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Hello all.

Let $X_{p_0,\ldots,p_n}\subset\mathbb{P}^N=\mathbb{P}^N_{\mathbb{C}}$ be the sequence of blowing-ups $\pi:\mathrm{Bl}_{p_0,\ldots,p_n}\longrightarrow\mathbb{P}^n$ along $n+1$ points of $\mathbb{P}^n$ (maybe infinitely near points), embedded by very ample divisor $H_X\sim d\pi^{\ast}(H_{\mathbb{P}^4})-E_1-\cdots - E_n$ ($E_1,\ldots,E_n$ the exceptional divisors, $H_{\mathbb{P}^4}\in|\mathcal{O}_{\mathbb{P}^4}(1)|$, $d\geq 1$).

If $p_0' ,\ldots, p_n'$ is another list of $n+1$ points that produces $X_{p_0',\ldots,p_n'}$ as above, is true that $X_{p_0,\ldots,p_n}, X_{p_0',\ldots,p_n'}$ are projectively equivalent ?


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You might want to add some "general position" condition on the points, otherwise the two manifolds that you get are not even biholomorphic. For example take $n=2$: if the $3$ points are collinear then the resulting blowup is not Fano, but if they are not collinear it is.

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And if the points are in general position then the two varieties are isomorphic, since $Aut({\mathbb P}^n}$ acts transitively on sets of $n+1$ points in general position. – rita Oct 15 '11 at 16:22
This should be clear if the points are not infinitely near... If the points are not in general position then my statement should depend da $\dim(⟨p_0,…,p_n⟩)$. – gio Oct 15 '11 at 17:03

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