摘要翻译：
直线上八个有序点的射影不变量环是V上的多项式环的商，其中V是S8的14维表示，因此模五重(p^1)^8//GL(2)是Proj(sym*(V)/I_8)。我们证明了PV中存在唯一的三次超曲面S，其方程S是斜不变的，并且S的奇异轨迹是模五倍的。特别地，在Z[1/3]上，模的五倍被S的14个偏导数切掉。更好：这些方程产生I_8。在特征3中，需要立方s来生成理想。多尔加乔夫预言了这样一个立方体的存在。在Q上，我们恢复了Koike通过计算机计算找到的14个二次曲面，我们的方法产生了对表示的概念表示-理论描述。此外，我们还发现了任意特征的最小自由分辨率的分次Betti数。关于Q的证明是通过纯思想，利用李理论和交换代数。在Z以上，计算机的协助是必要的。这一结果将作为描述p^1上任意多个点的模空间方程的基本情况，在以后的论文中完成了我们上一篇论文（Duke Math.J.）的程序。由于Deligne-Mostow，Kondo和Freitag-Salvati Manni，模五重和相应的环已知有许多特殊的化身，例如分别作为球商或模形式的环。
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英文标题：
《The ring of projective invariants of eight points on the line via
  representation theory》
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作者：
Ben Howard, John Millson, Andrew Snowden, Ravi Vakil
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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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英文摘要：
  The ring of projective invariants of eight ordered points on the line is a quotient of the polynomial ring on V, where V is a fourteen-dimensional representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 // GL(2) is Proj(Sym* (V)/I_8). We show that there is a unique cubic hypersurface S in PV whose equation s is skew-invariant, and that the singular locus of S is the modular fivefold. In particular, over Z[1/3], the modular fivefold is cut out by the 14 partial derivatives of s. Better: these equations generate I_8. In characteristic 3, the cubic s is needed to generate the ideal. The existence of such a cubic was predicted by Dolgachev. Over Q, we recover the 14 quadrics found by computer calculation by Koike, and our approach yields a conceptual representation-theoretic description of the presentation. Additionally we find the graded Betti numbers of a minimal free resolution in any characteristic.   The proof over Q is by pure thought, using Lie theory and commutative algebra. Over Z, the assistance of a computer was necessary. This result will be used as the base case describing the equations of the moduli space of an arbitrary number of points on P^1, with arbitrary weighting, in a later paper, completing the program of our previous paper (Duke Math. J.). The modular fivefold, and corresponding ring, are known to have a number of special incarnations, due to Deligne-Mostow, Kondo, and Freitag-Salvati Manni, for example as ball quotients or ring of modular forms respectively.
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PDF链接：
https://arxiv.org/pdf/0809.1233