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Faithful actions from hyperplane arrangements

Yuki Hirano and Michael Wemyss

Geometry & Topology 22 (2018) 3395–3433

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.

derived category, flopping contraction, Deligne groupoid
Mathematical Subject Classification 2010
Primary: 18E30
Secondary: 14E30, 14F05, 14J30, 20F36
Received: 13 January 2017
Revised: 14 November 2017
Accepted: 31 December 2017
Published: 23 September 2018
Proposed: Richard Thomas
Seconded: Jim Bryan, Dan Abramovich
Yuki Hirano
Department of Mathematics
Kyoto University
Michael Wemyss
School of Mathematics and Statistics
University of Glasgow
United Kingdom