16

# Problem

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?

# Results

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.

# Code

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