On a class of first order congruences of lines

Abstract

We study a class of new examples of congruences of lines of order one, i.e. the congruences associated to the completely exceptionalMonge-Ampere equations. We prove that they are in general not linear, and that through a general point of the focal locus there passes a planar pencil of lines of the congruence. In particular, the completely exceptional Monge-Ampere equations are of Temple type. Introduction In [AF01], Agafonov and Ferapontov introduce and study congruences of lines associated to hyperbolic systems of conservation laws. They prove that in PN, N ≤ 4, these families of lines, if the systems are of Temple type, are, in fact, linear congruences. Successively, they consider, for all N ∈ N, the completely exceptional MongeAmpere equations, studied in [Boi92], and state that these systems are of Temple type, and for N ≥ 5, the associated congruence of lines is not linear. In [DM07], we have studied the first interesting case, i.e. N = 5, giving a geometrical construction of the congruence of lines BMA in P 5 in this way obtained: it results to be a first order congruence and a smooth Fano 4-fold in P11 in the Plucker embedding; its focal locus is a sextic threefold X such that the lines of BMA through a general point of X form a (planar) pencil. This confirms the fact that the considered system is of Temple type. ∗This research was partially supported by MiUR, project “Geometria delle varieta algebriche e dei loro spazi di moduli” for both authors, and by Indam project “Birational geometry of projective varieties” for the second one 2000Mathematics Subject Classification : Primary 14M15; Secondary 53A25, 14M07.