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Generic coverings of the plane with $A$-$D$-$E$-singularities
V. S. Kulikova,
Vik. S. Kulikovb a Moscow State Academy of Printing Arts
b Steklov Mathematical Institute, Russian Academy of Sciences
Abstract:
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
$m\geqslant 5$.
MSC: 14E20 Received: 27.07.1999
DOI:
10.4213/im312
Fulltext:
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