J functions, non-nef toric varieties and equivariant local mirror symmetry of curves

We provide a straightforward computational scheme for the equivariant local mirror symmetry of curves, i.e. mirror symmetry for O(k) circle plus O(-2-k) -> P-1 for k >= 1, and detail related methods for dealing with mirror symmetry of non-nef toric varieties, based on the theorems of Refs. 2 and 13. The basic tools are equivariant I functions and their Birkhoff factorization.