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

Bollettino dell'Unione Matematica Italiana (2013)

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

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topBastianelli, 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 -

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