Derived categories of Quot schemes of zero-dimensional quotients on curves

TL;DR

This paper establishes semiorthogonal decompositions for derived categories of Quot schemes on curves, linking them to symmetric products and advancing categorical understanding of these geometric objects.

Contribution

It provides a categorical analogue of a known class formula for Quot schemes, extending to a broader relative dimension case and connecting to wall-crossing phenomena.

Findings

Semiorthogonal decompositions for Quot schemes on curves.

Relation to derived categories of symmetric products.

Applications to categorical wall-crossing and Donaldson-Thomas invariants.

Abstract

We prove the existence of semiorthogonal decompositions of derived categories of Quot schemes of zero-dimensional quotients on curves in terms of derived categories of symmetric products of curves. The above result is a categorical analogue of a similar formula for the class of Quot schemes in the Grothendieck ring of varieties by Bagnarol-Fantechi-Perroni. It is a special case of a more general Quot formula of relative dimension one, which is regarded as a Bosonic counterpart of the Quot formula conjectured by Jiang and proved by the author. The proof involves categorical wall-crossing formula for framed one loop quiver, which itself is motivated and has applications to categorical wall-crossing formula of Donaldson-Thomas invariants.