Stability of pullback of orbifold bundles

TL;DR

This paper investigates how the stability of vector bundles on orbifold curves behaves under pullback via finite morphisms, establishing conditions for stability preservation and categorical equivalences in the orbifold setting.

Contribution

It introduces a notion of slope stability for vector bundles on formal orbifold curves and proves stability preservation under pullback via genuinely ramified morphisms.

Findings

Stability of vector bundles is preserved under pullback in orbifold settings.

Categorical equivalence between formal orbifold curves and Deligne-Mumford orbifold curves.

Conditions for genuinely ramified morphisms analogous to classical curve cases.

Abstract

In this article, we study the behavior of the stability of pullback of a vector bundle under a finite morphism from a (not necessarily smooth) stacky curve to an orbifold curve. We establish a categorical equivalence between proper formal orbifold curves and proper orbifold curves in the sense of Deligne-Mumford stacks. Using this identification, we define the notion of slope $P$-(semi)stability of vector bundles on proper formal orbifold curves $(X,P)$. We establish some equivalent conditions for a stacky genuinely ramified morphism, analogous to the case of curves. Finally, we show that for a cover of an orbifold curve arising as a cartesian pullback via a genuinely ramified morphism of smooth projective connected curves, the orbifold slope stability is preserved under the pullback.