Faithful actions from hyperplane arrangements

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.

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Keywords

derived category, flopping contraction, Deligne groupoid

Mathematical Subject Classification 2010

Primary: 18E30

Secondary: 14E30, 14F05, 14J30, 20F36

References

Publication

Received: 13 January 2017

Revised: 14 November 2017

Accepted: 31 December 2017

Published: 23 September 2018

Proposed: Richard Thomas

Seconded: Jim Bryan, Dan Abramovich

Authors

Yuki Hirano
Department of Mathematics Kyoto University Kyoto Japan

Michael Wemyss
School of Mathematics and Statistics University of Glasgow Glasgow United Kingdom

Volume 22, issue 6 (2018)

Yuki Hirano and Michael Wemyss

Abstract