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Let $\pi \colon X\to \mathbb{P}^4$ be the blow-up of a smooth surface $S\subset \mathbb{P}^4$. Is there a formula to compute $(K_X)^4$ ? (which should be dependent on invariants of $S$).

In dimension $3$, the formula is $(K_X)^3=62-8d+2g$, where $d$ is the degree of the curve and $g$ is its genus.

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  • $\begingroup$ Related: mathoverflow.net/questions/162839/… $\endgroup$
    – user5117
    Commented Feb 3, 2015 at 13:41
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    $\begingroup$ The cohomology algebra of a blow up can be described in a completely explicit way, see for instance Griffiths-Harris, pp. 605 and following. Getting from this the formula you are asking for is just a (somewhat lengthy...) exercise. $\endgroup$
    – abx
    Commented Feb 3, 2015 at 15:27

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