Deformations of semi-smooth varieties

Abstract

For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T 1 X := Ext 1 (ΩX , OX ). A variety is semi-smooth if its singularities are étale locally the product of a double crossing point (uv = 0) or a pinch point (u 2 − v 2 w = 0) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf T 1 X for a semi-smooth variety X in terms of the gluing data.