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].