Logarithmic bundles of deformed Weyl arrangements of type $A_2$

TL;DR

This paper investigates the sheaves of logarithmic vector fields of deformed Weyl arrangements of type A2, revealing they are Steiner bundles and identifying unstable lines, with implications for the shift isomorphism problem.

Contribution

It explicitly describes the sheaves of logarithmic vector fields for deformed Weyl arrangements of type A2 and finds they are Steiner bundles, providing new insights and counter-examples.

Findings

Sheaves are Steiner bundles for all deformations studied.

Explicit determination of unstable lines.

Counter-examples to the shift isomorphism problem.

Abstract

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.