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Let $X$ be an irreducible, generically smooth, closed subscheme of $\mathbb{P}^N=\mathbb{P}^N_ {\mathbb{C}}$ with saturated ideal $I_{X,\mathbb{P}^N}$ generated by quadrics, and let $\widetilde{\mathbb{P}^N}=\mathrm{Bl}_{X}(\mathbb{P}^N)\stackrel{f}{\longrightarrow}\mathbb{P}^N $ be the blow-up of $\mathbb{P}^N$ along $X$, $E=V( f^{-1}(\mathcal{I} _{X,\mathbb{P}^N})\cdot \mathcal{O} _{\widetilde{\mathbb{P}^N}})$ the exceptional divisor.

Also let $Q\in I_{X,\mathbb{P}^N}$ be a fixed smooth quadric, $\widetilde{Q}=\mathrm{Bl}_{X}(Q)$ the strict transform of $Q$, $F=E\cap \widetilde{Q}$ the exceptional divisor.

Moreover assume $\widetilde{Q}$ smooth and $F$ reduced and irreducible.

My question is: Is it true that $$\exists\ \alpha\in \mathbb{Z}\ : \ f^{\ast}(Q)\sim \widetilde{Q}+\alpha\ E$$

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Do you know if $E$ is irreducible? If it is, what you want is certainly true. – Sándor Kovács Jul 14 '11 at 18:12
Unfortunately I do not know about – gio Jul 14 '11 at 19:55

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