From formal smoothings to geometric smoothings

TL;DR

This paper investigates conditions under which formal smoothability of a singular algebraic scheme guarantees its geometric smoothability, bridging a gap in deformation theory with implications for algebraic geometry.

Contribution

It provides sufficient conditions on a scheme that ensure formal smoothability implies geometric smoothability, extending Tziolas' criteria.

Findings

Formal smoothability implies geometric smoothability under certain conditions.

Provides criteria linking formal and geometric smoothings.

Enhances understanding of deformation theory for singular schemes.

Abstract

Let X be a projective, equidimensional, singular scheme over an algebraically closed field. Then the existence of a geometric smoothing (i.e. a family 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 guarantee 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.