TY - JOUR

T1 - A Greedy Algorithm to Compute Arrangements of Lines in the Projective Plane

AU - Cuntz, Michael

N1 - Funding Information: The computations required for the results of this paper were performed on a computer cluster funded by the DFG, project number 411116428.

PY - 2022/7

Y1 - 2022/7

N2 - We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.

AB - We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.

KW - math.CO

KW - 20F55, 52C35, 14N20

KW - Reflection group

KW - Matroid

KW - Simplicial arrangement

UR - http://www.scopus.com/inward/record.url?scp=85121430085&partnerID=8YFLogxK

U2 - 10.1007/s00454-021-00351-y

DO - 10.1007/s00454-021-00351-y

M3 - Article

VL - 68

SP - 107

EP - 124

JO - Discrete & computational geometry

JF - Discrete & computational geometry

SN - 0179-5376

IS - 1

ER -