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On the stability of the universal quotient bundle restricted to congruences of low degree of \(\mathbb G (1, 3)\). (English) Zbl 1202.14038
Denote by \(G\) the Grassmannian of lines in \(\mathbb{P}^3\), which is a quadric in \(\mathbb{P}^5\), and contains two different families of planes. Any smooth surface contained in \(G\) is a congruence of lines. The bidegree \((a,b)\) of a congruence is defined in terms of its intersection with planes of the two families. It is not completely known for which numbers \(a,b\) there exists a congruence of bidegree \((a,b)\) in \(G\). Dolgachev and Reider conjectured that the restriction of the universal rank \(2\) quotient bundle \(Q\), to any congruence \(S\), is semi-stable. The conjecture, together with Bogomolov’s inequality on the Chern classes of semi-stable bundles, implies that the bidegree of a congruence must satisfy \(a\leq 3b\). The authors attack the conjecture by studying the highest slope of subbundles of \(Q_{|S}\). Their technique makes use of degeneration arguments, so their results apply to the conjecture only in the range in which the irreducibility of the Hilbert schemes, parameterizing congruence, is settled, i.e. for small degrees. The main result shows that Dolgachev-Reider’s conjecture holds for congruences of degree \(\leq 10\).
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M15 Grassmannians, Schubert varieties, flag manifolds
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