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Extendability of the $B_2$-arrangement

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Pith's one-line read The paper proves that the Coxeter arrangement of type $B_2$ carries infinitely many free multiarrangements—one for each $k\ge 4$—that admit no free extension, and that the obstruction passes to higher-rank Coxeter arrangements.

desk verdict First infinite family of non-extendable free multiplicities on B2, but the proof rests on two sketchy arguments (Lemma 5.1 and the graph analysis) that need to be written out. read the letter →

arxiv 2506.02512 v1 pith:OLGQ7JL7 submitted 2025-06-03 math.CO

Torsten Hoge , Shota Maehara , Sven Wiesner This is my paper
classification math.CO MSC 52C3532S2251F15
keywords freearrangementmultiarrangementlogarithmicderivationmoduleCoxetertypeB2restrictionofarrangementsextensionmixedproductsinductively
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the Coxeter arrangement of type $B_2$ carries infinitely many free multiarrangements that cannot be extended to a free simple arrangement: for every $k\ge 4$, the multiarrangement with defining polynomial $x^2y^k(x-y)(x+y)^k$ has no free extension. This matters because a free extension would let the exponents of the multiarrangement be read off from a purely combinatorial invariant of a larger arrangement, a route that works for type $A_2$ but now fails for $B_2$. The obstruction is local: any higher-rank Coxeter arrangement containing this $B_2$-slice as a rank-two localization also admits no free extension, and an explicit infinite family on $B_3$ is constructed.

What carries the argument

The load-bearing device is the comparison between the locally mixed product $\mathrm{LMP}(\mathscr{E})$, a sum over rank-two flats of the products of local exponents, and the virtually globally mixed product $\mathrm{VGMP}(\mathscr{E},H)$ computed from the restriction to $H$; Lemma 4.3 reformulates the freeness criterion to say that a central 3-arrangement is free exactly when these two numbers agree. Corollary 5.4 turns this equality into interval bounds on the number of hyperplanes in a restriction. The second engine is a uniqueness lemma: over characteristic zero, every free extension of the simpler multiarrangement $x^ky^k(x-y)^2$ is, after a change of basis, the canonical extension whose deconing produces a staircase of triple points that can never revisit a point. This forced geometry yields the contradictory lower bound in the first branch of Theorem 1.1, while the second branch is excluded by a graph whose sources and sinks must both lie on one line.

What would settle it

Find, for any $k\ge 4$, an explicit free 3-arrangement whose restriction to some hyperplane equals $(A(B_2),(2,k,1,k))$; presenting such an arrangement would disprove Theorem 1.1. A smaller target is to exhibit a free extension of $x^ky^k(x-y)^2$ over a characteristic-zero field that is not basis-equivalent to the canonical extension, which would break Lemma 5.1 and with it the lower-bound contradiction.

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Extended reading notes

Core claim

A free multiarrangement $(\mathscr{A},m)$ is a hyperplane arrangement with integer multiplicities whose module of logarithmic derivations is free; a free extension is a simple free arrangement $\mathscr{E}$ with a hyperplane whose restriction, with multiplicities recording how many hyperplanes of $\mathscr{E}$ meet each intersection, equals $(\mathscr{A},m)$. The paper's central claim is that the type-$B_2$ Coxeter arrangement with multiplicities $(2,k,1,k)$, $k\ge 4$, is free but admits no free extension. The proof works by deriving two incompatible constraints on any hypothetical extension: a count of intersection points on one hyperplane forces $|\mathscr{E}^H|\ge k+3$, while the freeness criterion expressed through mixed products forces $k+1\le |\mathscr{E}^H|\le k+2$. A separate argument rules out the borderline case $|\mathscr{E}^H|=k+1$ by tracking a directed graph of triple points on the two copies of the line $x=0$.

Load-bearing premise

The whole first branch of the proof rests on a uniqueness lemma claiming that, over the rationals, every free extension of the simpler multiarrangement $x^ky^k(x-y)^2$ has the same intersection pattern as the canonical extension; if a genuinely different free extension existed, the contradiction that kills the $k\ge 4$ case would not go through.

