# Blowing-up along few points and embedding

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 ?

Thanks.

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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.
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