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On Fano manifolds, which are \({\mathbb{P}}^ k\)-bundles over \({\mathbb{P}}^ 2\). (English) Zbl 0728.14037
Studied in earlier works are Fano threefolds and Fano manifolds \(X={\mathbb{P}}({\mathcal E})\) where \({\mathcal E}\) is a rank 2 bundle on a surface. Using Mori theory, the authors classify (in an imprecise way) Fano manifolds \(X={\mathbb{P}}({\mathcal E})\), where \({\mathcal E}\) is a rank r bundle on \({\mathbb{P}}^ 2\), according to Chern classes \(c_ 1, c_ 2\) and a description of X. Thus, for example, if \(c_ 1=c_ 2=2\), one classifies according to:
(i) \(r=2\) when X is a blow up of a cone over a line; or
(ii) \(r\geq 3\) when X is the blow up of a cone over a smooth quadric along a linear subspace \({\mathbb{P}}^{r-1}\) containing the vertex \(\cong {\mathbb{P}}^{r-2}.\)
As a precursor to the proof of this result, the authors show that \(X={\mathbb{P}}({\mathcal E})\) is Fano if and only if \({\mathcal E}\) is spanned.

MSC:
14J45 Fano varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
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