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A note on degenerations of del Pezzo surfaces. (Une remarque sur dégénérescences de surfaces de del Pezzo.) (English. French summary) Zbl 1330.14063
The paper under review deals with singular del Pezzo surfaces \(X\) that admit a smoothing, that is, that can appear as central fibre in a family over the unit disk \(\pi:\mathcal{X}\rightarrow\Delta\) where \(\pi^{-1}0=X\) and \(\pi^{-1}t\) is smooth for any \(t\neq 0\). More precisely, the author studies del Pezzo surfaces with only quotient singularities that admit a \(\mathbb{Q}\)-Gorenstein smoothing. Thanks to J. Kollár and N. I. Shepherd-Barron [Invent. Math. 91, No. 2, 299–338 (1988; Zbl 0642.14008)], the singularities of such an \(X\) are either Du Val or cyclic quotient singularities.
The main result is a bound on the number \(s(X)\) of non Du Val points: \(s(X)\leq \rho(X)+2\) and if \(s(X)= \rho(X)+2\) then \(X\) is toric, if \(s(X)= \rho(X)+1\) then \(X\) admits a \(\mathbb{C}^{\ast}\) action.
The case \(\rho(X)=1\) has been worked out in [P. Hacking and Y. Prokhorov, Compos. Math. 146, No. 1, 169–192 (2010; Zbl 1194.14054)] and the case \(\pi^{-1}t\cong\mathbb{P}^2\) is the object of M. Manetti [J. Reine Angew. Math. 419, 89–118 (1991; Zbl 0719.14023)]. The proof relies on a thorough description of the birational morphisms that appear in a \(K\)-MMP for a suitable pair.

MSC:
14J10 Families, moduli, classification: algebraic theory
14E30 Minimal model program (Mori theory, extremal rays)
14J26 Rational and ruled surfaces
14J17 Singularities of surfaces or higher-dimensional varieties
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