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.

The gonality theorem of Noether for hypersurfaces

DE POI, Pietro
2014

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/871778
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