2
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – ulrich Dec 14 '11 at 8:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.