We consider plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. The object of this work is to study the link 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. We reveal 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 we emphasize also the role of one single highest multiplicity. In this vein we describe classes of Cremona maps for large and small values of the highest virtual multiplicity. We also deal with the delicate question as to when is the base ideal non-saturated and consider the structure of its saturation.

This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, where the entries of the corresponding presentation matrix are general linear forms. The main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal. A full description of the corresponding symbolic Rees algebra is given in some cases. An application is an affirmative solution of a conjecture of Eisenbud-Mazur in [11, Section 2].

One considers certain degenerations of the generic symmetric matrix over a field k of characteristic zero and the main structures related to the determinant f of the matrix, such as the ideal generated by its partial derivatives, the polar map defined by these derivatives and its image V (f ), the Hessian matrix, the ideal and the map given by the cofactors, and the dual variety of V (f ). A complete description of these structures is obtained.

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 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.

One deals with catalectic codimension two perfect ideals and certain degenerations thereof, with a view towards the nature of their symbolic powers. In the spirit of [10] one considers linearly presented such ideals, only now in the situation where the number of variables is sufficiently larger than the size of the matrix, yet still stays within reasonable bounds.