On projective manifolds admitting 3-gonal or 4-gonal curve sections

Abstract

Let X be a projective manifold, of dimension n≥3, and L a very ample line bundle on X. In this paper we investigate the pairs (X,L) with the following property: there exists a transversal intersection of n-1 members of |L| which is a smooth d-gonal curve (d=3,4). We prove, under suitable assumptions, that n≤4 or 5 (resp. if d=3 or 4) and (X,L) is a fibration over P1 with genera; fibre a rational cubic (resp. quartic) scroll.