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Smoothability of non normal stable Gorenstein Godeaux surfaces

Barbara Fantechi
Wednesday, June 22, 2022 - 16:30

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 (i.e., not normal) which are Gorenstein (i.e., not too singular).  The simplest possible singularities are semi-smooth, ie a double curve with finitely many pinch points. We prove that stable Godeaux semi-smooth surfaces correspond to a point in the moduli that 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 other cases, and to more general moduli as well.

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