Forms of the affine line and their genus
type: Item
instance of: scholarly article
title: Forms of the affine line and their genus
zbMATH Open document ID: 0576.14028
DOI: 10.1016/0022-4049(86)90138-6
author: Cornelius Greither
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: This paper studies forms of the affine line \({\mathbb{A}}^ 1\) over a field k in the following sense: K/k is inseparable of degree \(p=char(k)\), and X is a form of \({\mathbb{A}}^ 1\) if \(K\otimes X\cong {\mathbb{A}}^ 1_ K.\) Continuing work of Kambayashi, Miyanishi, and Takeuchi, who studied forms of arithmetical genus 0 and 1, we describe the numbers g which occur as genera of forms. Two wit, g occurs iff \(g=(p-1)\cdot j/2\) with \(j\neq - 1(p).\) \textit{P. Russell} [Pac. J. Math. 32, 527-539 (1970; Zbl 0199.245)] described by generators and relations those forms which carry a group structure. This is generalized to a wider class of forms called ''parallelizable''. X is parallelizable iff \(\Omega_ X\) is free cyclic as an \({\mathcal O}_ X\)-module. By considering the genus one sees that the parallelizable forms still constitute a small subclass of the class of all forms.
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 12F15
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 3920655
zbMATH Keywords: desingularization of curves
zbMATH Keywords: inseparable field extension
zbMATH Keywords: affine line
zbMATH Keywords: arithmetical genus
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90138-6
OpenAlex ID: W2091208192
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description: scientific article; zbMATH DE number 3920655