The universal rank-(n − 1) bundle on G(1, n) restricted to subvarieties

Abstract

The author has, in several articles, studied varieties in the Grassmannian G(k, n) of kplanes in projective n-space, that are projections from a variety in G(k,N). In the case k = 1 the varieties of dimension n−1 in G(1, n) that are projections from G(1,N) were studied by E. Arrondo and I. Sols [“On congruences of lines in the projective space”, M´em. Soc. Math. Fr., Nouv. S´er. 50 (1992; Zbl 0804.14016)] and solved for n = 3 by E. Arrondo [J. Algebr. Geom. 8, No. 1, 85-101 (1999; Zbl 0945.14030)]. In the paper under review the author studies the other extreme k = n−1, n−2. The case k = n−1 is solved completely, and in the case k = n−2 it is shown that if Y is a smooth variety of dimension s in G(1, n) whose dual Y in G(n − 2, n) is a non-trivial projection from G(n − 2, n + 1), then s = n − 1 and Y is completely classified. The methods are from classical projective geometry and based upon results by E. Rogora [Manuscr. Math. 82, No. 2, 207-226 (1994; Zbl 0812.14038)] and B. Segre.