Classification of jets of surfaces in projective 3-space via central projection
type: Item
instance of: scholarly article
title: Classification of jets of surfaces in projective 3-space via central projection
zbMATH Open document ID: 1384.58028
author: Peng Zhang
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2018-01-16T00:00:00Z
review text: This paper is devoted to explore a local classification of smooth surfaces in \(\mathbb{P}^{3}\) in terms of the singularity types (of codimension \(\leq 4\)) of their central projections to a plane which is associated with a classification of jets of surfaces in \(\mathbb{P}^{3}\) via projective transformations as a natural extension of results in [\textit{O. A. Platonova}, Russ. Math. Surv. 36, No. 1, 248--249 (1981; Zbl 0478.14027)] and [\textit{E. E. Landis}, Funct. Anal. Appl. 15, 103--114 (1981; Zbl 0493.58005)]. The authors identify \(\mathbb{R}^{3}\) in the paper with an open chart of the projective space, \(\{[x : y : z : 1]\} \subset\mathbb{P}^{3}\), and consider germs of surfaces in \(\mathbb{R}^{3}\) at the origin given by Monge forms \(z = f (x, y)\) with \(f (0) = 0\) and \(d f (0) = 0\). Projective transformations preserving the origin and the \(xy\)-plane form a 10-dimensional subgroup of \(\mathrm{PGL}(4)\), which acts on the space of jets of Monge forms of order \(p\), denoted by \(V_{p}\). The authors concern with an invariant stratification of the jet space \(V_{p}\). The main objective of the paper is that in order to stratify the higher codimensional part of \(V_{p}\), they make use of criteria of singularities appearing in central projections which are studied in [the second author, Topology Appl. 202, 216--238 (2016; Zbl 1365.57033)]. Their stratification of the jet space \(V_{p}\) up to codimension 4 yields a classification of Monge forms appearing in generic two-parameter families of embedded surfaces; normal forms are shown in Tables 2 and 3. Moreover, the authors provide exact normal forms to surface germs at transition moments on bifurcations with respect to parabolic curves and flecnodal curves. They characterize precisely moduli parameters in their corresponding Monge form for each type of Uribe-Vargas's classification. For computation and simulation in the paper, Mathematica software is utilized by authors.
reviewed by: Mohammad Sajid
Mathematics Subject Classification ID: 58K05
Mathematics Subject Classification ID: 53A20
Mathematics Subject Classification ID: 58A20
zbMATH DE Number: 6826666
zbMATH Keywords: singularities of smooth maps
zbMATH Keywords: projection of surfaces
zbMATH Keywords: projective differential geometry
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: s00574-017-0036-x
OpenAlex ID: W2606689132
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DOI: 10.1007/S00574-017-0036-X
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description: scientific article; zbMATH DE number 6826666