\(\mathbb Q\)-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.
Reviewer: Eleonora Palmieri (Roma)
MSC:
14D06 | Fibrations, degenerations in algebraic geometry |
14J17 | Singularities of surfaces or higher-dimensional varieties |
14J26 | Rational and ruled surfaces |