I have seen this many times but never know a rigorous proof. In a flat family of stable curves, if an elliptic tail is contracted then we get a cuspidal curve, if an elliptic bridge is contracted, we get a tacnodal curve. I understand the reason behind this ( arithmetic genus & Euler characteristic ). How do you know that if we contracted a subcurve in the central fiber (assume the other fibers are smooth), then the resulting family is still flat ? What condition guarantees this?
