Arrangements of lines with tree resolutions (Q807696)

An arrangement is a set \({\mathcal L}=\{L_ 1,...,L_ n\}\) of lines in the complex projective plane \({\mathbb{P}}^ 2\). A point \(x\in \cup_{i}L_ i \) is said to have \(valence\quad r\) if it is contained in exactly r of the lines \(L_ i\). Let b: \(X\to {\mathbb{P}}^ 2\) be the blowing-up of \({\mathbb{P}}^ 2\) along the set N of points of valence \(\geq 3\). The set S consisting of the proper transforms of the lines \(L_ i\) and of the exceptional curves \(b^{-1}(x)\), \(x\in N\), has the following interesting property: to any subset of S one can associate a dual graph whose vertices correspond to the curves and the edges correspond to their non- empty intersections. The authors define an arrangement to have a tree resolution if it possible to associate to it in a technical way in line with the above a connected graph with a suitable number of cycles. The technicalities of the definition are motivated by the applications to the study of the homology planes made by the first author [J. Fac. Sci., Univ. Tokyo, Sect. I A 37, No.1, 33-69 (1990; Zbl 0715.14003)]. The paper is devoted to prove the following finiteness result for arrangements of this kind. If \({\mathcal L}\) is an arrangement with no point of valence \(\geq n-2\), having a tree resolution, then \(n\leq 13\). This offers the possibility of a projective classification of all arrangements having tree resolutions.