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

SIGMA 17 (2021), 067, 14 pages      arXiv:2009.09854

A New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves

Peter H. van der Kamp
Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

Received January 15, 2021, in final form July 02, 2021; Published online July 13, 2021

For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to construct integrable maps of the plane which leave invariant a cubic pencil.

Key words: integrable map of the plane; Manin transformation; Bertini involution; invariant; pencil of cubic curves.

pdf (436 kb)   tex (110 kb)  


  1. Bayle L., Beauville A., Birational involutions of ${\bf P}^2$, Asian J. Math. 4 (2000), 11-17, arXiv:math.AG/9907028.
  2. Bertini E., Ricerche sulle trasformazioni univoche involutorie nel piano, Ann. Mat. Pura Appl. 8 (1877), 244-286.
  3. Caudrelier V., van der Kamp P.H., Zhang C., Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings, Int. Math. Res. Not., to appear, arXiv:2009.00412.
  4. Diller J., Favre C., Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135-1169.
  5. Duistermaat J.J., Discrete integrable systems. QRT maps and elliptic surfaces, Springer Monographs in Mathematics, Springer, New York, 2010.
  6. Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2016.
  7. Iatrou A., Roberts J.A.G., Integrable mappings of the plane preserving biquadratic invariant curves. II, Nonlinearity 15 (2002), 459-489.
  8. Jogia D., Roberts J.A.G., Vivaldi F., An algebraic geometric approach to integrable maps of the plane, J. Phys. A: Math. Gen. 39 (2006), 1133-1149.
  9. Manin Yu.I., The Tate height of points on an Abelian variety, its variants and applications, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1363-1390, English transl.: AMS Translations Ser. 2, Vol. 59, Amer. Math. Soc. (AMS), Providence, RI, 1966, 82-110.
  10. Maple (2021), Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.
  11. Moody E.I., Notes on the Bertini involution, Bull. Amer. Math. Soc. 49 (1943), 433-436.
  12. Papageorgiou V.G., Nijhoff F.W., Capel H.W., Integrable mappings and nonlinear integrable lattice equations, Phys. Lett. A 147 (1990), 106-114.
  13. Petrera M., Suris Yu.B., Wei K., Zander R., Manin involutions for elliptic pencils and discrete integrable systems, Math. Phys. Anal. Geom. 24 (2021), 6, 26 pages, arXiv:2008.08308.
  14. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations, Phys. Lett. A 126 (1988), 419-421.
  15. Quispel G.R.W., Roberts J.A.G., Thompson C.J., Integrable mappings and soliton equations. II, Phys. D 34 (1989), 183-192.
  16. Roberts J.A.G., Quispel G.R.W., Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep. 216 (1992), 63-177.
  17. Tsuda T., Integrable mappings via rational elliptic surfaces, J. Phys. A: Math. Gen. 37 (2004), 2721-2730.
  18. van der Kamp P.H., McLaren D.I., Quispel G.R.W., Generalised Manin transformations and QRT maps, J. Computat. Dyn. 8 (2021), 183-211, arXiv:1806.05340.
  19. van der Kamp P.H., Quispel G.R.W., The staircase method: integrals for periodic reductions of integrable lattice equations, J. Phys. A: Math. Theor. 43 (2010), 465207, 34 pages, arXiv:1005.2071.
  20. Veselov A.P., Integrable mappings, Russian Math. Surveys 46 (1991), no. 5, 1-51.

Previous article  Next article  Contents of Volume 17 (2021)