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1 June 2016 Noncommutative deformations and flops
Will Donovan, Michael Wemyss
Duke Math. J. 165(8): 1397-1474 (1 June 2016). DOI: 10.1215/00127094-3449887


We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence, we associate to every such curve a noncommutative deformation algebra Acon. This new invariant extends and unifies known invariants for flopping curves in 3-folds, such as the width of Reid and the bidegree of the normal bundle. It also applies in the settings of flips and singular schemes. We show that the noncommutative deformation algebra Acon is finite-dimensional, and give a new way of obtaining the commutative deformations of the curve, allowing us to make explicit calculations of these deformations for certain (3,1)-curves.

We then show how our new invariant Acon also controls the homological algebra of flops. For any flopping curve in a projective 3-fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the 3-fold by twisting around a universal family over the noncommutative deformation algebra Acon, and prove that this autoequivalence is an inverse of Bridgeland’s flop-flop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 3-fold and, thus, the Bridgeland stability manifold.


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Will Donovan. Michael Wemyss. "Noncommutative deformations and flops." Duke Math. J. 165 (8) 1397 - 1474, 1 June 2016.


Received: 28 May 2014; Revised: 18 June 2015; Published: 1 June 2016
First available in Project Euclid: 23 March 2016

zbMATH: 1346.14031
MathSciNet: MR3504176
Digital Object Identifier: 10.1215/00127094-3449887

Primary: 14D15
Secondary: 14E30 , 14F05 , 16S38 , 18E30

Keywords: autoequivalence , birational geometry , complex $3$-fold , deformation theory , derived category , homological algebra , noncommutative deformation

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 8 • 1 June 2016
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