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Examples of blown up varieties having projective bundle structures. (English) Zbl 1445.14003

Asking under which conditions a projective variety blown-up along a projective subvariety is isomorphic to a projective bundle over some projective variety is a natural question. For instance, this happens for the projective space \(\mathbb P^n\) blown-up along a linear subspace \(\mathbb P^{r-1}\), in which case the resulting variety is the \(\mathbb P^r\)-bundle over \(\mathbb P^{n-r}\) given by \(\mathbb P(\mathcal E)\), where \(\mathcal E = \mathcal O_{\mathbb P^{n-r}}(1) \oplus \mathcal O_{\mathbb P^{n-r}}^{\oplus r}\).
In the paper under review the author produces examples where the same property holds for some \(\mathbb P^n\) blown-up along a non-linear subvariety. Let \(X\) be the Segre product \(\mathbb P^1 \times \mathbb P^2\) embedded in \(\mathbb P^5\). The author shows that \(\widetilde{\mathbb P_X^5}\), the blow-up of \(\mathbb P^5\) along \(X\), is a \(\mathbb P^3\)-bundle over \(\mathbb P^2\), and describes the vector bundle \(E\) such that \(\widetilde{\mathbb P_X^5}=\mathbb P(E)\) as the cokernel of an injective homomorphism \(\mathcal O_{\mathbb P^2}(-1)^{\oplus 2} \to \mathcal O_{\mathbb P^2}^{\oplus 6}.\) Then, by taking the section of \(X\) with one or two general hyperplanes of \(\mathbb P^5\), the author replicates the construction showing that the corresponding variety obtained by blowing-up \(\mathbb P^4\) or \(\mathbb P^3\), respectively, is a \(\mathbb P^{3-i}\)-bundle (\(i=1,2\)) over \(\mathbb P^2\) and describes the corresponding vector bundle. Finally, the author describes the nef cone of all such varieties.

MSC:

14A10 Varieties and morphisms
14C20 Divisors, linear systems, invertible sheaves
14C22 Picard groups
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References:

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