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I recently starded studying the book "Orbifolds and Stringy Topology" by Adem, Leida and Ruan and I'm trying to see if there is a relation between the singularites of two orbifolds when there is a smooth map from one to another. More precisely:

Let $X$ and $Y$ be smooth orbifolds and $f:X \rightarrow Y$ a smooth map. What would be a sufficient condition under which the image of a singular point is also a singular point?

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If your orbifolds are complex varieties, then $f$ being a (Euclidean-) local isomorphism is a sufficient condition. Being a local isomorphism implies that the differentials of $f$ are isomorphisms of tangent spaces. A singular point in $X$ will have a tangent space of non-generic dimension, meaning that the same will be true of the tangent space to the image point in $Y$. So, the image point will also be singular.

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As always with orbifolds this might depend on your defintions and what smooth maps of orbifolds are for you. Some answers (for a certain definition of orbifold morphism which is modelled to represent the Lie groupoid morphisms in a local charts setting) are contained in https://arxiv.org/pdf/1301.5551.pdf

For example orbisections (=vector fields on orbifolds, Section 3.2) and orbifold diffeomorphisms (Section 2.1) preserve the type of singularities (the last one is of course a no brainer if diffeomorphism should mean anything). Further orbifold geodesics either enter a singular stratum at one point (and then immediately leave again) or remain in it for all time (see Appendix F, it is somewhat hidden in the proofs there)

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