  In this paper, we examine the space of initial values for integrable lattice
equations, which are lattice equations classified by Adler {\em et al} (2003),
known as ABS equations. By considering the map which iterates the solution
along particular directions on the lattice, we perform resolutions of
singularities for several examples of ABS equations for the first time. Our
geometric observations lead to new Miura transformations and reductions to
ordinary difference equations.
