Simplicity of the universal quotient bundle restricted to congruences of lines in \(\mathbb P^3\).

*(English)*Zbl 1112.14050From 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.

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 |

Full Text:
DOI

##### References:

[1] | Arrondo I, S.) (50) pp 96– (1992) |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.