Symplectic reduction is a technique that can be used to decrease the dimension of Hamiltonian manifolds. Unfortunately, this only works under strong assumptions on the group action, and in general, even for a compact Lie group, the reduction at a coadjoint orbit that is transverse to the moment map will only yield a symplectic orbifold.
In this article, we show how to construct resolutions of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularize generic symplectic quotients for compact Lie group actions. More precisely, if a point in the Lie coalgebra is regular, that is, its stabilizer is a maximal torus, then we may apply our desingularization result. Regular elements of the Lie coalgebra are generic in the sense that the singular strata have codimension at least three.
Additionally, we show that even though the result of a symplectic cut is an orbifold, it can be modified in an arbitrarily small neighborhood of the cut hypersurface to obtain a smooth symplectic manifold.