Generic coverings of the plane with $A$-$D$-$E$-singularities

V. S. Kulikov ; Vik. S. Kulikov

Izvestiya. Mathematics , Tome 64 (2000) no. 6, pp. 1153-1195

Résumé

We investigate representations of an algebraic surface $X$ with $A$-$D$-$E$-singularities as a generic covering $f\colon X\to\mathbb{P}^2$, 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 [6], 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 $m\geqslant 5$.