Simplicial Arrangements with up to 27 Lines

Abstract

Abstract. We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Grünbaums catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines, respectively. As a byproduct we classify simplicial arrangements of pseudolines with up to 27 lines. In particular, we disprove Grünbaums conjecture about unstretchable arrangements with at most 16 lines, and prove the conjecture that any simplicial arrangement with at most 14 pseudolines is stretchable.

Simplicial arrangements revisited

“…The enumeration of simplicial arrangements of pseudolines in [3] shows that all simplicial arrangements with at most 14 pseudolines are stretchable, thus confirming a conjecture made in [8]. The computer-assisted enumeration in [3] uses "wiring diagrams" introduced Goodman in [5], and elaborated in Goodman and Pollack [6] and other publications, together with innovative arguments to reduce the computational effort. The results, in particular, disprove another conjecture in [8]: Namely, that there is a single unstretchable simplicial arrangement of 15 pseudolines and four of 16 pseudolines.…”

“…In [3], Cuntz first enumerates simplicial arrangements of at most 27 pseudolines, and then investigates their stretchability, that is, the isomorphism to linear arrangements. The bound 27 is due to limitations of the computing power available, but even with this bound several notable results are obtained and several conjectures of the present writer are resolved.…”

“…The results, in particular, disprove another conjecture in [8]: Namely, that there is a single unstretchable simplicial arrangement of 15 pseudolines and four of 16 pseudolines. In the paper [3] Cuntz establishes that there are precisely two such arrangements with 15, and precisely seven with 16 pseudolines. The second 15-pseudoline arrangement is shown in Figures 7 and 8 in two forms.…”

Milnor fibers of real line arrangements

“…More generally, using Theorem 2.3 and Theorem 3.21, we can prove that A(6m, 1) is pure-tone. All other examples except for A(6m, 1) in the catalogue [8] (and [5]) satisfy H 1 (F A ) =1 = 0. It seems natural to pose the following.…”

“…Then the projective arrangement cA = A ∪ {H ∞ } in the real projective plane RP 2 is called simplicial if each chamber is a triangle. Grünbaum [8] presents a catalogue of known simplicial arrangements with up to 37 lines (see [5] for additional information).…”