Generalized Chisini's Conjecture

Vik. S. Kulikov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 122-131

Résumé

Chisini's Conjecture claims that a generic covering of the plane of degree $\geq 5$ is determined uniquely by its branch curve. A generalization (to the case of normal surfaces) of Chisini's Conjecture is formulated and considered. The generalized conjecture is checked in the following two cases: when the maximum of degrees of two generic coverings $\geq 12$ and when it $\leq 4$. Conditions on the number of singular points of a cuspidal curve $B$ necessary for $B$ to be the branch curve of a generic covering of given degree are found. In particular, it is shown that, if $B$ is a pure cuspidal curve (i.e. all its singular points are ordinary cusps), then $B$ can be the branch curve only of a generic covering of degree $\leq 5$.