Arithmetic families of smooth surfaces and equisingularity of embedded schemes

This article deals with the foundations of a theory of equisingularity for families of zero-dimensional sheaves of ideals on smooth algebraic surfaces , in the arithmetic context, i.e., where one works with schemes defined ove r Dedekind rings. Here, different equisingularity conditions are analyzed a nd compared, based on one of the following requirements: 1) each member of the the family has the same desingularization tree, 2) the family admits a simultaneous desingularization, 3) a naturally associated family of curves is equisingular. Similar conditions had been investigated, in the context o f Complex Local Analytic Geometry, by J. J. Risler.