The eigenpoints of $f$ are the fixed points of the gradient map $\nabla f \colon \mathbb{P}^3 \dashrightarrow \mathbb{P}^3$. For generic cubics $f$, there are $15$ eigenpoints. They form the eigenconfiguration of the $4\times 4 \times 4$ tensor $f$. Which configurations of $15$ points in $\mathbb{P}^3$ arise as eigenpoints of a cubic surface?


Please see the following manuscript which is based on the problem; by Turku Ozlum Celik, Francesco Galuppi, Avinash Kulkarni and Miruna-Stefana Sorea. The abstract is as follows.

We show that the eigenschemes of $4 × 4 × 4$ symmetric tensors are parametrized by a linear subvariety of the Grassmannian $Gr(3,\mathbb{P}^{14})$. 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.


Macaulay2 and Magma scripts accompanying the manuscript are available here.
Experimental sage code available here.

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