# Infinitesimal deformations of semi-smooth varieties

*Barbara Fantechi (SISSA)*

**16-Oct-2020, 19:00-20:00 (12 months ago)**

**Abstract: **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) {\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.

algebraic geometry

Audience: researchers in the topic

( video )

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

**Stanford algebraic geometry seminar **

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