Homaloidal nets and ideals of fat points II: subhomaloidal nets

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.