  We find a decomposition formula of the local Bayer-Macr\`i map for the nef
line bundle theory on the Bridgeland moduli space over surface. If there is a
global Bayer-Macr\`i map, such decomposition gives a precise correspondence
from Bridgeland walls to Mori walls. As an application, we compute the nef cone
of the Hilbert scheme $S^{[n]}$ of $n$-points over special kinds of fibered
surface $S$ of Picard rank two.

Algebraic Geometry