Forthcoming seminars
Seminar calendar
List of seminars
Archive by years
Register a seminar

Forthcoming seminars

You may need the following programs to see the files

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

Number of views:
This page:90

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 $S_R$.

SHARE: FaceBook Twitter Livejournal
Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2017