Classification of Hamiltonian $S^1$-actions on compact symplectic orbifolds with isolated cyclic singular points in dimension four

In this paper, we classify Hamiltonian $S^1$-actions on compact, four dimensional symplectic orbifolds that have isolated singular points with cyclic orbifold structure groups, thus extending the classification due to Karshon to the orbifold setting. To such a space, we associated a combinatorial invariant, a labeled multigraph, that determines the isomorphism type of the space. Moreover, we show that any such space can be obtained by applying finitely many equivariant weighted blow-ups to a minimal space, i.e., one on which no equivariant weighted blow-down can be applied.