Bayer-Macr\`i decomposition on Bridgeland moduli spaces over surfaces

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.