Smoothings of schemes with nonisolated singularities

Abstract

In this paper we study the deformation and Q-Gorenstein deformation theory of schemes with non-isolated singularities. We obtain obstruction spaces for the existence of deformations and also for local deformations to exist globally. Finally we obtain explicit criteria in order for a pure and reduced scheme of finite type over a field k to have smoothings and Q-Gorenstein smoothings.

Smoothings of Fano Varieties With Normal Crossing Singularities

“…Proof . The proof of the second part is given in [15,Theorem 12.5]. We proceed to show the first part.…”

“…The author does not know if the condition that T 1 X is finitely generated by its global sections is a necessary condition, too, for X to be smoothable. X is certainly not smoothable if H 0 (T 1 X ) = 0 [15]. In all the cases of the previous theorem, the condition finitely generated by global sections implies that Def(X) is smooth.…”

Deformations of semi-smooth varieties

“…In a more recent paper [Tzi10], Tziolas proves that if we assume that X has lci singularities then a formal smoothing exists, provided that T 1 X is generated by global sections and that H 1 (X, T 1 X ) = H 2 (X, T X ) = 0 (in this case the deformations are also unobstructed).…”

Smoothing semi-smooth Stable Godeaux surfaces

“…We verify for such surfaces the assumptions of Tziolas's formal smoothability criterion (Theorem 12.5 in [Tzi10]).…”

“…The key theorem we use is the following result of Tziolas guaranteeing the existence of a formal smoothing (Definition 11.6 of [Tzi10]).…”