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\(\mathbb Q\)-Gorenstein deformations of nonnormal surfaces. (English) Zbl 1162.14005

The paper under review describes the \(\mathbb Q\)-Gorenstein smoothings of a germ \(\Delta\subset H\) of a nonnormal surface with semi-log-canonical singularities along a proper curve with smooth irreducible components. In particular, the sheaf of first order \(\mathbb Q\)-Gorenstein deformations \(T^1_{qG}(H)\) is described: its 1-dimensional part \(p(T^1_{qG}(H))\) is a rank 1 sheaf on \(\Delta\) which is locally free at all points except degenerate cusps of embedding dimension 4. Moreover \(p(T^1_{qG}(H))\) fits into an exact sequence \[ 0 \longrightarrow L \longrightarrow p(T^1_{qG}(H)) \longrightarrow \oplus_P k(P) \longrightarrow 0 \] where the sum \(\oplus_P k(P)\) is taken over all the degenerate cusps of embedding dimension 4 which correspond to singular points of \(\Delta\). In the case \(\Delta\) is Gorenstein and rational an explicit formula for \(p(T^1_{qG}(H))\) in terms of \(L\) is given.

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14J26 Rational and ruled surfaces