From formal smoothings to geometric smoothings

Let X be a projective, equidimensional, singular scheme over an al- gebraically closed field. Then the existence of a geometric smoothing (i.e. a fam- ily of deformations of X over a smooth base curve whose generic fibre is smooth) implies the existence of a formal smoothing as defined by Tziolas. In this paper we address the reverse question giving sufficient conditions on X that guaran- tee the converse, i.e. formal smoothability implies geometric smoothability. This is useful in light of Tziolas’ results giving sufficient criteria for the existence of formal smoothings.