On infinitesimal deformations of singular varieties I

The deformation theory of singular varieties plays a central role in understanding the geometry and moduli of algebraic varieties. For a variety $X$ with possibly singular points, the space of first-order infinitesimal deformations is given by $T^1_X = \operatorname{Ext}^1_{\mathcal{O}_X}(\Omega_X, \mathcal{O}_X),$ which measures the Zariski tangent space to the deformation functor of $X$. When $T^1_X = 0$, the variety is said to be \emph{rigid}; otherwise, nonzero elements of $T^1_X$ correspond to nontrivial first-order deformations. We investigate the structure of $T^1_X$ for singular varieties and provide cohomological and geometric criteria ensuring non-rigidity. In particular, we show that if the sheaf of tangent fields $T_X$ possesses nonvanishing cohomology $H^1(X, T_X)$ or if the local contributions $\mathcal{E}xt^1(\Omega_X, \mathcal{O}_X)$ are supported on a positive-dimensional singular locus, then $T^1_X \neq 0$. For hypersurface singularities $X = \{ f = 0 \} \subset \mathbb{C}^{n+1}$, we recover the Jacobian criterion, $$T^1_X \cong \frac{\mathbb{C}[x_0, \dots, x_n]}{(f, \partial f / \partial x_0, \dots, \partial f / \partial x_n)},$$ where the positivity of the Tjurina number $\tau(X)$ characterizes the existence of nontrivial deformations. Moreover, non-rigidity arises when $X$ % appears as a cone over a projectively nonrigid variety. These criteria provide effective tools for detecting non-rigidity in both local and global settings, linking the vanishing of Ext and cohomology groups to the deformation behavior of singularities. The results contribute to a deeper understanding of the interplay between singularity theory, moduli, and the rigidity properties of algebraic varieties.