Torsion Trajectories from Local Discriminants to Global Obstructions

TL;DR

This paper investigates how local torsion invariants of surface singularities influence global obstruction theories, revealing that finite discriminant torsion is a codimension-two phenomenon with implications for Hodge theory.

Contribution

It tracks the trajectory of torsion from local singularity data to global obstructions, providing example-driven computations and revealing a codimension-two nature of finite discriminant torsion.

Findings

Surface A1 singularities have local Z/2-torsion.

Threefold ordinary double points have torsion-free links.

Finite discriminant torsion is a codimension-two phenomenon.

Abstract

For a normal surface singularity, the discrepancy between the ordinary and dual middle-perversity intersection complexes over \(\mathbb Z\) is measured by a finite group \(E\). In previous work, \(E\) was identified with link torsion, the exceptional-lattice discriminant group \(\Lambda^\vee/\Lambda\), a resolution-neighborhood boundary quotient, and, in the hypersurface case, \(\operatorname{coker}(T-\mathrm{id})_{\mathrm{tors}}\). This paper tracks the trajectory of this torsion from local singularity data to global obstruction theory. We follow the discriminant package \((E,q)\) through support cohomology, excision, global torsion, Brauer comparison, Bloch--Ogus residues, and rationalization. The method is example-driven: trajectory tables are computed for \(A_1\), \(A_k\), \(D_4\), \(E_8\), a non-ADE Brieskorn singularity, the threefold ordinary double point, nodal threefolds, nodal quintics, and the Benoist--Ottem benchmark. The computations reveal a sharp distinction: a surface \(A_1\) singularity has local \(\mathbb Z/2\)-torsion, whereas a threefold ordinary double point has torsion-free link \(S^2\times S^3\) and contributes free vanishing-cycle data instead. Thus finite discriminant torsion is naturally a codimension-two phenomenon, not a generic feature of nodes. The resulting pattern motivates a specialization problem: whether the Enriques \(2\)-torsion in Benoist--Ottem integral Hodge counterexamples is genuinely global, or can arise after degeneration from transverse \(A_1\)-type discriminant data along codimension-two strata.