Authors: Dragović, Vladimir 
Radnović, Milena
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: Beyond Poncelet porisms
Journal: Advances in Mathematics
Volume: 219
Issue: 5
First page: 1577
Last page: 1607
Issue Date: 1-Dec-2008
Rank: M21a
ISSN: 0001-8708
DOI: 10.1016/j.aim.2008.06.021
The thirty years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper. Starting with the observation of the billiard nature of some classical constructions and configurations, we construct the billiard algebra, that is a group structure on the set T of lines simultaneously tangent to d - 1 quadrics from a given confocal family in the d-dimensional Euclidean space. Using this tool, the related results of Reid, Donagi and Knörrer are further developed, realized and simplified. We derive a fundamental property of T: any two lines from this set can be obtained from each other by at most d - 1 billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: s-skew lines in T and s-weak Poncelet trajectories, s = - 1, 0, ..., d - 2. The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in this paper enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results: Cayley's theorem, Weyr's theorem, Griffiths-Harris theorem and Darboux theorem.
Keywords: Billiard | Cayley's theorem | Closed billiard trajectories | Griffiths-Harris theorem | Hyperelliptic curve | Hyperelliptic Jacobian | Pencils of quadrics | Poncelet theorem | Weyr's theorem
Publisher: Elsevier
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 

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