摘要翻译：
给定一个有单位的有限结合环$R$,由$R$的$uni$模($n+1$)-元组生成的任意自由（左）循环子模(FCS)表示$R$上的$n$维射影空间的一个点。假设$R$也具有由($n+1$)-元组生成的FCSs，这些元组不是$单模的：什么样的几何学可以归因于这样的FCSs？这里，当$R$是（唯一的）八阶非交换环时，我们（部分地）回答了$n=2$的问题。相应的几何学被称为“法诺雪花”，因为它的图形外观和它的中心包含法诺平面的事实。事实上，存在着两个这样的构型--每一个都与环的两个最大理想中的任何一个相联系--它们具有共同的Fano平面，因此可以看作是孪生的。本文还概述了这些值得注意的构型与量子信息理论和弦黑洞的潜在相关性。
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英文标题：
《Twin "Fano-Snowflakes" Over the Smallest Ring of Ternions》
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作者：
Metod Saniga (ASTRINSTSAV), Hans Havlicek (TUW), Michel Planat
  (FEMTO-ST), Petr Pracna (JH-Inst)
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最新提交年份：
2008
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分类信息：

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
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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：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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一级分类：Physics        物理学
二级分类：Quantum Physics        量子物理学
分类描述：Description coming soon
描述即将到来
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英文摘要：
  Given a finite associative ring with unity, $R$, any free (left) cyclic submodule (FCS) generated by a $uni$modular ($n+1$)-tuple of elements of $R$ represents a point of the $n$-dimensional projective space over $R$. Suppose that $R$ also features FCSs generated by ($n+1$)-tuples that are $not$ unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for $n=2$ when $R$ is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a "Fano-Snowflake" due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations -- each being tied to either of the two maximal ideals of the ring -- which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined.
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PDF链接：
https://arxiv.org/pdf/0803.4436