Generators for arrangements of type G(r,r,ell)

Establish that, for every simple connected graph G on ell vertices, the derivation module D(A_G) of the arrangement of type G(r,r,ell) is generated by the vector field x_1^{r-1}x_2^{r-1}\cdots x_ell^{r-1}\cdot\eta_{-1}, the derivations \eta_0,\eta_1,\ldots,\eta_{\kappa(G)}, the vertex derivations \theta_v^{N(v)} for v\in V(G), and a finite set of derivations \theta_C^{T,1} associated with separators T and components C\in\mathscr C(T).

Background

The paper proves a finite-generation result for the derivation module D(A_G) when the arrangement A_G is associated with G(r,p,ell) and p<r. In that case, coordinate hyperplanes are present, and separator-based derivations together with the standard derivations \eta_k generate D(A_G).

The case p=r is different because the coordinate hyperplanes are absent. For the complete graph K_ell, a known basis consists of x_1{r-1}x_2{r-1}\cdots x_ell{r-1}\cdot\eta_{-1}, \eta_0,\ldots,\eta_{ell-2}. Motivated by this complete-graph basis and observations about derivations for general graphs, the authors conjecture a corresponding generating set for all connected graphs, augmented by the vertex derivations \theta_v{N(v)} and finitely many separator-based derivations.

References

By observation of derivations of $A_G$ of type $G(r, r ,\ell)$, we pose the following conjecture. Given a simple connected graph $G$. The vector fields $x_1{r-1} x_2{r-1} \dots x_\ell{r-1}~\cdot~\eta_{-1}, \eta_0, \eta_1, \dots, \eta_{\kappa(G)}$, $\theta_v{N(v)}$ (for $v\in V(G) $), and a finite set of vector fields $\theta_{C}{T,1}$ (for separators $T$ and $C\in \mathscr{C}(T)$) form a generating set for derivation module $D(A_G)$ of arrangement $A_G$ of type $G(r, r, \ell)$.

— The separator-based derivations for arrangements of monomial groups  (2610.06098 - Wang et al., 5 Oct 2026) in Conjecture in Section 4, “Discussion” (Section \ref{sec:conjecture})