Every curve is a Hurwitz space
type: Item
instance of: scholarly article
title: Every curve is a Hurwitz space
zbMATH Open document ID: 0712.14013
DOI: 10.1215/S0012-7094-89-05933-4
author: David Harbater
author: Steven P. Diaz
author: Ron Donagi
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The main result of this paper is that any branched covering map \(\pi\) : \(C\to {\mathbb{P}}^ 1\) ramified only over 0, 1 and \(\infty\), arises from a Hurwitz space. That is, C is the closure of an irreducible component of a Hurwitz scheme H parametrizing a family of branched covers of \({\mathbb{P}}^ 1\) branched only over 0, 1, \(\infty\), and \(\lambda\), where \(\lambda\) is allowed to vary. As applications it is shown that a nonsingular complete curve is a Hurwitz curve iff it is definable over \({\bar {\mathbb{Q}}}\), and that there exist branched covers of \({\mathbb{P}}^ 1\) branched at 4 points for which the Hurwitz number (the number of components of H) is arbitrarily large.
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14H10
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 4171089
zbMATH Keywords: branched covering map
zbMATH Keywords: Hurwitz space
zbMATH Keywords: Hurwitz curve
MaRDI profile type: Publication
cites work: ON GALOIS EXTENSIONS OF A MAXIMAL CYCLOTOMIC FIELD
cites work: Q3943063
cites work: Hurwitz families and arithmetic Galois groups
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cites work: ALGEBRAIC CURVES OVER FUNCTION FIELDS. I
full work available at URL: s0012-7094-89-05933-4
OpenAlex ID: W2076543581
description: scientific article; zbMATH DE number 4171089