Real and complex supersolvable line arrangements in the projective plane
- PDF / 491,920 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 45 Downloads / 253 Views
Real and complex supersolvable line arrangements in the projective plane Krishna Hanumanthu1 · Brian Harbourne2 Received: 13 March 2020 / Accepted: 30 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract If we regard a set of s lines in P2 over either the reals or the complex numbers as an algebraic plane curve, then it is an open problem to classify for all s those for which the number t2 of points of multiplicity 2 satisfies t2 < s/2. By the Sylvester–Gallai theorem, there are no nontrivial (i.e., not a pencil or a near pencil) real arrangements with t2 = 0, but there are complex arrangements with t2 = 0 and it is an open problem to classify them. In this paper, we initiate a classification of an interesting class of line arrangements called the supersovable line arrangements and give a partial classification for them over the reals or the complex numbers. In particular, we show that a complex line arrangement which is nontrivial cannot have more than 4 modular points and we completely describe those with 3 or 4 modular points. Keywords Dirac–Motzkin conjecture · Homogeneous supersolvable line arrangements · Modular points · Double points
1 Introduction Line arrangements have provided useful insight in studying a range of recent problems in algebraic geometry. They have played a fundamental role for studying the containment problem (see [9,10]), for the bounded negativity problem and H constants [4]
The first author was partially supported by a Grant from Infosys Foundation and by DST SERB MATRICS Grant MTR/2017/000243. The second author was partially supported by Simons Foundation Grant #524858. We also thank T. Szemberg, J. Szpond and S. ¸ Tohˇaneanu for some helpful comments.
B
Krishna Hanumanthu [email protected] Brian Harbourne [email protected]
1
Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103, India
2
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA
123
Journal of Algebraic Combinatorics Fig. 1 A supersolvable line arrangement with 2 modular points (shown as white dots)
and for unexpected curves [6,7]. The supersolvable arrangements are a particularly tractable subclass of line arrangements which have played a role in the study of unexpected curves [6,7]. Understanding supersolvable arrangements better should make them even more useful. Thus, the goal of the present paper is to pin down, as much as currently possible, properties of real and complex supersolvable line arrangements. A line arrangement is simply a finite set of s > 1 distinct lines L = {L 1 , . . . , L s } in the projective plane. A point p is a modular point for L if it is a crossing point (i.e., a point where two or more of the lines meet) with the additional property that whenever q is any other crossing point, then the line through p and q is L i for some i. We say L is supersolvable if it has a modular point (see Fig. 1). If the s lines of L are concurrent (i.e., all meet at a point), then L is supersolvable since it
Data Loading...
