Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 031, 16 pages      arXiv:1710.02863

Cartan Prolongation of a Family of Curves Acquiring a Node

Susan Jane Colley a and Gary Kennedy b
a) Department of Mathematics, Oberlin College, Oberlin, Ohio 44074, USA
b) Ohio State University at Mansfield, 1760 University Drive, Mansfield, Ohio 44906, USA

Received October 27, 2017, in final form April 03, 2018; Published online April 07, 2018

Using the monster/Semple tower construction, we study the structure of the Cartan prolongation of the family $x_1x_2 = t$ of plane curves with nodal central member.

Key words: curve families; nodal singularity; vector distributions; prolongation.

pdf (379 kb)   tex (24 kb)


  1. Brieskorn E., Knörrer H., Plane algebraic curves, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1986.
  2. Bryant R.L., Nine lectures on exterior differential systems, Graduate Summer Workshop on Exterior Differential Systems at the Mathematical Sciences Research Institute, 1999, available at
  3. Bryant R.L., Hsu L., Rigidity of integral curves of rank $2$ distributions, Invent. Math. 114 (1993), 435-461.
  4. Cartan E., Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bull. Soc. Math. France 42 (1914), 12-48.
  5. Castro A., Colley S.J., Kennedy G., Shanbrom C., A coarse stratification of the monster tower, Michigan Math. J. 66 (2017), 855-866, arXiv:1606.07931.
  6. Colley S.J., Kennedy G., A higher-order contact formula for plane curves, Comm. Algebra 19 (1991), 479-508.
  7. Colley S.J., Kennedy G., Triple and quadruple contact of plane curves, in Enumerative Algebraic Geometry (Copenhagen, 1989), Contemp. Math., Vol. 123, Amer. Math. Soc., Providence, RI, 1991, 31-59.
  8. Colley S.J., Kennedy G., The enumeration of simultaneous higher-order contacts between plane curves, Compositio Math. 93 (1994), 171-209, alg-geom/9212003.
  9. Gherardelli G., Sul modello minimo della varietà degli elementi differenziali del $2^\circ$ ordine del piano projettivo, Atti Accad. Italia. Rend. Cl. Sci. Fis. Mat. Nat. (7) 2 (1941), 821-828.
  10. Giaro A., Kumpera A., Ruiz C., Sur la lecture correcte d'un résultat d'Élie Cartan, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), A241-A244.
  11. Hartshorne R., Algebraic geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York - Heidelberg, 1977.
  12. Kollár J., Lectures on resolution of singularities, Annals of Mathematics Studies, Vol. 166, Princeton University Press, Princeton, NJ, 2007.
  13. Kumpera A., Ruiz C., Sur l'équivalence locale des systèmes de Pfaff en drapeau, in Monge-Ampère Equations and Related Topics (Florence, 1980), Ist. Naz. Alta Mat. Francesco Severi, Rome, 1982, 201-248.
  14. Lejeune-Jalabert M., Chains of points in the Semple tower, Amer. J. Math. 128 (2006), 1283-1311.
  15. Montgomery R., Swaminathan V., Zhitomirskii M., Resolving singularities with Cartan's prolongation, J. Fixed Point Theory Appl. 3 (2008), 353-378.
  16. Montgomery R., Zhitomirskii M., Geometric approach to Goursat flags, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 459-493.
  17. Montgomery R., Zhitomirskii M., Points and curves in the Monster tower, Mem. Amer. Math. Soc. 203 (2010), x+137 pages.
  18. Mormul P., Exotic moduli of Goursat distributions exist already in codimension three, in Real and Complex Singularities, Contemp. Math., Vol. 459, Amer. Math. Soc., Providence, RI, 2008, 131-145.
  19. Semple J.G., Some investigations in the geometry of curve and surface elements, Proc. London Math. Soc. 4 (1954), 24-49.

Previous article  Next article   Contents of Volume 14 (2018)