Symbolic powers of perfect ideals of codimension 2 and birational maps

TL;DR

This paper explores the structure of symbolic powers of codimension two perfect ideals in polynomial rings, linking birational geometry with algebraic properties to describe their symbolic Rees algebra in specific cases.

Contribution

It introduces a novel approach using birational theory to analyze symbolic powers and provides detailed descriptions of the symbolic Rees algebra for certain ideals.

Findings

Deep interlacing between generators and inverse map factors

Full description of symbolic Rees algebra in specific cases

Connection between birational maps and symbolic powers

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.