On some examples of spherical functors related to flops

TL;DR

This paper explores specific examples of spherical functors related to flops, providing explicit descriptions of their source categories and connecting geometric autoequivalences with formal spherical twist decompositions.

Contribution

It offers detailed analysis of source categories for spherical functors in standard and Mukai flops, linking geometric autoequivalences with spherical twist decompositions.

Findings

Source categories respect known flop-flop autoequivalence decompositions

Explicit descriptions for standard and Mukai flops

Connections between geometric autoequivalences and spherical twists

Abstract

In arXiv:2007.14415 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.