  Families of Bridgeland stability conditions induce families of stability
data, wall-crossing structures and scattering diagrams on the motivic Hall
algebra. These structures can be transferred to the quantum torus if the
stability conditions of the family have global dimension at most 2. We show
that geometric stability conditions on surfaces with nef anticanonical bundle
have global dimension 2 and we study the resulting family of stability data. We
formulate a conjecture relating this family to the family of stability data
associated with a quiver with potential and we verify this conjecture for the
projective plane.
