Primitive arcs on curves
type: Item
instance of: scholarly article
title: Primitive arcs on curves
zbMATH Open document ID: 1368.14025
author: Julien Sebag
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2017-02-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1480993581
review text: The arc space \(X_\infty\) of an algebraic variety \(X\) carries important information about its singular locus \(X_{\mathrm{sing}}\), in particular about its numerical invariants, and about the essential divisors over \(X\). The formal neighborhood of an arc \(\alpha\in X_\infty\) could be seen as a space parametrizing its formal deformations in the space of arcs. The theorem of \textit{V. Drinfeld} [``On the Grinberg-Kazhdan formal arc theorem'', \url{arXiv:math/0203263}] and \textit{M. Grinberg} and \textit{D. Kazhdan} [Geom. Funct. Anal. 10, No. 3, 543--555 (2000; Zbl 0966.14002)] describes the formal neighborhood of any non-constant closed arc in \(X_\infty\), whose image is not entirely contained in \(X_{\mathrm{sing}}\). This result has been further developed in author's papers [the author and \textit{D. Bourqui}, ``The Drinfeld-Grinberg-Kazhdan theorem is false for singular arcs'', J. IMJ (to appear); ``The Drinfeld-Grinberg-Kazhdan theorem for formal schemes and singularity theory'', (to appear); J. Singul. 16, 130--140 (2017; Zbl 1366.14019)]. In the article under review is introduced the notion of a primitive arc at a point on curve (which is unique if exists). It is used to investigate the smoothness at closed point \(x\) of analytically irreducible curve \(X\) in terms of the formal neighborhood of the constant arc \(\alpha_x\in X_\infty\) at \(x\). The main result of the article provides criteria, each one equivalent to the existence of a primitive arc at \(x\). Among them are the following: the smoothness of the curve at x, and, that the formal neighborhood of \(\alpha_x\) is isomorphic to \(\mathbb D^\infty=Spf kX_i,i\in\mathbb N\). By taking the normalization morphism \(\nu:\overline X\to X\), we have induced morphism \(\hat\nu\) between the formal neighborhood of \(\alpha_{\bar x}\in\bar X_\infty\), where \(\bar x=\nu^{-1}(x)\), and the formal neighborhood of \(\alpha_x\in X_\infty\). The theorem claims further that the existence of a primitive…
reviewed by: Peter Petrov
Mathematics Subject Classification ID: 14E18
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 14B20
zbMATH DE Number: 6684726
zbMATH Keywords: arc scheme
zbMATH Keywords: analytically irreducible curve
zbMATH Keywords: primitive arc
zbMATH Keywords: formal neighborhood
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6684726