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?



Orbifold singularities over a smooth map
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.


