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, 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].

Homaloidal determinants

“…The congruence translates into the existence of a uniquely defined form D ∈ R such that, using a short vector notation, g(f ) = D · (x). In [26] D has been dubbed the inversion factor of the map F or, more precisely, its source inversion factor. Its degree is deg(F) deg(G) − 1.…”

“…where the rightmost matrix is the syzygy K in (26), viewed as a column vector, where l = n − 1. By reasoning as in the argument that ensues (27), one gets that the ith entry of g 4 is −r n−i , 1 ≤ i ≤ n − 1.…”

“…. , y n ], where g i = 2x n−i y 0 + 2i − 1 i x n−i+1 y 1 + · · · + x n y i (1 ≤ i ≤ n − 1) and g n = (n − 1)x 0 y 0 + (n − 2)x 1 y 1 + · · · + x n y n , g n+1 = n−1 j=1 r j y j + x n−1 n−1 y n generate L (from (30)), with r i being as in (26). Decompose further h i = h ′ i + y n h ′′ i , with h ′ i ∈ R[y 0 , .…”

Determinacy of determinantal varieties

“…Linear determinantal varieties have been intensively studied by many authors, see for instance [6], [8] and [13]. We discuss here a method to get examples of homogeneous EIDS of arbitrarily high degree from a special class of linear EIDS.…”

Linearly presented perfect ideals of codimension 2 in three variables

“…The case of reduction number 2 is specially dealt with. Section 3 calls upon a second main character, namely, a certain submatrix of the socalled Jacobian dual matrix developed in various sources ( [5], [16], [19]) and originally introduced in [23]. This submatrix undergoes several stages of virtual cloning throughout the text, all satisfying the property of being 1-generic in the sense of [6] and [10,Chapter 9].…”