Secants, Socles and Stability

TL;DR

The paper develops a stability-driven framework that connects the geometry of secants to Veronese embeddings with the algebra of Gorenstein quotients by viewing the projective space of symmetric tensors as a moduli space of socles. It introduces parity-dependent stability conditions in the derived category ${ m D}^b({ m P}^n)$, yielding Harder-Narasimhan filtrations that stratify ${ m P}^n_d$ by the rank and structure of socle maps and their associated Betti tables. The authors work out explicit low-degree, low-dimensional examples (notably in ${ m P}^2$) to illustrate how the strata align with palindromic Hilbert functions and Betti-table patterns, including the role of skew-symmetric maps and corank conditions. They provide a systematic approach (even/odd degree) to partition the space of symmetric tensors via stability data, with a view toward guiding secant-variety geometry and offering a lattice of possible Betti patterns; the framework integrates Beilinson-type hearts, Eagon–Northcott complexes, and Hilbert-scheme stratifications to achieve this categorification of secant phenomena.

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.