Lines on Brieskorn-Pham surfaces
type: Item
instance of: scholarly article
title: Lines on Brieskorn-Pham surfaces
zbMATH Open document ID: 0982.14023
author: Guangfeng Jiang
author: M. Oka
author: Duc Tai Pho
author: Dirk Siersma
published in: Kodai Mathematical Journal
publication date: 2002-03-21T00:00:00Z
review text: Motivated by the problem posed by \textit{J. F. Nash jun.} in 1968 [see Duke Math. J. 81, 31-38 (1995; Zbl 0880.14010)] in order to study the space of arcs in an algebraic or analytic variety over \(\mathbb C\), and since the problem had already been solved for minimal surface singularities by \textit{A. J. Reguera} [Manuscr. Math. 88, 321-333 (1995; Zbl 0867.14012)], \textit{G. Gonzalez-Springberg} and \textit{M. Lejeune-Jalabert} proposed a paper [Ann. Pol. Math. 67, 179-190 (1997; Zbl 0894.14017)] trying to understand how the property of the existence of smooth curves in the minimal surface singularities helps to solve the problem. They characterized the existence of smooth curves in a surface singularity \((X,0)\) by the fact that the maximal cycle \({\mathcal Z}_X\) for its minimal desingularization \(\pi: {\widetilde X} \rightarrow X\) (i.e. the underlying cycle of the ideal sheaf \({\mathfrak m} {\mathcal O}_{\widetilde X}\), where \(\mathfrak m\) is the maximal ideal of \({\mathcal O}_{X,0}\)) has at least one reduced component. They also gave a natural partition of the set \(\mathcal L\) of smooth arcs in \((X,0)\) in families \({\mathcal L}_E\), each \({\mathcal L}_E\) defined by a reduced component of \({\mathcal Z}_E\). And they showed that there is no wedge centered at smooth curves of two different families. In the paper under review, the authors study smooth curves on the Brieskorn-Pham type surfaces \(G(p,q,r)\), i.e. defined in \(\mathbb C^3\) by \(x^p+y^q+z^r=0\), for some positive integers \(p \leq q \leq r\). The minimal desingularization of such a surface can be obtained by making a regular subdivision of the Newton fan defined by its equation. Using this, the authors characterize the existence of smooth curves in \(G(p,q,r)\) as follows: If \(s=\text{gcd}(p,q,r)\) then, either \(r> \frac {pq} {\text{gcd}(p,q)}\), or else two of the three integers \(\frac ps\), \(\frac qs\), \(\frac rs\) are coprime and the other is divisible by at least one of …
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 32S25
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 1547125
zbMATH Keywords: Brieskorn-Pham surface
zbMATH Keywords: toric environment
zbMATH Keywords: minimal surface singularities
zbMATH Keywords: smooth curves in a surface singularity
zbMATH Keywords: Newton fan
reviewed by: Ana J. Reguera-López
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044211
OpenAlex ID: W1988130781
cites work: Q4297430
cites work: Families of smooth curves on surface singularities and wedges
cites work: Local embeddings of lines in singular hypersurfaces
cites work: Normal Two-Dimensional Singularities. (AM-71)
cites work: Q4237350
cites work: Q3758954
DOI: 10.2996/KMJ/1138044211
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description: scientific article; zbMATH DE number 1547125