Question 4: resultant of a cubic


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 $\mathbb{P}^3$.


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).

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
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