Deformations of Semi-Smooth Varieties

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-X(1) := epsilon(l)(xt) (Omega(X), O-X). A variety is semi-smooth if its singularities are etale 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(X)1 for a semi-smooth variety X in terms of the gluing data.