On some examples of spherical functors related to flops

Abstract

In [Bar20b] we proved that the "flop-flop" autoequivalence can be realized as the spherical twist around a spherical functor whose source category arises naturally from the geometry.In this companion paper we study in detail some examples so to explicitly describe what the source category looks like. In some cases we are able to prove that the source category respects the known decomposition of the flop-flop autoequivalence, and therefore we tie up our geometric description with formal results which appear in the literature about gluing and splitting of spherical twists around spherical functors.The examples we treat completely are standard flops (both in the local model and in the family version), and Mukai flops. We also discuss the cases of Grassmannian flops, and the Abuaf flop. Contents 14 2.3. Mukai flops 19 2.4. Other examples 31 Appendix A. Computations 34 A.1. Standard flops (local model) 34 A.2. Standard flops (family case) 37 A.3. Mukai flops 48 References 64