Relating derived equivalences for simplices of higher-dimensional flops

TL;DR

This paper explores the relationships between derived equivalences of crepant resolutions of certain higher-dimensional singularities, using simplices to model flop transformations and their associated functor relations.

Contribution

It introduces a geometric framework linking flop functors to simplices, revealing new relations among derived equivalences in higher-dimensional algebraic geometry.

Findings

Faces of the simplex correspond to specific relations between flop functors.

Each dimension n has resolutions associated with vertices of an (n-2)-simplex.

Flops correspond to edges of the simplex, encoding functor relations.

Abstract

I study a sequence of singularities in dimension 4 and above, each given by a cone of rank 1 tensors of a certain signature, which have crepant resolutions whose exceptional loci are isomorphic to cartesian powers of the projective line. In each dimension n, these resolutions naturally correspond to vertices of an (n-2)-simplex, and flops between them correspond to edges of the simplex. I show that each face of the simplex may then be associated to a certain relation between flop functors.