On the geometry of real or complex supersolvable line arrangements. (English) Zbl 1335.52031
Summary: Given a rank 3 real arrangement \(\mathcal{A}\) of \(n\) lines in the projective plane, the Dirac-Motzkin conjecture (proved by B. Green and T. Tao [Discrete Comput. Geom. 50, No. 2, 409–468 (2013; Zbl 1309.51002)]) states that for \(n\) sufficiently large, the number of simple intersection points of \(\mathcal{A}\) is greater than or equal to \(n / 2\). With a much simpler proof we show that if \(\mathcal{A}\) is supersolvable, then the conjecture is true for any \(n\) (a small improvement of the original conjecture). The Slope problem (proved by P. Ungar [J. Comb. Theory, Ser. A 33, 343–347 (1982; Zbl 0496.05001)]) states that \(n\) non-collinear points in the real plane determine at least \(n - 1\) slopes; we show that this is equivalent to providing a lower bound on the multiplicity of a modular point in any (real) supersolvable arrangement. In the second part we find connections between the number of simple points of a supersolvable line arrangement, over any field of characteristic 0, and the degree of the reduced Jacobian scheme of the arrangement. Over the complex numbers even though the Sylvester-Gallai theorem fails to be true, we conjecture that the supersolvable version of the Dirac-Motzkin conjecture is true.
MSC:
| 52C30 | Planar arrangements of lines and pseudolines (aspects of discrete geometry) |
Software:
Macaulay2References:
| [1] | Aigner, M.; Ziegler, G., Proofs from THE BOOK (2010), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 1185.00001 |
| [2] | Björner, A.; Ziegler, G., Broken circuit complexes: factorizations and generalizations, J. Combin. Theory Ser. B, 51, 1, 96-126 (1991) · Zbl 0753.05019 |
| [3] | Crowe, D. W.; McKee, T. A., Sylvester’s problem on collinear points, Math. Mag., 41, 30-34 (1968) · Zbl 0169.52202 |
| [4] | Csima, J.; Sawyer, E. T., There exist \(6 n / 13\) ordinary points, Discrete Comput. Geom., 9, 187-202 (1993) · Zbl 0771.52003 |
| [5] | Dickson, L., The points of inflexion of a plane cubic curve, Ann. of Math., 16, 50-66 (1914-1915) · JFM 45.0838.02 |
| [6] | Elkies, N.; Pretorius, L.; Swanepoel, K., Sylvester-Gallai theorems for complex numbers and quaternions, Discrete Comput. Geom., 35, 361-373 (2006) · Zbl 1091.51501 |
| [7] | Gallai, T., Solution to problem number 4065, Amer. Math. Monthly, 51, 169-171 (1944) |
| [8] | Grayson, D.; Stillman, M., Macaulay2, a software system for research in algebraic geometry, available at |
| [9] | Green, B.; Tao, T., On sets defining few ordinary lines, Discrete Comput. Geom., 50, 409-468 (2013) · Zbl 1309.51002 |
| [10] | Grünbaum, B., The importance of being straight, (Proc. 12th Internat. Sem. Canad. Math. Congress. Proc. 12th Internat. Sem. Canad. Math. Congress, Vancouver, 1969 (1970), Canadian Mathematical Congress: Canadian Mathematical Congress Montreal), 243-254 · Zbl 0225.50016 |
| [11] | Jambu, M.; Terao, H., Free arrangements of hyperplanes and supersolvable lattices, Adv. Math., 52, 248-258 (1984) · Zbl 0575.05016 |
| [12] | Jamison, R.; Hill, D., A catalogue of slope-critical configurations, Congr. Numer., 40, 101-125 (1983) · Zbl 0539.52011 |
| [13] | Kelly, L., A resolution of the Sylvester-Gallai problem of J.-P. Serre, Discrete Comput. Geom., 1, 101-104 (1986) · Zbl 0593.51002 |
| [14] | Kelly, L.; Moser, W., On the number of ordinary lines determined by \(n\) points, Canad. J. Math., 10, 210-219 (1958) · Zbl 0081.15103 |
| [15] | Orlik, P.; Terao, H., Arrangements of Hyperplanes, Grundlehren Math. Wiss., Bd. 300 (1992), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0757.55001 |
| [16] | Oxley, J., Matroid Theory, Oxford Science Publications (1992), Oxford University Press: Oxford University Press New York · Zbl 0784.05002 |
| [17] | Pearson, K., Cohomology of OS algebras for quadratic arrangements, Lect. Mat., 22, 103-134 (2001) · Zbl 1069.05019 |
| [18] | Scott, P., On the sets of directions determined by \(n\) points, Amer. Math. Monthly, 77, 502-505 (1970) · Zbl 0192.57603 |
| [19] | Stanley, R., Supersolvable lattices, Algebra Universalis, 2, 214-217 (1972) · Zbl 0256.06002 |
| [20] | Sylvester, J., Mathematical question 11851, Educational Times, 46, 156 (March 1893) |
| [21] | Tohaneanu, S., On freeness of divisors on \(P^2\), Comm. Algebra, 41, 2916-2932 (2013) · Zbl 1301.13016 |
| [22] | Tohaneanu, S., A computational criterion for supersolvability of line arrangements, Ars Combin., 117, 217-223 (2014) · Zbl 1340.52027 |
| [23] | Ungar, P., \(2N\) noncollinear points determine at least \(2N\) directions, J. Combin. Theory Ser. A, 33, 343-347 (1982) · Zbl 0496.05001 |
| [24] | Ziegler, G., Matroid representations and free arrangements, Trans. Amer. Math. Soc., 320, 525-541 (1990) · Zbl 0727.05019 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
