The gonality of curves on a Hirzebruch surface
type: Item
instance of: scholarly article
title: The gonality of curves on a Hirzebruch surface
zbMATH Open document ID: 0872.14022
DOI: 10.1007/BF01197600
author: Gerriet Martens
published in: Archiv der Mathematik
publication date: 1996-11-24T00:00:00Z
review text: Let \(X_e\) be a Hirzebruch surface, that is a geometrically ruled rational surface of invariant \(e\geq 0\). The main theorem in this paper shows that for \(C\) a reduced and irreducible curve on \(X_e\), which is not a fibre, the minimal degree \(k\) of a morphism from \(C\) to \(\mathbb{P}^1\) is computed (with one exception) by a ruling of \(X_e\). In the case where \(C\) is smooth \(k\) is the gonality of \(C\), that is the smallest degree of a linear pencil on \(C\). The author applies his main result to calculate the gonality of smooth irreducible curves \(C\) on \(X_e\). The author uses an inequality of Serrano [cf. \textit{F. Serrano}, Math. Ann. 277, 395-413 (1987; Zbl 0595.14005)] and \textit{R. Paoletti}, Math. Ann. 303, No. 1, 109-123 (1995; Zbl 0835.14005)] to prove his main result.
reviewed by: Aigli Papantonopoulou
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 938612
zbMATH Keywords: Hirzebruch surface
zbMATH Keywords: ruled rational surface
zbMATH Keywords: morphism
zbMATH Keywords: gonality
MaRDI profile type: Publication
cites work: Q4143433
cites work: Q4749943
cites work: Free pencils on divisors
cites work: Extension of morphisms defined on a divisor
full work available at URL: https://doi.org/10.1007/bf01197600
OpenAlex ID: W1963799233
description: scientific article; zbMATH DE number 938612