Suppose we have a proper (or projective) singular variety $X_0$ over a field of characteristic $p$, $k$, and with dual graph $\Gamma$. We may suppose that the components of $X_0$ are smooth and reduced, irreducible (if you wish a simple normal crossing variety).

I wonder about the lifting problem:

There exist a flat proper scheme $\mathcal X$ over the spectrum of some local domain of mixed characteristic $(o,p)$ such that the special fibre is isomorphic to $X_0$ and whose generic fibre is singular with the same dual graph of $X$?

The question is: What is known about the obstruction to have such a lift.

  • 2
    This is already false when $X_0$ is smooth... – abx Nov 8 at 20:33
  • You are right, but if the deformations are unobstructed you can lift it. So I am not asking if this is true always, I am asking about the problem, that is, what is known about it, and under which conditions we can lift it. What are the obstuctions etc. – Rogelio Yoyontzin Nov 8 at 22:04
  • For the existence of an "equisingular" lift of a normal crossing variety, lifting should be equivalent to lifting all the components separately together with their pairwise intersections as closed subschemes of the components. So (with a proper formulation) usual deformation theory for smooth schemes and their smooth subschemes should be enough. – Piotr Achinger Nov 9 at 13:03

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.