Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms

TL;DR

This paper advances the understanding of higher-dimensional Poncelet problems by developing a billiard algebra on lines tangent to confocal quadrics, leading to new generalizations of classical theorems involving hyperelliptic Jacobians.

Contribution

It constructs a billiard algebra on tangent lines to confocal quadrics and generalizes classical genus 1 results to higher dimensions and genera.

Findings

Established a group structure on tangent lines to quadrics

Derived a property that any two lines can be connected by a finite number of billiard reflections

Generalized classical theorems to higher-dimensional, higher-genus contexts

Abstract

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 in $R^d$ simultaneously tangent to d-1 quadrics from a given confocal family. Using this tool, the related results of Reid, Donagi and Knoerrer 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: the Cayley's theorem, the Weyr's theorem, the Griffiths-Harris theorem and the Darboux theorem.