The Cantor-Bendixson rank of certain Bridgeland-Smith stability conditions (Q2413063)

This paper studies meromorphic quadratic differentials over Riemann surfaces using the spaces of Bridgeland stability conditions. The study is motivated by the work of \textit{T. Bridgeland} and \textit{I. Smith} [Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)] and the work of \textit{D. Gaiotto} et al. [Adv. Math. 234, 239--403 (2013; Zbl 1358.81150)]. The main result is that given a meromorphic quadratic differential with at least one pole of order at least two, the set of directions admitting a saddle connection has finite Cantor-Bendixson rank (Theorem 1). Actually an explicit upper bound is given on the Cantor-Bendixson rank, and a family of surfaces having all possible ranks is realized (\S7, \S8).