Faithful actions from hyperplane arrangements

Abstract

We show that if $X$ is a smooth quasiprojective $3$–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 (or mutation functor). This allows us to relate compositions of the flop functor (or 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 a result of Brav and Thomas (Math. Ann. 351 (2011) 1005–1017) for Kleinian singularities.