  One considers plane Cremona maps with proper base points and the {\em base
ideal} generated by the linear system of forms defining the map. The object of
this work is the interweave between the algebraic properties of the base ideal
and those of the ideal of these points fattened by the virtual multiplicities
arising from the linear system. One reveals conditions which naturally regulate
this association, with particular emphasis on the homological side. While most
classical numerical inequalities concern the three highest virtual
multiplicities, here one emphasizes also the role of one single highest
multiplicity. In this vein one describes classes of Cremona maps for large and
small value of the highest virtual multiplicity. One also deals with the
delicate property as to when the base ideal is non-saturated and the structure
of its saturation.
