On Wahl's proof of $\mu(6)=65$

TL;DR

This paper provides a simple proof that the maximum number of nodes on a sextic hypersurface in projective three-space is 65, by analyzing associated linear codes and their dimensions.

Contribution

It offers an elementary proof of Wahl's bound on the dimension of certain codes related to sextic hypersurfaces, clarifying the connection to nodes and correcting previous assertions.

Findings

Confirmed the maximum nodes on a sextic hypersurface is 65.

Established the dimension bound for codes with weights in {24,32,40,56}.

Integrated Wahl's ideas into a simplified proof approach.

Abstract

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.