16

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?

Active problem!

Francesco Galuppi, Hanieh Keneshlou and Avinash Kulkarni and Miruna-Stefana Sorea are working on this problem. Experimental sage code available here.