Configurations of eigenpoints

TL;DR

This paper investigates the configurations of eigenpoints of cubic surfaces in projective 3-space, establishing conditions under which general eigenschemes correspond to complete intersections on determinantal surfaces and can be extended to eigenschemes of partially symmetric tensors.

Contribution

It proves that a general eigenscheme in projective n-space is a complete intersection on a determinantal surface and characterizes eigenpoint configurations in projective 3-space, answering a specific open question.

Findings

General eigenschemes are complete intersections of determinantal curves.

The converse holds for n=3, linking eigenschemes to eigenpoints of cubic surfaces.

Any general set of points in P^3 can be extended to an eigenscheme of a partially symmetric tensor.

Abstract

This note is motivated by the Question 16 of http://cubics.wikidot.com: Which configurations of 15 points in the projective 3-space arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in the projective n-space is the complete intersection of two suitable smooth determinantal curves on a smooth determinantal surface. Moreover, we prove that the converse result holds if n=3, providing an answer in any degree to the cited question. Finally, we show that any general set of points in the projective 3-space can be enlarged to an eigenscheme of a partially symmetric tensor.