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Let $M$ be a smooth close $4-$dimensional manifold and let $p\in M$ be a point. Denote by $\hat{M}$ the blow-up of $M$ at $p$ and let $E \cong \mathbb{C}P^{1}$ denote the exceptional sphere which is embedded. How can one show that the self-intersection number of $E$ is $-1$, i.e. $E \cdot E = -1$?

Phillip
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  • Do you know any reference where the answer of Ashwath Rabindranath from your link is explained in more detail? I have to admit that I am new in this field. – Phillip Oct 22 '17 at 07:01
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    For general 4-manifolds, this is essentially by definition. The blowup replaces a ball with a punctured $\overline{\Bbb CP^2}$. The exceptional sphere is just the sphere representing a line in $\Bbb CP^2$. There is no (as far as I know) other reasonable extension of the algebraic geometric definition. – PVAL-inactive Oct 23 '17 at 23:17

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