On the eigenpoints of cubic surfaces

TL;DR

This paper investigates the structure of eigenschemes of symmetric tensors, revealing their parametrization within a Grassmannian and analyzing their decomposition into zero eigenvalue components, with detailed categorization of degrees and dimensions.

Contribution

It introduces a parametrization of eigenschemes of symmetric tensors via a linear subvariety of a Grassmannian and studies their decomposition and classification.

Findings

Eigenschemes are parametrized by a linear subvariety of Grassmannian Gr(3, P^{14})

Decomposition of eigenschemes into zero eigenvalue subschemes and residues analyzed

Degrees and dimensions of eigenscheme components categorized

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.