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We completely characterize the covers of connected orbifold curves whichpreserve slope stability of vector bundles under the pullback morphism. Moreprecisely, given a cover $f \colon (Y,Q) \longrightarrow (X,P)$ of connectedorbifold curves, we show that the maximal destabilizing sub-bundle of thepushforward sheaf $f_*\mathcal{O}_{(Y,Q)}$ defines the maximal \'{e}talesub-cover of $f$. The cover $f$ is said to be genuinely ramified if $f$ doesnot factor through any non-trivial \'{e}tale sub-cover. Our main result statesthat the class of covers $f$ that preserves the stable bundles under a pullbackare precisely the class of genuinely ramified covers $f$. Further, we establishequivalent conditions for the cover $f$ to be genuinely ramified, generalizingearlier works on covers of curves. We thoroughly study the slope stabilityconditions of bundles on an orbifold curve, their properties under thepushforward and pullback maps under covers with a stand point ofDeligne-Mumford stacks, hence giving a solid foundation of the subject. As aconsequence, we also answer the question of descent of stable bundles undergenuinely ramified covers.

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