# Infinitesimal deformations of semi-smooth varieties

*Barbara Fantechi (SISSA)*

**16-Oct-2020, 19:00-20:00 (19 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 **

**Series comments: **This seminar requires both advance registration, and a password.
Register at stanford.zoom.us/meeting/register/tJEvcOuprz8vHtbL2_TTgZzr-_UhGvnr1EGv
Password: 362880

If you have registered once, you are always registered for the seminar, and can join any future talk using the link you receive by email. If you lose the link, feel free to reregister. This might work too: stanford.zoom.us/j/95272114542

More seminar information (including slides and videos, when available): agstanford.com

Organizer: | Ravi Vakil* |

*contact for this listing |