Geodesic
Generic coverings of the plane with $A$-$D$-$E$-singularities
Izvestiya. Mathematics, Tome 64 (2000) no. 6, pp. 1153-1195 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $m\geqslant 5$. 
@article{IM2_2000_64_6_a2,
     author = {V. S. Kulikov and Vik. S. Kulikov},
     title = {Generic coverings of the plane with $A$-$D$-$E$-singularities},
     journal = {Izvestiya. Mathematics},
     pages = {1153--1195},
     year = {2000},
     volume = {64},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2000_64_6_a2/}
}
TY  - JOUR
AU  - V. S. Kulikov
AU  - Vik. S. Kulikov
TI  - Generic coverings of the plane with $A$-$D$-$E$-singularities
JO  - Izvestiya. Mathematics
PY  - 2000
SP  - 1153
EP  - 1195
VL  - 64
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/IM2_2000_64_6_a2/
LA  - en
ID  - IM2_2000_64_6_a2
ER  - 
%0 Journal Article
%A V. S. Kulikov
%A Vik. S. Kulikov
%T Generic coverings of the plane with $A$-$D$-$E$-singularities
%J Izvestiya. Mathematics
%D 2000
%P 1153-1195
%V 64
%N 6
%U http://geodesic.mathdoc.fr/item/IM2_2000_64_6_a2/
%G en
%F IM2_2000_64_6_a2
V. S. Kulikov; Vik. S. Kulikov. Generic coverings of the plane with $A$-$D$-$E$-singularities. Izvestiya. Mathematics, Tome 64 (2000) no. 6, pp. 1153-1195. http://geodesic.mathdoc.fr/item/IM2_2000_64_6_a2/

[1] Arnold V. I., “Indeksy osobykh tochek 1-form na mnogoobrazii s kraem, svorachivanie invariantov grupp, porozhdennykh otrazheniyami, i osobye proektsii gladkikh poverkhnostei”, UMN, 34:2 (1979), 3–38 | MR | Zbl

[2] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii. Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982 | MR

[3] Barth W., Peters C., Van de Ven A., Compact complex surfaces, Springer, Berlin, 1984 | MR | Zbl

[4] Catanese F., “On a Problem of Chisini”, Duke Math. J., 53:1 (1986), 33–42 | DOI | MR | Zbl

[5] Griffits F., Kharris Dzh., Printsipy algebraicheskoi geometrii, Mir, M., 1982 | MR

[6] Kulikov Vik. S., “O gipoteze Kizini”, Izv. RAN. Ser. matem., 63:6 (1999), 83–116 | MR | Zbl

[7] Moishezon B., Complex Surfaces and Connected Sums of Complex Projective Planes, LNM, 603, Springer, 1977 | MR | Zbl

[8] Zariski O., Algebraic surfaces, Springer-Verlag, Berlin, 1971 | MR | Zbl