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Simplicity of the universal quotient bundle restricted to congruences of lines in \(\mathbb P^3\). (English) Zbl 1112.14050
From the introduction: Let \(S\) be a congruence of \(\mathbb{G}(1,3)\), that is, a smooth surface in the Grassmannian \(\mathbb{G}(1,3)\). I. Dolgachev and I. Reider [in: Algebraic geometry Lect. Notes Math. 1479, 39–49 (1991; Zbl 0766.14031)] conjectured that, if \(S\) is not contained in a hyperplane of \(\mathbb{P}^5\) (the Plücker ambient space of \(\mathbb{G}(1,3))\), then the restriction \(Q|_S\) of the universal quotient bundle \(Q\) of \(\mathbb{G}(1,3)\) is semistable. The somehow most special case was studied by E. Arrondo and I. Sols [“On congruences of lines in the projective space.” Mém. Soc. Math. France 50 (1992; Zbl 0804.14015)], where there is a classification of those congruences for which \(Q|_S\) splits as a direct sum of two line bundles.
In this paper we improve the result of [loc. cit.] by classifying those congruences \(S\) for which \(Q|_S\) is not simple. It turns out that the list coincides with the one in [loc. cit.]. The main idea is to consider two possibilities: if \(Q|_S\) has more than four independent sections, then the corresponding congruence is projected from a surface of \(\mathbb{G}(1,4)\); such congruences are classified in [loc. cit.]. On the other hand, if \(Q|_S\) has exactly four independent sections and is not simple then there exists a fundamental line (i.e., a line that meets all the lines of the congruence), and the congruences with this property are classified in [loc. cit.], too.
MSC:
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M15 Grassmannians, Schubert varieties, flag manifolds
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[1] Arrondo I, S.) (50) pp 96– (1992)
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