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Rational mappings of Del Pezzo surfaces, and singular compactifications of two-dimensional affine varieties. (English) Zbl 0546.14005

Let F be an algebraically closed field, denote by \({\mathbb{P}}^ 2(F)\) the projective plane over F, and let X be a singular Gorenstein surface over F with first Betti number equal to 1 and with negative canonical line bundle. In this paper we give a complete classification of all rational mappings \(f: X\to {\mathbb{P}}^ 2(F)\) for which the singularities of f can be resolved simultaneously with the singularities of X. The technique involves analyzing the configurations of negatively embedded curves on non-singular Del Pezzo surfaces in terms of collections of extended Coxeter graphs, with some special results in the case of fields of characteristic 2. As a consequence we find all compactifications of a particularly nice type of affine subvarieties of \({\mathbb{P}}^ 2({\mathbb{C}}).\)
[These results have been announced by the authors in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 1, 145- 151 (1983; Zbl 0525.14018)].

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
32J05 Compactification of analytic spaces
14E05 Rational and birational maps
14M07 Low codimension problems in algebraic geometry

Citations:

Zbl 0525.14018
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References:

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