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We show that on any compact K\"ahler surface existence of solutions to theZ-critical equation can be characterized using a finite number of effectiveconditions, where the number of conditions is bounded above by the Picardnumber of the surface.This leads to a first PDE analogue of the locally finitewall-chamber decomposition in Bridgeland stability.  As an application we characterize optimally destabilizing curves forDonaldson's J-equation and the deformed Hermitian Yang-Mills equation, prove anon-existence result for optimally destabilizing test configurations foruniform J-stability, and remark on improvements to convergence results forcertain geometric flows.

The set of destabilizing curves for deformed Hermitian Yang-Mills and Z-critical equations on surfaces