Secants, Socles and Stability

TL;DR

This paper explores the stability conditions of symmetric tensors viewed as Gorenstein rings, using Harder-Narasimhan filtrations to classify their structure, with explicit examples in low-dimensional cases.

Contribution

It introduces explicit stability conditions for symmetric tensors as Gorenstein rings and connects these to Harder-Narasimhan filtrations, providing detailed classifications in specific cases.

Findings

Partition of tensor space by stability types

Explicit stability conditions for even and odd degrees

Comparison with Betti tables of Gorenstein rings

Abstract

The projective space of symmetric tensors of degree d can be reinterpreted as a projective space of finite, graded Gorenstein rings with socle in degree d. Via a pair of explicit stability conditions (one for even values of d and one for odd values), the space of symmetric tensors is partitioned by Harder-Narasimhan filtration type. This is worked out explicitly for low degree examples in dimension three (the projective plane) and compared with the betti tables of the Gorenstein rings.