The depth of the Rees algebra of three general binary forms

Abstract

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\ge 5$ 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.

Keywords:

• Almost Cohen–Macaulay
• Hilbert function
• Huckaba–Marley test
• Ratliff–Rush filtration
• Rees algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A02
• 13A30
• 13C15
• 13D02
• 13D40

Acknowledgements

The authors thank Stefan Tohǎneanu for early discussions in many passages and basic calculations, especially in the first two sections.

Additional information

Funding

This work was supported by CNPq [grant number 302298/2014-2].