Barbara Fantechi (Oct. 16, 2020): 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 algebraically closed field of characteristic not 2) {\em 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 S\"onke) corresponds to a smooth point of the KSBA moduli space, in the closure of the open locus of smooth surfaces.