The gonality theorem of Noether for hypersurfaces

TL;DR

This paper extends Noether's gonality theorem from plane curves to higher-dimensional hypersurfaces, establishing the degree of irrationality for certain smooth hypersurfaces and classifying exceptions.

Contribution

It generalizes gonality to the degree of irrationality for hypersurfaces, providing new bounds and classifications for surfaces and threefolds in projective spaces.

Findings

Degree of irrationality for hypersurfaces of large degree is d-1.

Classified exceptions where degree of irrationality is d-2.

Improved understanding of congruences of lines in projective spaces.

Abstract

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