On the eigenpoints of cubic surfaces

Abstract

We show that the eigenschemes of $4 \times 4 \times 4$ symmetric tensors are parametrized by a linear subvariety of the Grassmannian $\operatorname{Gr}(3,\mathbb{P}^{14})$. We also study the decomposition of the eigenscheme into the subscheme associated to the zero eigenvalue and its residue. In particular, we categorize the possible degrees and dimensions.