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 sonic 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 Knorrer 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 - I billiard reflections at some quadrics front 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. (c) 2008 Elsevier Inc. All rights reserved.
