On the projective dimension of some deformations of Weyl arrangements

TL;DR

This paper investigates the projective dimension of the logarithmic derivation module of deformed Weyl arrangements, providing explicit resolutions and analyzing their geometric properties for specific root systems.

Contribution

It establishes the projective dimension for deformations of Weyl arrangements and explicitly constructs minimal free resolutions for types A3 and B2, also analyzing their vector bundles.

Findings

Projective dimension is one for certain deformations of Weyl arrangements.

Explicit minimal free resolutions are provided for types A3 and B2.

The study distinguishes modules with identical Betti numbers using jumping lines.

Abstract

We show that the logarithmic derivation module of (the cone of) the deformation A of a Weyl arrangement associated with a root system of simply laced type has projective dimension one if the deforming parameter ranges from -j to j+2. In addition, we give an explicit minimal free resolution when the root system is of type A3 and B2. Moreover, in the second case, we determine the jumping lines of maximal jumping order of the associated vector bundle. When the deforming parameter of A (respectively A') ranges from -k to k+j (respectively, from -k' to k'+j), with k different from k' and j at least 3, this allows to distinguish D0(A) from D0(A') shifted by 4(k'-k), even though these modules have the same graded Betti numbers.