Unstable blueprints can be shared

TL;DR

This paper explores how certain face rings of 2-spheres lacking the Lefschetz property can be decomposed along an equator, revealing symmetries in non-rigid sphere triangulations using algebraic and geometric methods.

Contribution

It introduces a novel approach combining toric perturbation and biased pairing theory to analyze face rings of 2-spheres and their decompositions.

Findings

Face rings of 2-spheres without Lefschetz property can be cut along a flat equator.

Reveals symmetry in non-rigid sphere triangulations.

Connects algebraic properties with geometric decompositions.

Abstract

This expository note illustrates toric perturbation and biased pairing theory to show that Artinian reductions of face rings of $2$-spheres that do not satisfy the Lefschetz property can be cut along a flat equator. This complements classical work of Bricard and Connelly, and exhibits a fundamental symmetry in non-rigid triangulations of spheres.