The gonality theorem of noether for hypersurfaces

It is well known since Noether that the gonality of a smooth curve C C P2 of degree d ≥ 4 is d - 1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is, the minimum degree of a dominant rational map X → Pk. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in Pn in terms of degree of irrationality. We prove that both surfaces in P3 and threefolds in P4 of sufficiently large degree d have degree of irrationality d - 1, except for finitely many cases we classify, whose degree of irrationality is d - 2. To this aim we use Mumford's technique of induced differentials and we shift the problem to study first order congruences of lines of Pn. In particular, we also slightly improve the description of such congruences in P4 and we provide a bound on the degree of irrationality of hypersurfaces of arbitrary dimension.