Root stacks and periodic decompositions

TL;DR

This paper investigates the periodicity of semiorthogonal decompositions in derived categories of root stacks associated with divisors, revealing new spherical functors and employing GIT variations for their analysis.

Contribution

It demonstrates that the semiorthogonal decomposition of root stacks is 2n-periodic and constructs higher spherical functors for n>2, extending known results.

Findings

Decomposition is 2n-periodic for root stacks.

Constructs higher spherical functors for n>2.

Uses GIT variation for root stack analysis.

Abstract

For an effective Cartier divisor D on a scheme X we may form an nth root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is 2n-periodic. For n=2 this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For n>2 we find a higher spherical functor in the sense of recent work of Dyckerhoff, Kapranov and Schechtman. We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.