MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'd like to know a notion for a morphism between algebraic varieties in characteristic $p$ that plays the role of a $C^{\infty}$-fiber bundle. It should be, in particular, flat. I'm not assuming the morphism to be smooth; e.g. a family of nodal curves $X_t$ with $p_a=2$ and such that the $j$-invariant of the normalizations $\widetilde{X}_t$ is not constant in $t,$ would certainly qualify.

Motivation: If $f:X\to Y$ is a proper morphism of complex algebraic varieties, then by Morse theory, there exists a "stratification of $f";$ in particular, over each stratum of $Y,\ f$ is a $C^{\infty}$-fiber bundle. I wonder if this could be true in char. $p,$ but the first thing is to have an analogous notion.

share|cite|improve this question
Hey Shenghao, maybe check out this MO question...… – Kevin H. Lin Aug 9 '11 at 9:25
Thanks, Kevin. I had a question about the suggested definition, and have just posted it there. – shenghao Aug 9 '11 at 12:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.