# Lifting singularities to zero characteristic

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.

• This is already false when $X_0$ is smooth... – abx Nov 8 '18 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 '18 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 '18 at 13:03