摘要翻译：
D.Jaffe和D.Ruberman在1997年证明了$\MathBB{P}^3$中的一个六次超曲面最多有65个节点（Barth构造的界是尖锐的）。几乎同时，J.Wahl对同样的结果提出了一个更短的证明，证明了权值在$24,32,40中的线性码$V\子集\F^{66}$具有维数$\dim(V)\leq12$。他声称Jaffe-Ruberman定理是一个推论，因为与具有n个结点的六次元相关联的码的维数至少为$n-53$，Casnati和Catanese所陈述的一个不正确的结果断言六次元上偶数结点集的可能基数仅为24、32和40。最近Catanese和Tonoli表明，在一个六角形上偶数节点集的可能基数正好是24，32，40，56。根据上述基数，将Jaffe和Ruberman的定理归结为：设$V\子集\F^{66}$为权值为$24,32,40,56}$的码。然后$\dim(V)\leq12$。在这篇短文中，我们利用并综合了Wahl的思想，给出了这个定理的初等证明。
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英文标题：
《On Wahl's proof of $\mu(6)=65$》
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作者：
Roberto Pignatelli and Fabio Tonoli
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最新提交年份：
2007
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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：
  D. Jaffe and D. Ruberman proved in 1997 that a sextic hypersurface in $\mathbb{P}^3$ has at most 65 nodes (the bound is sharp by Barth's construction).   Almost at the same time, J. Wahl proposed a much shorter proof of the same result, by proving that a linear code $V\subset \F^{66}$ with weights in $\{24,32,40\}$ has dimension $\dim(V)\leq12$.   He claimed that Jaffe-Ruberman's theorem follows as a corollary since the code associated to a sextic with n nodes has dimension at least $n-53$ and an incorrect result stated by Casnati and Catanese asserted that the possible cardinalities of an even set of nodes on a sextic were only 24, 32 and 40.   Recently Catanese and Tonoli showed that the possible cardinalities of an even set of nodes on a sextic are exactly 24, 32, 40, 56. According to the above cardinalities, the theorem of Jaffe and Ruberman reduces to the following:   Let $V\subset \F^{66}$ be a code with weights in $\{24,32,40,56\}$. Then $\dim(V)\leq12$.   In this short note we give an elementary proof of this theorem using and integrating Wahl's ideas.
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PDF链接：
https://arxiv.org/pdf/0706.4358