Projects per year
Abstract
We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3fold 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 3folds,
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 finitedimensional, 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 3fold with only Gorenstein terminal singularities, we construct an autoequivalence of the derived category of the 3fold by twisting around a universal family over the noncommutative deformation algebra Acon, and prove that this autoequivalence is an inverse of Bridgeland’s flopflop functor. This demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 3fold and, thus, the Bridgeland stability manifold.Original language  English 

Pages (fromto)  13971474 
Journal  Duke Mathematical Journal 
Volume  165 
Issue number  8 
Early online date  23 Mar 2016 
DOIs  
Publication status  Published  1 Jun 2016 
Fingerprint
Dive into the research topics of 'Noncommutative Deformations and Flops'. Together they form a unique fingerprint.Projects
 2 Finished


RIGID STRUCTURE IN NONCOMMUTATIVE, GEOMETRIC & COMBINATORIAL PROBLEMS
1/09/08 → 30/06/14
Project: Research