally). We know $E_1.E_2 = 0$ since the exceptional divisors are disjoint. And we know $E_1^2 = E_2^2 = -1$. So $$ L^2 + 2 \cdot 2 + (-1 + 0 -1) = 1 $$ which gives $L^2 = -1$.

Zach Teitler
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  • Let's assume there is a singularity $R$ on $E_{1}$ which I blow up such that $R$ is replaced by the exceptional line $E_{3}$. But then, say, $R$ is also on the proper transform $L'$ of $L$. So, the self-intersection number of the proper transform $E_{1}'$ of $E_{1}$ is $-2$ and that of $L'$ is $-2$? How do we relate this to an equation similar to $L$, $E_{1}$ and $E_{2}$? – Radz Oct 05 '17 at 13:09
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    Right. Say $X$ is a smooth surface, $C$ is a curve on $X$, $P$ is a point on $C$ of multiplicity $m$ ($m=1$ in the ones you are asking about), and $C'$ is the strict transform of $C$ in the blowup of $X$ at $P$. Say the exceptional divisor is $E$. Then we have $E^2=-1$, $C'.E = m$, the total transform is $C'+mE$, and $(C'+mE)^2 = C^2$... – Zach Teitler Oct 05 '17 at 14:28