Discrete Integrable Systems

TLDR: 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.

Abstract: 10.1007/978-1-4419-9126-3 Copyright owner: Springer Science+Buisness Media, LLC, 2010 Data set: Springer Source Springer Monographs in Mathematics The rich subject matter in this book brings in mathematics from different domains, especially from the theory of elliptic surfaces and dynamics.The material comes from the authorâ€TMs insights and understanding of a birational transformation of the plane derived from a discrete sine-Gordon equation, posing the question of determining the behavior of the discrete dynamical system defined by the iterates of the map. The aim of this book is to give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based. Readers with a good knowledge of algebraic geometry will be interested in Kodairaâ€TMs theory of elliptic surfaces and the collection of nontrivial applications presented here. While prerequisites for some readers will demand their knowledge of quite a bit of algebraicand complex analytic geometry, different categories of readers... more Identifiers series ISSN : 1439-7382 ISBN 978-1-4419-7116-6 e-ISBN 978-1-4419-9126-3 DOI Authors Additional information Publisher Springer New York book Read online Download Add to read later Add to collection Add to followed Share Export to bibliography J.J. Duistermaat Utrecht University, Department of Mathematics, Utrecht, Netherlands Terms of service Accessibility options Report an error / abuse © 2015 Interdisciplinary Centre for Mathematical and Computational Modelling Discrete integrable systems independent: it neither relies on nor used in the proof of integrability. Section 6 is not used in the proof of integrability. It discusses more specic discrete cluster integrable systems, assuming Theorem 3.7. Proof of part i) Take a pair of matchings (M1, M2) on Γ. Let us assign to them another pair of matchings (M1, M2) on Γ. Observe that [M1] − [M2] is a 1-cycle.

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Springer Monographs in Mathematics
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Discrete Integrable Systems
QRT Maps and Elliptic Surfaces
Johannes J. Duistermaat

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ISSN 1439-7382
Springer New York Dordrecht Heidelberg London
© Springer Science+Business Media, LLC 2010
Library of Congress Control Number:
e-ISBN 978-0-387-72923-7
ISBN 978-1-4419-7116-6
Johannes J. Duistermaat (deceased, March 2010)
2010934229
DOI 10.1007/978-0-387-72923-7

Contents
Preface ............................................................ vii
1 The QRT Map ................................................. 1
1.1 The Rational Formula for the QRT Map ........................ 1
1.2 Indeterminacy of the QRT Map ............................... 3
1.3 Reconstruction ............................................. 4
2 The Pencil of Biquadratic Curves in P
1
× P
1
...................... 9
2.1 Complex Analytic Geometry ................................. 10
2.2 Complex Projective Varieties ................................. 28
2.3 Elliptic Curves ............................................. 33
2.4 Biquadratic Curves ......................................... 49
2.5 The QRT Mapping on a Smooth Biquadratic Curve .............. 60
2.6 Real Points ................................................ 74
3 The QRT surface ............................................... 85
3.1 The surface in P
1
×(P
1
×P
1
) ................................ 86
3.2 Blowing Up ............................................... 93
3.3 Blowing Up P
1
×P
1
at the Base Points ........................105
3.4 The QRT Map on the QRT Surface ............................119
4 Cubic Curves in the Projective Plane .............................129
4.1 From P
1
×P
1
to P
2
and Back ................................129
4.2 Manin Transformations ......................................134
4.3 Manin QRT Automorphisms .................................138
4.4 Aronhold’s Invariants .......................................143
4.5 Pencils of Cubic Curves with Only One Base Point ..............148
5 The Action of the QRT Map on Homology ........................157
5.1 The Action of the QRT Map on Homology Classes ..............157
5.2 QRT Transformations of Finite Order ..........................164
6 Elliptic Surfaces ...............................................179
6.1 Fibrations .................................................179
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