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On first order congruences of lines in \(\mathbb P^{4}\) with generically non-reduced fundamental surface. (English) Zbl 1147.14025

A congruence of lines in the projective space \(\mathbb P^n\) is a subvariety \(B\) of the Grassmannian \(\mathbb G(1,n)\) of dimension \(n-1\). The order is the number of lines of \(B\) passing through a general point of \(\mathbb P^n\). Congruences of order one have proved to be interesting not only in projective algebraic geometry but also in the applications, for instance to the theory of systems of conservation laws [see S. I. Agafonov, E. V. Ferapontov, Manuscr. Math. 106, No. 4, 461–488 (2001; Zbl 1149.35385)]. The classification of these congruences in \(\mathbb P^3\) has been performed classically by Kummer and concluded by Z. Ran [J. Reine Angew. Math. 368, 119–126 (1986; Zbl 0601.14042)].
The article under review is part of series devoted by the author to the classification of congruences of order one in \(\mathbb P^4\). After classifying those with a fundamental curve [P. De Poi, Manuscr. Math. 106, No. 1, 101–116 (2001; Zbl 1066.14062)] and those with irreducible fundamental surface [P. De Poi, Math. Nachr. 278, No. 4, 363–378 (2005; Zbl 1070.14045)], here he gives a complete description of those whose focal locus \(F\), which carries a natural structure of scheme, is a generically non-reduced surface. If \(F\) is irreducible, a general line of \(B\) either intersects \(F\) in a fat point of length \(3\) or in \(2\) points, one of which is a fat point of length \(2\). In the first case \(F_{red}\) results to be a plane, and in the second one a non-degenerate cubic scroll. If \(F\) is reducible, then it splits in two components \(F_1\) and \(F_2\), the first one non-reduced and the other reduced, and there are the following possibilities: either \((F_1)_{red}\) is a plane and \(F_2\) is a rational non-degenerate surface, or \(F_2\) is a plane and \((F_1)_{red}\) is a non-degenerate rational scroll. Examples are given and the geometry of the family of lines \(B\) is explained in all cases.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
51N35 Questions of classical algebraic geometry
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