In toric geometry, we can describe manifolds by specifying a fan, which is roughly a collection of cones (of various dimensions) in an $n$-dimensional lattice. Many topological properties can be easily read off from the fan, so that many computations are greatly simplified for manifolds that can be described in this way,
The particular aspect I am interested in is the following. Consider such a (Kähler) manifold that is described using toric geometry. We can imagine trying to tune one (or more) of the Kähler moduli (parameters), perhaps because we want to shrink some subspace to zero volume and find the resulting space.
I am wondering if there is a nice 'toric' way to determine whether/which walls in the Kähler cone we must pass through as we try to go along this path in Kähler moduli space, and if we must pass through a wall, I wonder if there is a nice way to determine how the triangulation of the fan must change (which 'flops' must occur) in order for us to move into a new sensible geometry.
(The very manual way, to determine these wall crossings and so on, would be to first write down the volumes of all curves, surfaces, etc in terms of the Kähler moduli, and observe what happens as we tune. When some volume becomes negative, we would try to find a way to fix the triangulation in order delete troublesome subspaces and replace them with ones with positive volumes. We would then iterate this.)
