Abstract
We present a local classification of smooth surfaces in \({\mathbb {P}}^3\) in terms of the singularity types (of codimension \(\le \)4) of their central projections to a plane. Based on our classification result, we also give exact normal forms to surface germs at transition moments on bifurcations with respect to parabolic curves and flecnodal curves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
As an exception, if \(c_{40}=1\), we see by the same manner that \(x^3y\) can not be killed via any projective transformations, so the normal form is \(y^2+x^2y+x^4+c_{31}x^3y\).
References
Arnold, V.I.: Catastrophe Theory, 3rd edn. Springer (2004)
Arnold, V.I., Goryunov, V.V., Lyashko, O.V., Vasil’ev, V.A.: Singularity Theory II, Classification and Applications. In: Arnold, V.I. (ed.) Encyclopaedia of Mathematical Sciences, Dynamical System VIII, vol. 39 (translation from Russian version). Springer (1993)
Bruce, J.W.: Projections and reflections of generic surfaces in \({{\mathbb{R}}}^3\). Math. Scand. 54(2), 262–278 (1984)
Bruce, J.W., Fletcher, G.J., Tari, F.: Bifurcations of implicit differential equations. Proc. R. Soc. Edinb. A 130, 485–506 (2000)
Deolindo-Silva, J.L., Kabata, Y.: Projective classification of jets of surfaces in \({\mathbb{P}}^4\) (2016). arXiv:1601.06255
Gaffney, T., Ruas, M.A.S.: (Unpublished work, 1977)
Goryunov, V.V.: Singularities of projections of complete intersections. J. Sov. Math. 27, 2785–2811 (1984)
Kabata, Y.: Recognition of plane-to-plane map-germs. Topol. Appl. 202, 216–238 (2016)
Landis, E.E.: Tangential singularities. Funt. Anal. Appl. 15, 103–114 (1981) (translation)
Mond, D.M.Q.: Classification of certain singularities and applications to differential geometry. Ph.D. thesis, University of Liverpool (1982)
Olver, P.J.: Equivalence, Invariants and Symmetry. Cambridge Univ. Press, Cambridge (1995)
Panov, D.A.: Special points of surfaces in the three-dimensional projective space. Funct. Anal. Appl. 34, 276–287 (2000)
Platonova, O.A.: Singularities of the mutual disposition of a surface and a line. Uspekhi Mat. Nauk 36, 221–222 (1981)
Platonova, O.A.: Projections of smooth surfaces. J. Sov. Math. 35(6), 2796–2808 (1986)
Rieger, J.H.: Families of maps from the plane to the plane. J. Lond. Math. Soc. 36, 351–369 (1987)
Rieger, J.H.: Versal topological stratification and the bifurcation geometry of map-germs of the plane. Math. Proc. Camb. Philos. Soc. 107(1), 127–147 (1990)
Rieger, J.H.: The geometry of view space of opaque objects bounded by smooth surfaces. Artif. Intell. 44, 1–40 (1990)
Salmon, G.: A treatise on the analytic geometry of three dimensions. Hodges, Smith, Dublin (1862)
Tresse, A.: Sur les invariants des groupes continus de transformations. Acta Math. 18, 1–88 (1894)
Uribe-Vargas, R.: Surface evolution, implicit differential equations and pairs of Legendrian fibrations (2002)
Uribe-Vargas, R.: A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve. Mosc. Math. J. 6, 731–772 (2006)
West, J.: The Differential Geometry of the Cross-Cap. Ph.D. thesis, University of Liverpool (1995)
Wilczynski, E.J.: Projective Differential Geometry of Curves and Ruled Surfaces, vol. VIII. B. G. Teubner, Leipzig (1906)
Yoshida, T., Kabata, Y., Ohmoto, T.: Bifurcations of plane-to-plane map-germs of corank \(2\). Q. J. Math. 66, 369–391 (2015)
Yoshida, T., Kabata, Y., Ohmoto, T.: Bifurcations of plane-to-plane map-germs of corank \(2\) of parabolic type. RIMS Kokuyroku Bessatsu B55, 239–258 (2016)
Acknowledgements
The authors would like to thank Takashi Nishimira and Farid Tari for organizing the JSPS-CAPES international cooperation project in 2014-2015. In fact, the second and third authors are supported by the project for their stays in ICMC-USP and Hokkaido University, respectively. The authors appreciate Ricardo Uribe-Vargas for letting them take attention to his paper Uribe-Vargas (2002) and Panov’s (2000) and the referee for valuable comments in revising earlier versions of the present paper. The authors are partly supported by JSPS KAKENHI Grant Numbers 24340007 and 15K13452.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Sano, H., Kabata, Y., Silva, J.L.D. et al. Classification of Jets of Surfaces in Projective 3-Space Via Central Projection. Bull Braz Math Soc, New Series 48, 623–639 (2017). https://doi.org/10.1007/s00574-017-0036-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00574-017-0036-x