On Wahl's proof of $μ(6)=65$
| dc.creator | Pignatelli, Roberto | |
| dc.creator | Tonoli, Fabio | |
| dc.date | 2007-06-29 | |
| dc.date.accessioned | 2026-07-07T08:13:06Z | |
| dc.date.available | 2026-07-07T08:13:06Z | |
| dc.description | 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. | |
| dc.description | 5 pages | |
| dc.identifier | https://arxiv.org/abs/0706.4358 | |
| dc.identifier | http://arxiv.org/abs/0706.4358 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132683 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 94B05,14N25,14M99,14Q10 | |
| dc.title | On Wahl's proof of $μ(6)=65$ | |
| dc.type | text |