Non-commutative deformations of perverse coherent sheaves and rational curves

We consider non-commutative deformations of sheaves on algebraic varieties. We develop some tools to determine parameter algebras of versal non-commutative deformations for partial simple collections and the structure sheaves of smooth rational curves. We apply them to universal flopping contractions of length 2 and higher. We confirm Donovan-Wemyss conjecture in the case of deformations of Laufer’s flops.