Algebraic-geometry approach to integrability of birational plane mappings. Integrable birational quadratic reversible mappings. I. (English) Zbl 0917.14008
The author studies plane Cremona transformation which admit an invariant pencil of curves. He gives necessary and sufficient conditions for such a transformation in terms of the decomposition of the set of its fundamental points in orbits of the cyclic group generated by the inverse transformation. The existence of an invariant pencil allows one to find explicit first integrals for the autonomous dynamical system in \(\mathbb{C}^2\) with discrete time defined by \(x_i(n+1)= \varphi_i(x_1(n), x_2(n),1)/ \varphi_3 (x_1(n), x_2(n),1)\), \(i=1,2\), where \((\varphi_1, \varphi_2, \varphi_3)\) are the homogeneous polynomials defining the Cremona transformation. The paper ends with a list of nine quadratic transformations which admit an invariant pencil of lines or conics.
Reviewer: I.Dolgachev (MR 98k:14015)
MSC:
| 14E07 | Birational automorphisms, Cremona group and generalizations |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
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