Noether's Theorem on Gonality of Plane Curves for Hypersurfaces

Bastianelli, Francesco

Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 3, pp. 781-791

Résumé

A well-known theorem of Max Noether asserts that the gonality of a smooth curve $C \subset \mathbb{P}^{2}$ of degree $d \geq 4$ is $d - 1$, and any morphism $C \to \mathbb{P}^{1}$ of minimal degree is obtained as the projection from one point of the curve. The most natural extension of gonality to n-dimensional varieties $X$ is the degree of irrationality, that is the minimum degree of a dominant rational map $X -- \to \mathbb{P}^{n}$. This paper reports on the joint work [4] with Renza Cortini and Pietro De Poi, which aims at extending Noether's Theorem to smooth hypersurfaces $X \subset \mathbb{P}^{n+1}$ in terms of degree of irrationality. We show that both generic surfaces in $\mathbb{P}^{3}$ and generic threefolds in $\mathbb{P}^{4}$ of sufficiently large degree $d$ have degree of irrationality $d - 1$, and any dominant rational map of minimal degree is obtained as the projection from one point of the variety. Furthermore, we classify the exceptions admitting maps of minimal degree smaller than $d - 1$, and we show that their degree of irrationality is $d - 2$.