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Generic coverings of the plane with A-D-E-singularities. (English. Russian original) Zbl 1012.14004

Izv. Math. 64, No. 6, 1153-1195 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 6, 65-106 (2000).
Summary: We investigate representations of an algebraic surface \(X\) with \(A\)-\(D\)-\(E\)-singularities as a generic covering \(f:X\to\mathbb{P}^2\), that is, a finite morphism which has at most folds and pleats apart from singular points and is isomorphic to the projection of the surface \(z^2 = h(x, y)\) onto the plane \(x, y\) near each singular point, and whose branch curve \(B\subset\mathbb{P}^2\) has only nodes and ordinary cusps except for singularities originating from the singularities of \(X\). It is regarded as folklore that a generic projection of a non-singular surface \(X\subset\mathbb{P}^r\) is of this form.
In this paper we prove this result in the case when the embedding of a surface \(X\) with \(A\)-\(D\)-\(E\)-singularities is the composite of the original one and a Veronese embedding. We generalize the results of Vik. S. Kulikov [Izv. Math. 63, No. 6, 1139-1170 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 6, 83-116 (1999; Zbl 0962.14005)], which considers Chisini’s conjecture on the unique reconstruction of \(f\) from the curve \(B\). To do this, we study fibre products of generic coverings. We get the main inequality bounding the degree of the covering in the case when there are two inequivalent coverings with branch curve \(B\). This inequality is used to prove Chisini’s conjecture for \(m\)-canonical coverings of surfaces of general type for \(m\geqslant 5\).

MSC:

14E22 Ramification problems in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14E20 Coverings in algebraic geometry

Citations:

Zbl 0962.14005
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