Moduli stacks of Serre stable representations in tilting theory

TL;DR

This paper introduces the Serre stable moduli stack, a new geometric object that unifies aspects of representation theory and algebraic geometry by connecting derived equivalences with moduli of sheaves.

Contribution

It defines the Serre stable moduli stack and demonstrates its role in reinterpreting classical derived equivalences via the universal sheaf, linking algebraic and geometric structures.

Findings

Reinterpreted derived equivalences using the Serre stable moduli stack

Connected weighted projective lines with moduli of sheaves

Provided a new perspective on moduli problems in algebraic geometry

Abstract

We introduce a new moduli stack, called the Serre stable moduli stack, which corresponds to studying families of point objects in an abelian category with a Serre functor. This allows us in particular, to re-interpret the classical derived equivalence between most concealed-canonical algebras and weighted projective lines by showing they are induced by the universal sheaf on the Serre stable moduli stack. We explain why the method works by showing that the Serre stable moduli stack is the tautological moduli problem that allows one to recover certain nice stacks such as weighted projective lines from their moduli of sheaves. As a result, this new stack should be of interest in both representation theory and algebraic geometry.