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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
##### Keywords:
spannedness; Mori theory; Fano manifolds; Chern classes
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##### References:
 [1] Proc. AMS Summer Inst. Bowdoin College. Proc. Symp. Pure Math. 46 pp 3– (1987) [2] DOI: 10.1007/BF01389057 · Zbl 0554.14001 · doi:10.1007/BF01389057 [3] Compositio Math. 76 pp 295– (1990) [4] DOI: 10.2969/jmsj/03240709 · Zbl 0474.14017 · doi:10.2969/jmsj/03240709 [5] DOI: 10.1007/BF01170131 · Zbl 0478.14033 · doi:10.1007/BF01170131 [6] Proc. Sendai Conf. Adv. in Pure Math. 10 (1987) [7] Vector Bundles on Complex Projective Spaces (1981)
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