Simple geometric characterization of supersolvable arrangements (Q5940635)

An arrangement of hyperplanes is a finite collection of \(\mathbb{C}\)-linear subspaces of dimension \(d-1\) in \(\mathbb{C}^d.\) Let \(A\) be an arrangement in \(\mathbb{C}^3\) and \(A^*\) be the natural projective arrangements in \(\mathbb{C}\mathbb{P}^2\) associated to it. Let \(L(A)\) be the (rank three geometric) lattice of intersections of elements of \(A,\) ordered by reversed inclusion. A point \(x\) (intersection of lines) of \(A^*\) is called a center of \(A^*\) if for each intersection point \(y\) of \(A^*\) there is a line connecting \(x\) and \(y.\) The authors prove: NEWLINENEWLINENEWLINEMain Theorem. \(L(A)\) has a maximal chain of modular elements (is supersolvable) if and only if \(A^*\) has a center. NEWLINENEWLINENEWLINE There are one-to-one maps between the atoms [resp. the rank two elements] of \(L(A)\) and the lines [resp. points] of \(A^*.\) Then the theorem is just equivalent to the statement that ``\(L(A)\) is supersolvable iff there is a modular rank two flat'', a trivial and well-known result in lattice theory.