Faithful Actions from Hyperplane Arrangements

TL;DR

This paper demonstrates that the fundamental group of a hyperplane arrangement associated with a 3-fold flop acts faithfully on the derived category, using torsion pairs to analyze the flop functor's iterations.

Contribution

It introduces a novel use of torsion pairs to track objects under flop functors and relates compositions of flops to Deligne normal form, simplifying previous proofs.

Findings

Fundamental group acts faithfully on derived categories of 3-folds.

Torsion pairs effectively track objects under flop functors.

Provides a simplified proof for Kleinian singularities case.

Abstract

We show that if $X$ is a smooth quasi-projective $3$-fold admitting a flopping contraction, then the fundamental group of an associated simplicial hyperplane arrangement acts faithfully on the derived category of $X$. The main technical advance is to use torsion pairs as an efficient mechanism to track various objects under iterations of the flop functor (respectively, mutation functor). This allows us to relate compositions of the flop functor (respectively, mutation functor) to the theory of Deligne normal form, and to give a criterion for when a finite composition of $3$-fold flops can be understood as a tilt at a single torsion pair. We also use this technique to give a simplified proof of the result of Brav-Thomas for Kleinian singularities.