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In this paper we study the normal bundle of the embedding of subvarieties of dimension  n  – 1 in the Grassmann variety of lines in   . Making use of some results on the geometry of the focal loci of congruences ([Arrondo, Bertolini and Turrini, Asian J. of Math. 5: 535–560, 2001] and [Arrondo, Bertolini and Turrini, Asian J. of Math. 9: 449–472, 2005]), we give some criteria to decide whether the normal bundle of a congruence is ample or not. Finally we apply these criteria to the line congruences of small degree in   .

On the ampleness of the normal bundle of line congruences