On adjacent complete ideals above a given complete ideal

Given a complete m-primary ideal J in a local regular two-dimensional ring (R, m), we describe every adjacent complete ideal above J as the integral closure of some ideal (f, g) for suitable f, g associated to J. We also provide a geometrical procedure that gives its base points, thus determining its equisingularity class. We decompose the set I(J) of these adjacent ideals in terms of the Rees valuations of J. As a consequence, we obtain a geometrical characterization of the finiteness of I(J).