Symbolic powers of perfect ideals of codimension 2 and birational maps

Abstract

This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, assuming that 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. One application is an affirmative solution of a conjecture of Eisenbud–Mazur in [11, Section 2].