Elimination and nonlinear equations of Rees algebra

TL;DR

This paper introduces a novel method for computing the Rees algebra of a base ideal using approximation complexes and torsion analysis, revealing new insights into the nonlinear equations defining the algebra.

Contribution

It provides a detailed torsion analysis of the symmetric algebra, leading to free resolutions of the Rees algebra in certain degrees and connecting torsion to non Koszul syzygies.

Findings

Torsion contributes to the first free module in the resolution.

Half of the degrees with non-vanishing torsion are understood in general cases.

The approach links torsion to nonlinear equations via upgrading maps.

Abstract

A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, half of the degrees where the torsion does not vanish are understood.