On the Covers of Orbifold Curves Preserving the Slope Stability under Pullback

TL;DR

This paper characterizes orbifold curve covers that preserve slope stability of vector bundles under pullback, identifying genuinely ramified covers as the key class and providing conditions for their stability-preserving properties.

Contribution

It provides a complete characterization of covers preserving slope stability on orbifold curves, extending previous results and establishing new conditions for genuinely ramified covers.

Findings

Genuinely ramified covers are exactly those that preserve slope stability.

Equivalent conditions for a cover to be genuinely ramified are established.

The study is grounded in the theory of Deligne-Mumford stacks and orbifold curves.

Abstract

We completely characterize the covers of connected orbifold curves which preserve slope stability of vector bundles under the pullback morphism. More precisely, given a cover $f \colon (Y,Q) \longrightarrow (X,P)$ of connected orbifold curves, we show that the maximal destabilizing sub-bundle of the pushforward sheaf $f_*\mathcal{O}_{(Y,Q)}$ defines the maximal \'{e}tale sub-cover of $f$. The cover $f$ is said to be genuinely ramified if $f$ does not factor through any non-trivial \'{e}tale sub-cover. Our main result states that the class of covers $f$ that preserves the stable bundles under a pullback are precisely the class of genuinely ramified covers $f$. Further, we establish equivalent conditions for the cover $f$ to be genuinely ramified, generalizing earlier works on covers of curves. We thoroughly study the slope stability conditions of bundles on an orbifold curve, their properties under the pushforward and pullback maps under covers with a stand point of Deligne-Mumford stacks, hence giving a solid foundation of the subject. As a consequence, we also answer the question of descent of stable bundles under genuinely ramified covers.