The depth of the Rees algebra of three general binary forms

TL;DR

This paper proves that the Rees algebra of three general binary forms of degree at least five has depth one, using Ratliff-Rush filtration and matrix rank properties, and proposes a conjecture for a shorter proof.

Contribution

It establishes the depth property of the Rees algebra for three general binary forms and introduces a conjecture that could simplify the proof.

Findings

Rees algebra of three general binary forms has depth one for degree ≥ 5

Behavior of Ratliff-Rush filtration is crucial in the proof

Certain large matrices of quadratic forms have maximal rank

Abstract

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that certain large matrices whose entries are quadratic forms have maximal rank. One also conjectures a shorter result that implies the main theorem of the paper.