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

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