It is well known since Noether that the gonality of a smooth curve C ? P~2 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 → P~k. In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in P~n 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 the degree of irrationality of hypersurfaces of arbitrary dimension.
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