Perverse schobers, stability conditions and quadratic differentials I

We develop a unified approach for identifying spaces of stability conditions of triangulated categories arising from weighted marked surfaces with moduli spaces of quadratic differentials. This identification is based on the use of perverse schobers (perverse sheaves of triangulated categories) and a notion of positive arc system kit on a perverse schober $\mathcal F$, which provides a systematic way of assigning to a graded curve on the surface a global section of $\mathcal F$. This assignment allows us to identify mixed-angulations and their flips with finite-length hearts and their tilts. As an application we obtain a generalization of the results of Bridgeland--Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential).