# Cuspidal stable curves

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?

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Any dominant morphism $f:X \to Y$ with $Y$ a smooth curve and $X$ integral is flat. This follows from the fact that a module over a dvr is flat iff it is torsion free. –  ulrich Dec 14 '11 at 8:09