# Noether's Theorem on Gonality of Plane Curves for Hypersurfaces

• Volume: 6, Issue: 3, page 781-791
• ISSN: 0392-4041

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## Abstract

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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 $d-2$.

## How to cite

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Bastianelli, Francesco. "Noether's Theorem on Gonality of Plane Curves for Hypersurfaces." Bollettino dell'Unione Matematica Italiana 6.3 (2013): 781-791. <http://eudml.org/doc/294044>.

@article{Bastianelli2013,
abstract = {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 $d - 2$.},
author = {Bastianelli, Francesco},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {781-791},
publisher = {Unione Matematica Italiana},
title = {Noether's Theorem on Gonality of Plane Curves for Hypersurfaces},
url = {http://eudml.org/doc/294044},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Bastianelli, Francesco
TI - Noether's Theorem on Gonality of Plane Curves for Hypersurfaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/10//
PB - Unione Matematica Italiana
VL - 6
IS - 3
SP - 781
EP - 791
AB - 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 $d - 2$.
LA - eng
UR - http://eudml.org/doc/294044
ER -

## References

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