  This paper is a natural sequel to [22] in that it tackles problems of the
same nature. Here one aims at the ideal theoretic and homological properties of
a class of ideals of general plane fat points whose second symbolic powers hold
virtual multiplicities of proper homaloidal types. For this purpose one carries
a detailed examination of their linear systems at the initial degree, a good
deal of the results depending on the method of applying the classical
arithmetic quadratic transformations of Hudson--Nagata. A subsidiary guide to
understand these ideals through their initial linear systems has been supplied
by questions of birationality with source $\mathbb{P}^2$ and target higher
dimensional spaces. This leads, in particular, to the retrieval of birational
maps studied by Geramita--Gimigliano--Pitteloud, including a few of the
celebrated Bordiga--White parameterizations.
