Manin Involutions for Elliptic Pencils and Discrete Integrable Systems
TLDR
In this paper, a geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), is given.Abstract:
We contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.read more
Citations
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A New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves
van der Kamp,H Peter +1 more
TL;DR: In this article, the notion of an involution curve was introduced for cubic pencils, which is a curve which intersects each curve of the pencil in exactly one non-base point on the pencil and is used to construct integrable maps of the plane which leave invariant a cubic pencil.
Journal ArticleDOI
Generalised Manin transformations and QRT maps
TL;DR: In this paper, the authors generalize this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable.
Posted Content
Moufang Patterns and Geometry of Information.
TL;DR: In this paper, it was shown that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of information geometry, have the structure of a commutative Moufang loop.
Journal ArticleDOI
Moufang patterns and geometry of information
TL;DR: In this article , it was shown that the symmetries of spaces of probability distributions, endowed with their canonical Riemannian metric of geometry, have the structure of a commutative Moufang loop.
Posted Content
Involutions of Halphen Pencils of Index 2 and Discrete Integrable Systems
TL;DR: In this article, the authors constructed the birational mapping corresponding to the autonomous reduction of the elliptic Painleve equation for a Halphen pencil of index 2, and proved that the same pencil can be obtained as the composition of two such involutions.
References
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Rational surfaces associated with affine root systems and geometry of the Painlevé equations
TL;DR: In this article, a geometric approach to the theory of Painleve equations based on rational surfaces is presented, where a compact smooth rational surface X has a unique anti-canonical divisor D of canonical type.
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Integrable mappings and soliton equations II
TL;DR: In this article, it was shown that simple solutions of discrete soliton equations satisfy 2D mappings and that these belong to a recently introduced 18-parameter family of integrable reversible mappings of the plane.
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Discrete Integrable Systems
TL;DR: In this paper, the authors give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based, assuming Theorem 3.7.
Journal ArticleDOI
Geometric aspects of Painlevé equations
Book
Discrete Integrable Systems: QRT Maps and Elliptic Surfaces
TL;DR: The QRT Map as discussed by the authors is the pencil of biquadratic curves in the projective plane of the QRT surface and is used to measure the distance between two points.