Classification of jets of surfaces in projective 3-space via central projection
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.
scientific article; zbMATH DE number 6826666
scientific article