The equations defining blowup algebras of height three Gorenstein ideals

TL;DR

This paper determines the defining equations of Rees rings for height three Gorenstein ideals with linear presentation, using local cohomology and algebraic geometry techniques to analyze their structure.

Contribution

It provides explicit descriptions of Rees algebra equations for a class of Gorenstein ideals, connecting algebraic and geometric properties.

Findings

The defining equations are given by the unmixed part of maximal minors of a special matrix.

The degree of the parametrized variety is explicitly calculated.

The symmetric algebra and Rees algebra share the same defining image under certain conditions.

Abstract

We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An important step of the proof is the calculation of the degree of the variety parametrized by the forms generating the grade three Gorenstein ideal.