Editorial extensions

If this is right

  • For each $k\ge 4$, no free extension exists for $(A(B_2),(2,k,1,k))$, so the extendability question for free multiarrangements has a negative answer for infinitely many free multiplicities on a fixed Coxeter arrangement.
  • Any type-$B_n$ multiarrangement that has a rank-two localization isomorphic to this $B_2$ pair has no free extension, since localization of a free extension would produce a forbidden free extension of the $B_2$ slice.
  • There are inductively free $B_3$ multiarrangements with exponents $(5,k+1,k+2)$ that nevertheless admit no free extension.
  • Two free extensions of the same multiarrangement need not have isomorphic intersection lattices, so extendability is not captured by a single lattice invariant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same proof strategy should apply to other rank-two Coxeter and reflection arrangements: wherever a uniqueness lemma analogous to Lemma 5.1 holds and the mixed-product bounds force a gap, infinitely many non-extendable multiplicities will appear, so Question 5.6 likely has a negative answer for most types beyond $A_2$.
  • Since the no-revisit argument in Lemma 5.1 is characteristic dependent—the paper itself exhibits a free extension over $\mathbb{F}_9$—the main theorem should fail over suitable finite fields for the same multiplicities; a direct computer search for $k=4$ over small characteristics would test this.
  • The borderline case $k=3$ is extendable, which suggests that peak-point configurations mark the threshold; a classification of extendable multiplicities on $B_2$ may reduce to locating when a deletion creates a peak point.
  • For higher rank, one can seed new non-extendable families by embedding the $B_2$ slice in any arrangement whose free extension would localize; Corollary 6.2 is one instance, but any localization-closed class of arrangements would work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the extendability of free multiarrangements associated to the Coxeter arrangement of type B2. The main result (Theorem 1.1) asserts that for the multiplicity vector (2,k,1,k) with k ≥ 4, the multiarrangement (A(B2), m_k) admits no free extension. The proof combines an exponent classification for B2 multiplicities, a reformulation of Yoshinaga's freeness criterion in terms of locally and globally mixed products, and a detailed geometric analysis of a putative extension. As a corollary, the obstruction localizes to higher-rank Coxeter arrangements of type Bn, and an infinite family of inductively free multiplicities on B3 with no free extension is constructed.

Significance. If correct, the result provides the first infinite family of free multiplicities on a fixed Coxeter arrangement for which no free extension exists, answering a natural question of Yoshinaga in this setting. The localization argument in Corollary 6.2 is clean and gives a simple obstruction in higher rank. The paper is generally well organized, includes explicit examples, and makes effective use of recent classification results for B2 multiarrangements. The main theorem is falsifiable and the obstruction is concrete. However, the proof depends crucially on a geometric uniqueness lemma (Lemma 5.1) whose proof has significant gaps, and the counting and graph-theoretic arguments in the two branches of Theorem 1.1 require substantial additional justification.

major comments (3)
  1. [Section 5, Lemma 5.1] The proof of Lemma 5.1 asserts without derivation that |(E0 ∪ {H1})^H1| = k + 1. For a free 3-arrangement with exponents (1,k,k+1), the Ziegler restriction to H1 has total degree 2k+1, but the number of lines in the simple restriction is not forced to be k+1; this equality requires an argument showing that the slopes of the x-copies and y-copies are paired. The subsequent deconing argument ("Clearly, the next triple point must be P3=(1,1)") is a sketch rather than a proof of the uniqueness of the Yoshinaga pattern, and the conclusion that revisiting a point is impossible in characteristic zero is stated without proof. Because this lemma is the entire justification for identifying E' with the Yoshinaga extension in the first branch of Theorem 1.1, this gap is load-bearing.
  2. [Section 5, proof of Theorem 1.1, first branch] The bound |E^H| ≥ k+1 + ceil(k/2) depends on the assertion that at least half of the intersection points of H with the copies of y cannot lie on any copy of x+y. The coordinate computation shows that intersections with y-copies shift by integer offsets while intersections with x+y copies shift by half-integer offsets, but it is not explained why this implies the stated proportion. A precise counting or parity argument is needed to justify the contradiction with the upper bound |E^H| ≤ k+2 from Remark 5.5.
  3. [Section 5, proof of Theorem 1.1, second branch] The graph argument contains several unproved assertions: that (0,1) and (0,2) on H1 are connected via (1,1) on H, that (0,2) must be a sink in G(H1) because otherwise (1,2) would introduce another connected component on G(H2), and that the point (0,1/2) must be a triple point and forces the chain (0,1/4), (1,1/4), leading to a contradiction. These steps are not derived from the preceding definitions and appear to rely on additional geometric facts about the deconed arrangement; without a complete proof, the contradiction in this branch is not established.
minor comments (4)
  1. [Section 3] The word "wheter" in the opening sentence should be "whether".
  2. [Section 5, Lemma 5.1 proof] The word "arrrangement" appears in the first paragraph; it should be "arrangement".
  3. [Notation] The notation H1 is used for a copy of x-y in Lemma 5.1 but for a copy of x in the proof of Theorem 1.1; this dual usage may confuse readers.
  4. [Definition 2.14] In the definition of the Yoshinaga extension, the range for k is not explicitly restricted to integers, and the endpoints may be non-integers for even multiplicities; the intended integer ranges should be stated.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main non-existence theorem is derived from the paper's own lemmas and external theorems; the few self-citations are not load-bearing.

