Logarithmic bundles of deformed Weyl arrangements of type $A_2$

Takuro Abe (1) , Daniele Faenzi (2) , Jean Vallès (3)

1 Department of Mechanical Engineering and Science, Kyoto University
2 Algèbre et géométrie
3 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We consider deformations of the Weyl arrangement of type $A_2$, 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.

Mots clés: Root systems; Weyl arrangements; Line arrangements; Logarithmic sheaves

Domaines: Géométrie algébrique [math.AG]

Takuro Abe, Daniele Faenzi, Jean Vallès. Logarithmic bundles of deformed Weyl arrangements of type $A_2$. Bulletin de la société mathématique de France, 2016, 144 (4), pp.745-761. ⟨hal-00987021v3⟩