关于$μ(6)=65$的Wahl证明
On Wahl's proof of $μ(6)=65$
摘要：
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构造的边界是尖锐的）。几乎在同一时间，J. Wahl提出了一个更简短的证明，证明了一个线性码$V\subset \F^{66}$的重量在$\{24,32,40\}$有维数$\dim(V)\leq12$。他声称，Jaffe-Ruberman定理如下作为一个推论，因为代码相关的六次曲线与n个节点的尺寸至少$n-53$和一个不正确的结果所述Casnati和Catanese断言，可能的基数甚至一组节点的六次曲线只有24，32和40。最近卡坦尼斯和托诺利表明，可能的基数，甚至一组节点上的六次曲线正好是24，32，40，56。根据上述基数，Jaffe和Ruberman的定理简化为：设$V\subset \F^{66}$是在$\{24,32,40,56\}$中具有权重的码。然后是$\dim(V)\leq12$。在这个简短的说明中，我们给出了一个初等证明这个定理使用和整合Wahl的想法。
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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