no. 4
Logarithmic bundles of deformed Weyl arrangements of type A2
[Fibrés logarithmiques des arrangements de Weyl déformés de type A2]
Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 4, pp. 745-761

We consider deformations of the Weyl arrangement of type A2, which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.

Nous considérons des déformations des arrangements de Weyl de type A2, déformations dont les arrangements de Shi et de Catalan forment une classe particulière. Il est bien connu que ces derniers sont libres. Nous étudions les faisceaux de champs de vecteurs logarithmiques des autres arrangements déformés et montrons qu'ils sont des fibrés de Steiner. Nous déterminons explicitement leurs droites instables. Comme corollaire, des contres exemples du problème appelé « shift isomorphism » sont donnés.

Publié le :
DOI : 10.24033/bsmf.2726
Classification : 52C35, 14F05, 32S22
Keywords: Line arrangements, logarithmic sheaves, Weyl arrangements, root systems.
Mots-clés : Arrangements de droites, faisceaux logarithmiques, arrangements de Weyl, système de racines 
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     author = {Abe, Takuro and Faenzi, Daniele and Vall\`es, Jean},
     title = {Logarithmic bundles of deformed {Weyl} arrangements of type $A_2$},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {745--761},
     year = {2016},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {144},
     number = {4},
     doi = {10.24033/bsmf.2726},
     mrnumber = {3562611},
     zbl = {1358.52021},
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     url = {https://www.numdam.org/articles/10.24033/bsmf.2726/}
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Abe, Takuro; Faenzi, Daniele; Vallès, Jean. Logarithmic bundles of deformed Weyl arrangements of type $A_2$. Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 4, pp. 745-761. doi: 10.24033/bsmf.2726

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