×

The family of lines on the Fano threefold \(V_ 5\). (English) Zbl 0731.14025

Let \(V_ 5\subset P^ 6\) be the Del Pezzo Fano threefold of degree \( 5;\) \(V_ 5\) can be obtained as a section of the Grassmannian \(G(2,5)\subset P^ 9\) by three hyperplanes in general position. The authors analyse in detail the universal family for the two-dimensional family of lines on \(V_ 5\) and determine the hyperplane sections which can be the boundary of \({\mathbb{C}}^ 3\) in \(V_ 5\). In §1 the authors summarize some basic results about \(V_ 5\) obtained by V. A. Iskovskikh [Math. USSR, Izv. 11, 485-527 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 41, 516-562 (1977; Zbl 0363.14010)], T. Fujita [J. Math. Soc. Japan 33, 415-434 (1981; Zbl 0474.14018)] and T. Peternell and M. Schneider [Math. Ann. 280, No.1, 129-146 (1988; Zbl 0651.14025)]. In §2 they construct a \({\mathbb{P}}^ 1\)-bundle P(E) over \({\mathbb{P}}^ 2\) \(({\mathbb{P}}^ 2 \cong the\) base of the family of lines on \(V_ 5)\), where E is a locally free sheaf of rank 2 on \({\mathbb{P}}^ 2\), and a finite morphism \(\psi\) : P(E)\(\to V_ 5\hookrightarrow {\mathbb{P}}^ 6\); further, the authors show that P(E) is in fact the universal family for the family of lines on \(V_ 5\). In §3 they study the boundary of \({\mathbb{C}}^ 3\) in \(V_ 5\) and the set \(\{\) \(H\in | {\mathcal O}_{V_ 5}(1)|: V_ 5\setminus H\cong {\mathbb{C}}^ 3\}\).
Reviewer: A.Iliev (Sofia)

MSC:

14J45 Fano varieties
14J30 \(3\)-folds
14J10 Families, moduli, classification: algebraic theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.2307/2007050 · Zbl 0557.14021 · doi:10.2307/2007050
[2] pp 66– · Zbl 0239.13011
[3] DOI: 10.1007/BF01474185 · Zbl 0651.14025 · doi:10.1007/BF01474185
[4] DOI: 10.1070/IM1977v011n03ABEH001733 · Zbl 0382.14013 · doi:10.1070/IM1977v011n03ABEH001733
[5] 1016 pp 490– (1983)
[6] DOI: 10.2748/tmj/1178227726 · Zbl 0703.14025 · doi:10.2748/tmj/1178227726
[7] Nagoya Math. J. 104 pp 1– (1986) · Zbl 0612.14037 · doi:10.1017/S0027763000022649
[8] DOI: 10.2969/jmsj/03330415 · Zbl 0474.14018 · doi:10.2969/jmsj/03330415
[9] DOI: 10.2307/1969629 · Zbl 0056.16803 · doi:10.2307/1969629
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.