The gonality theorem of Noether for hypersurfaces

Bastianelli, F.; Cortini, R.; De Poi, P.

doi:10.1090/S1056-3911-2013-00603-7

Authors: F. Bastianelli, R. Cortini and P. De Poi
Journal: J. Algebraic Geom. 23 (2014), 313-339
DOI: https://doi.org/10.1090/S1056-3911-2013-00603-7
Published electronically: September 10, 2013
MathSciNet review: 3166393
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Abstract | References | Additional Information

Abstract: It is well known since Noether that the gonality of a smooth curve ${C\subset \mathbb {P}^2}$ of degree $d\geq 4$ 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 map ${X\dashrightarrow \mathbb {P}^k}$. 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 the degree of irrationality of hypersurfaces of arbitrary dimension.

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Additional Information

F. Bastianelli
Affiliation: Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy
MR Author ID: 878934
Email: francesco.bastianelli@unimib.it

R. Cortini
Affiliation: I. T. G. Fazzini-Mercantini, via Salvo D’Acquisto 30, 63013 Grottammare (AP) - Italy
Email: renzacortini@fazzinimercantini.it

P. De Poi
Affiliation: Dipartimento di Matematica e Informatica, Università degli Studi di Udine, via delle Scienze 206, 33100 Udine - Italy
MR Author ID: 621166
ORCID: 0000-0002-6741-6612
Email: pietro.depoi@uniud.it

Received by editor(s): February 28, 2011
Received by editor(s) in revised form: May 19, 2011
Published electronically: September 10, 2013
Additional Notes: The first author has been partially supported by Istituto Nazionale di Alta Matematica “F. Severi”; FAR 2010 (PV) “Varietà algebriche, calcolo algebrico, grafi orientati e topologici”, and INdAM (GNSAGA). The third author has been partially supported by MIUR, project “Geometria delle varietà algebriche e dei loro spazi di moduli”
Article copyright: © Copyright 2013 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.