Birational equivalences of vortex moduli

We construct a finite-dimensional Kähler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the τ-vortex moduli spaces (or τ-stable pairs). The Morse theory of the circle action induces natural birational maps between the reduced spaces for different values of τ which in the case of rank two bundles can be canonically resolved in a sequence of blow-ups and blow-downs.