Hessian discriminant

Question

Anna Seigal [see 1] identifies the Hessian discriminant as a locus where the complex rank of $f$ jumps from $5$ to $6$. This is a hypersurface of degree $120$ in $\mathbb{P}^{19}$, invariant under $\textrm{PGL}(3)$. How to write the Hessian discriminant in terms of fundamental invariants?

Solution

In [2], it is shown that the Hessian discriminant equals $I_{40}^3$, where $I_{40}$ is Salmon's invariant of degree 40.

Bibliography

1. A. Seigal, Ranks and Symmetric Ranks of Cubic Surfaces (2018), [https://arxiv.org/abs/1801.05377].

2. R. Dinu and T. Seynnaeve, The Hessian discriminant (2019), [https://arxiv.org/abs/1909.06681].