On the projective dimension of some deformations of Weyl arrangements

TL;DR

The paper investigates the projective dimension and resolutions of logarithmic derivation modules for deformations of Weyl arrangements. It proves that for simply-laced root systems, the coned deformation $D(c{\mathcal A}^{[-k,k+2]}_{\rootpositive})$ has projective dimension one, and it supplies explicit minimal free resolutions in types $A_3$ and $B_2$, along with stability and jumping-line analyses in the $B_2$ case. A key finding is that, although certain deformations share identical graded Betti numbers after degree shifts, they are not isomorphic as modules when their deformation parameters differ. The work combines deletion/addition techniques, SPOG theory, and vector-bundle methods on $\mathbf P^2$ to illuminate structure, splitting behavior, and geometric invariants of these deformations, with potential implications for distinguishing deformations beyond simple Betti-number data.

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.

Theorems & Definitions (49)

• Definition 1.1
• Conjecture 1.2
• Theorem 1.3: AFV
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1: Sa
• Proposition 2.2: Theorem 4.37, OT
• Theorem 2.3: Terao's deletion theorem, T1