Deformations of rational double points and simple elliptic singularities in characteristic \(p\) (Q1763042)

The main result of this paper is: Consider a fibration \(f: X\to C\) from a nonsingular threefold to a curve. We suppose that a general fiber of \(f\) is a normal surface. Then the following hold: There does not appear a simple elliptic singularity on a general fiber if \(p\geq 5\). Under the assumption that the anticanonical divisor of a fiber is ample, a general fiber is non singular if \(p\geq 11\), i.e. it is a del Pezzo surface. Under the assumption that the general fiber has a trivial dualizing sheaf and has only rational singularities, it is non singular if \(p\geq 23\), i.e; it is either an abelian surface or a \(K3\) surface. The main observation is that, if the general fiber has singularities isomorphic to hypersurface singularities, the dimension of the space \(T^1\) of first order infinitesimal deformations of the singularity is divisible by \(p\). The proof consist to check this last property for rational double points and simple elliptic singularities.