# Infinitesimal deformations of semi-smooth varieties

This is a report on joint work with Marco Franciosi and Rita Pardini. Generalizing the standard definition for surfaces, we call a variety X (over an alg closed field of char not 2) *semi-smooth* if its singularities are \'etale locally either uv=0 or u^2=v^2w (pinch point); equivalently, if X can be obtained by gluing a smooth variety (the normalization of X) along an involution (with smooth quotient) on a smooth divisor. They are the simplest singularities for non normal, KSBA-stable surfaces.

For a semi-smooth variety X, we calculate the tangent sheaf T_X and the infinitesimal deformations sheaf ${\mathcal T}^1_X:={\mathcal E}xt^1(\Omega_X,\mathcal O_X) which determine the infinitesimal deformations and smoothability of X.

As an application, we use Tziolas' formal smoothability criterion to show that every stable semi-smooth Godeaux surface (classified by Franciosi, Pardini and Sonke) corresponds to a smooth point of the KSBA moduli space, in the closure of the open locus of smooth surfaces.

The discussion for Barbara Fantechi’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2020-10-16-bf (and will be deleted after ~3-7 days).