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Suppose that $X$ is a smooth threefold, and $C \subset X$ a smooth curve. Let $Y$ be the blowup of $X$ along $C$, with exceptional divisor $E$. What is the intersection number $E^3$ on $Y$? (in terms of the genus and normal bundle of $C$, etc)

I assume that I could extract the answer from Theorem 6.7 of Fulton's book on intersection theory, were I better familiar with the contents of chapters one through five -- I'd be happy to hear either a direct method or a pointer about how to get it from Fulton!

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up vote 3 down vote accepted

The intersection number $E^3$ equals $-\deg N_{C|X}$ the negative of the degree of the normal bundle of $C$. Here, as usual, $\deg N_{C|X}=2g-2-K_X.C$. This statement and the proof can be found in Griffiths-Harris and in Iskovskikh-Prokhorov Algebraic Geometry V III. $\S$ 2.3.

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