作为二次曲面铅笔的台球代数的超椭圆Jacobian； 超越蓬松孔

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摘要翻译：
本文解决并完成了Griffiths和Harris三十年来关于理解Poncelet型问题的高维类比和综合方法的计划。从观察一些经典结构和构型的台球性质入手，我们构造了台球代数，即在给定共焦族的d-1二次曲面同时相切的线集T上的一个群结构。利用该工具，进一步发展、实现和简化了Reid、Donagi和Knoerrer的相关结果。我们导出了T的一个基本性质：从该集合中任意两条线可以通过在共焦族的某些二次曲面上至多d-1的台球反射而相互获得。我们引入了两个层次的概念：T中的s-斜线和s-弱Poncelet轨迹，s=-1,0,...,D-2。本文发展了台球动力学、二次曲面交线性子空间和超椭圆Jacobians之间的相互关系，使我们得到了几个经典属1结果的高维和高属推广：Cayley定理、Weyr定理、Griffiths-Harris定理和Darboux定理。
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英文标题：
《Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics:
  Beyond Poncelet Porisms》
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作者：
Vladimir Dragovic, Milena Radnovic
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Physics        物理学
二级分类：Mathematical Physics        数学物理
分类描述：Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
--
一级分类：Mathematics        数学
二级分类：Mathematical Physics        数学物理
分类描述：math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
math.mp是math-ph的别名。这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
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英文摘要：
  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.
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PDF链接：
https://arxiv.org/pdf/0710.3656

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关键词：Jacobi Jacob family lines 代数 定理 theorem