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Steklov Mathematical Institute Seminar
November 15, 2001, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

Generic covers of the plane

Vik. S. Kulikov

Abstract: It is known that for a non-singular projective surface $S$ embedded in the projective space $\mathbb{CP}^r$ the restriction $f$ to $S$ of a linear projection $\operatorname{pr}\colon\mathbb{CP}^r\to\mathbb{CP}^2$ that is generic with respect to $S$ has the following properties:
1) $f$ is a finite morphism of degree $d=\deg S$;
2) the discriminant curve (branch curve) $B\subset\mathbb{CP}^2$ is an irreducible cuspidal curve;
3) for a generic point $p\in B$ the number $#f^{-1}(p)$ of inverse images is equal to $d-1$.
A generic cover of the projective plane $\mathbb{CP}^2$, which is a natural generalization of the concept of a generic linear projection, is a morphism $f\colon S\to\mathbb{CP}^2$ of a non-singular surface $S$ for which the properties 1)–3) are valid.
Chisini's conjecture asserts that a generic cover $f\colon S\to\mathbb{CP}^2$ of degree$\deg f\ge5$ is uniquely determined by its discriminant curve.
In the first part of the report an outline is given of a proof of Chisini's conjecture (and also of a certain generalization of it) for almost all generic covers of the plane. The second part of the report is devoted to a discussion of the possibility of using the braid-monodromy invariants of the discriminant curve of a generic cover $f\colon S\to\mathbb{CP}^2$ as a complete set of discrete invariants determining a symplectic structure on a suitable four-dimensional real manifold $S_R$.