The gonality theorem of Noether for hypersurfaces

It is well known since M. Noether that the gonality of a smooth plane curve of degree d at least 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 to a k-dimensional projective space. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in the n-dimensional projective space in terms of degree of irrationality. We prove that both surfaces in P^3 and threefolds in P^4 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 P^n. In particular, we also slightly improve the description of such congruences in P^4 and we provide a bound on degree of irrationality of hypersurfaces of arbitrary dimension.