On infinitesimal deformations of singular varieties I

Mounir Nisse

Paper

On infinitesimal deformations of singular varieties I

Abstract

The deformation theory of singular varieties plays a central role in understanding the geometry and moduli of algebraic varieties. For a variety

with possibly singular points, the space of first-order infinitesimal deformations is given by \( T^1_X = \operatorname{Ext}^1_{\mathcal{O}_X}(Ω_X, \mathcal{O}_X), \) which measures the Zariski tangent space to the deformation functor of

. When

, the variety is said to be \emph{rigid}; otherwise, nonzero elements of

correspond to nontrivial first-order deformations. We investigate the structure of

for singular varieties and provide cohomological and geometric criteria ensuring non-rigidity. In particular, we show that if the sheaf of tangent fields

possesses nonvanishing cohomology

or if the local contributions

are supported on a positive-dimensional singular locus, then

. For hypersurface singularities

, 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

characterizes the existence of nontrivial deformations. Moreover, non-rigidity arises when

% 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.