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

TL;DR

This paper develops a decomposition formula for the Bayer-Macri map on Bridgeland moduli spaces over surfaces, linking wall structures and computing nef cones for specific fibered surfaces.

Contribution

It introduces a new decomposition formula for the local Bayer-Macri map and applies it to explicitly compute nef cones of Hilbert schemes on certain fibered surfaces.

Findings

Decomposition formula relates Bridgeland and Mori walls.

Explicit nef cone calculations for Hilbert schemes on fibered surfaces.

Establishment of a correspondence between Bridgeland and Mori wall structures.

Abstract

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.