Deformations of Semi-Smooth Varieties

Abstract

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 $\mathcal {T}^{1}_{X}:={{\mathcal {E}}xt}^{1}(\Omega _{X},\mathcal {O}_{X})$. A variety is semi-smooth if its singularities are étale locally the product of a double crossing point ($uv=0$) or a pinch point ($u^2-v^2w=0$) 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 $\mathcal {T}^{1}_{X}$ for a semi-smooth variety $X$ in terms of the gluing data.