Question 4: resultant of a cubic

Question

The discriminant of a quaternary cubic form is the resultant of its four partial derivatives. Can we express the resultant of four quaternary quadrics as the determinant of an 16×16 matrix whose entries are linear forms? This is the Chow form of the Veronese threefold in $\mathbb{P}^9$. Such a formula is derived from the unique rank 2 Ulrich bundle on $\mathbb{P}^3$.

Comments

In [1], Eisenbud and Schreyer give explicit determinantal and Pfaffian formulas for resultants. The most relevant for the question is for the resultant of 3 quadratic forms in three variables. This corresponds to the Chow form of the Veronese surface in $\mathbb{P}^5$. What would this look like for the Chow form of the Veronese embedding of $\mathbb{P}^3$ in $\mathbb{P}(H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(2))) = \mathbb{P}^9$?

Active Problem

This problem is currently being worked on by Dominic Bunnett and Hanieh Keneshlou (March 2019).

Bibliography
1. Eisenbud, D. and Schreyer, F.-O. and Weyman, J. "Resultants and Chow forms via exterior syzygies" Journal of the American Mathematical Society, Vol 16 (2003), 537—579
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License