This is joint work with Marco Franciosi and Rita Pardini.

Godeaux surfaces, with $K^2=1$ and $p_g=q=0$, are the (complex projective) surfaces of general type with the smallest possible invariants. A complete classification, i.e. an understanding of their moduli space, has been an open problem for many decades.

The KSBA (after Kollár, Sheperd-Barron and Alexeev) compactification of the moduli includes so called stable surfaces. Franciosi, Pardini and Rollenske classified all such surfaces in the boundary which are Gorenstein (i.e., not too singular).

We prove that most of these surfaces corresponds to a point in the moduli which is nonsingular of the expected dimension 8. We expect that the methods used (which include classical and recent infinitesimal deformation theory, as well as algebraic stacks and the cotangent complex) can be applied to all cases, and to more general moduli as well.

The talk is aimed at a non specialist mathematical audience, and will focus on the less technical aspects of the paper.