Supersolvability of Low-Exponent Free Hyperplane Arrangements

Definitions and mathematical statement

Let $K$ denote the intended base field, which the supplied record does not specify. Let $V$ be a finite-dimensional $K$-vector space of dimension $\ell$. A central hyperplane arrangement $\mathcal A$ in $V$ is a finite set of linear hyperplanes. Write $S=\operatorname{Sym}(V^*)$, and for each $H\in\mathcal A$, choose a nonzero linear form $\alpha_H$ with $H=\ker\alpha_H$. Its module of logarithmic derivations is \[ D(\mathcal A)=\{\theta\in\operatorname{Der}_K(S):\theta(\alpha_H)\in\alpha_H S\text{ for every }H\in\mathcal A\}. \] The arrangement is free if $D(\mathcal A)$ is a free $S$-module. For a homogeneous basis $\theta_1,\ldots,\theta_\ell$, its exponents are the multiset $\exp(\mathcal A)=\{d_1,\ldots,d_\ell\}$, where $d_i$ is the polynomial degree of the coefficients of $\theta_i$.

Let $L(\mathcal A)$ be the lattice of intersections of subfamilies of $\mathcal A$, including the empty intersection $V$, ordered by reverse inclusion. Set $\rho(X)=\operatorname{codim}_V X$. An element $X\in L(\mathcal A)$ is modular if \[ \rho(X)+\rho(Y)=\rho(X\wedge Y)+\rho(X\vee Y) \qquad\text{for every }Y\in L(\mathcal A), \] where meet and join are taken in this lattice. Put $T=\bigcap_{H\in\mathcal A}H$ and $r=\operatorname{codim}_V T$. The arrangement is supersolvable if there is a chain \[ V=X_0\supsetneq X_1\supsetneq\cdots\supsetneq X_r=T \] of modular elements of $L(\mathcal A)$ with $\rho(X_i)=i$.

The supplied conjecture asks whether, for every finite-dimensional $V$ and every central free arrangement $\mathcal A$ over the intended field $K$, \[ \left(d_i\in\{1,2,3\}\text{ for every }i,\quad \#\{i:d_i=3\}\leq 1\right) \quad\Longrightarrow\quad \mathcal A\text{ is supersolvable}. \] There is no rank bound in the supplied assertion. The intended base field, its characteristic, and whether the assertion quantifies over fields remain unresolved; the displayed formulation does not settle those missing qualifications. The exponents here are those of the ambient derivation module, with no zero exponents permitted by the stated hypothesis; the source does not explicitly discuss essentialization.

Short English statement

Must every central free hyperplane arrangement with all exponents equal to $1$ or $2$, except for at most one exponent equal to $3$, have a maximal chain of modular elements in its intersection lattice? The supplied conjecture has no rank bound, but its intended base field and characteristic require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004609$, source problem code catalogue entry, titled "Supersolvability of Low-Exponent Free Hyperplane Arrangements." The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2021, and supplies no individual proposer.
• Supplied source citation: Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications (2022), Oberwolfach Reports, pp. 225--300, DOI 10.4171/owr/2021/5. https://doi.org/10.4171/owr/2021/5. The record describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Tohaneanu, Free arrangements with low exponents. https://arxiv.org/abs/1707.07091.
• Supplied literature reference: Oberwolfach Report 5/2021. https://ems.press/content/serial-article-files/46883.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that the conjecture is proved in several ranks and for some exponent patterns, while the arbitrary-rank assertion remains open. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review confirming preservation of the mathematical target. These are supplied editorial qualifications, not independent verification of this draft or of the conjecture's mathematical status.