full rationale

The derivation of Theorem 1.1 is not circular. The non-existence of free extensions for (A(B2),(2,k,1,k)), k≥4, is shown by a case split: if the deletion E' is free, Lemma 5.1 (proved in the paper via Abe–Yamaguchi's free-path theorem) identifies E' with the Yoshinaga extension, and an explicit count of intersection points yields |E^H| ≥ k+1+⌈k/2⌉; this contradicts the upper bound |E^H| ≤ k+2 from Corollary 5.4 and Remark 5.5. The second branch uses a graph-theoretic argument that is entirely internal to the paper. The upper bound depends on Corollary 3.8, whose proof is given in the appendix: it derives the peak-point condition from the external result [FWY25, Thm. 2.3] and the paper's own Proposition 7.3, invoking [MN25] only to note agreement with an existing classification. Although [MN25] has overlapping authorship (Maehara is a coauthor), it is not load-bearing: the explicit condition m1=m2 and |m|∈4Z is derived from [FWY25] and Proposition 7.3, not imported from [MN25]. The other self-citation, [MRW24], supports only the auxiliary A2 extendability fact that is already due to Yoshinaga and is not used in the main proof. No prediction is fitted, no quantity is defined in terms of the target result, and no uniqueness theorem by the authors is invoked to force the conclusion. The proof does contain compressed steps—for example, the assertion in Lemma 5.1 that |(E0∪{H1})^H1| = k+1 is stated without a full derivation—but compression is a gap in justification, not a circular reduction of the theorem to its inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard theorems in arrangement theory and on two recent classification results by other authors, one coauthored by Maehara. No numerical constants are fitted and no new entities are posited; the multiplicity families are plain inputs. The main non-standard dependency is the peak-point classification used to determine which deletions are unbalanced or peak points.

assumptions (7)
  • domain assumption All arrangements are central over a field of characteristic zero.
    Stated in Section 2 and used in Theorem 3.4 and Lemma 5.1; characteristic zero is essential for the no-revisit argument in Lemma 5.1.
  • standard math Abe's exponent-gap bound, [A13, Thm. 1.6]: for a balanced 2-arrangement with |A|>2, |d1-d2| <= |A|-2.
    Invoked in Section 3 to constrain exponents of balanced B2 multiplicities in Corollary 3.6.
  • standard math Peak-point classification for B2 with minimum multiplicity 1, from [MN25, Thm. 3.3, Cor. 3.4] and [FWY25, Thm. 2.3].
    Used to prove Corollary 3.8, which controls when deleting a hyperplane leaves a peak point. [MN25] is coauthored by Maehara, so this is a self-citation dependency.
  • standard math Free path theorem, [AY23, Thm. 1.2].
    Used in Lemma 5.1 to justify deletion order and restriction sizes.
  • standard math Yoshinaga freeness criterion (Theorem 2.12) and Ziegler restriction theorem (Theorem 2.10).
    Reformulated as Lemma 4.3 and used throughout Corollary 4.7 and the proof of Theorem 1.1.
  • standard math Addition-Deletion theorem for simple arrangements, [T80].
    Used in Remark 5.5 to relate freeness of E' to |E^H|.
  • standard math Localization preserves freeness, [ANN09, Prop. 1.7].
    Used to transfer the B2 obstruction to Bn in Corollary 6.2.

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Pith. "Pith review of Extendability of the $B_2$-arrangement." pith.science (2026). https://pith.science/paper/OLGQ7JL7

@misc{pith2026250602512,
  author       = {Pith},
  title        = {Pith review of: Extendability of the $B_2$-arrangement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLGQ7JL7}},
  note         = {Machine review of arXiv:2506.02512}
}
abstract

Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always exists. In this work, we demonstrate that if $\mathscr{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathscr{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank.

Figures

Figures reproduced from arXiv: 2506.02512 by the authors.

Figure 1
Figure 1. The Yoshinaga-Extension E (A (B2),(3, 5, 2, 2)). Example 4.8. Let (A (B2), m) with m = (3, 5, 2, 2), then exp(A (B2), m) = (5, 7). Suppose there exists a free extension E with a hyperplane H ∈ E such that (E H, mH) = (A (B2), m). In this case, it holds that VGMP(E , H) = 47 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Free extensions of (A (B2),(3, 5, 2, 2)) with non-isomorphic lattices of intersection. Similarly, we define Hx,1, . . . , Hx,k ∈ E and Hy,1, . . . , Hy,k ∈ E as the copies of H˜ x and H˜ y, respectively. Note that the subarrangement E0 = E \ {H1, H2} is an extension of the Boolean multiar￾rangement (B, m0), defined by Q(B, m0) = x k y k . Hence, E0 is free with exp(E0) = (1, k, k). Furthermore, by Remark 3.5 (2), we… view at source ↗
Figure 3
Figure 3. Deconing (z = 1) of free extension of (A (A2),(2, 4, 4)). Proof. Let H0, H1, E be as in Lemma 5.3, and let H ∈ A such that m(H) = m1. Define the set CH := {H′ ∈ E | (H′ ∩ H0) = H} ⊂ E , which consists of m1 hyperplanes. Since E H0 = A , it follows that for any H′ ∈ CH that E(H0∩H1) ∩ E(H0∩H′) = ∅, as otherwise the hyperplanes would get identified in A . Furthermore, for any H′ ∈ CH, we have E(H0∩H′) = CH ∪ {H0}. In … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: H1 H2 0 H [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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