The Weizmann Institute of Science
Faculty of Mathematics and Computer Science
Algebraic Geometry and Representation Theory Seminar
Lecture Hall, Room 1, Ziskind Building
on Monday, May 21, 2012
Note the unusual time and place
will speak on
On ramified covers of algebraic surfaces, and combinatorics
of some infinite discrete groups.
Bernhard Riemann studied (complex) algebraic curves by considering ramified
covers of the projective line. In dimension two, one of the classical
approaches of the Italian school of algebraic geometry was to study (complex)
algebraic surfaces by considering ramified covers of the simplest surface --
the projective plane.
Such ramified covers are classified, in any dimensions, by suitable
representations of some (usually very large!) discrete group $\Gamma$
into a symmetric group $Sym_n$. Here $\Gamma$ is the fundamental group of a
complement to the branch divisor.
Now, a striking difference appears between dimensions one and two. It was
discovered by Enriques, Castelnuovo and, especially, Zariski.
First, any collection of points on a projective line is a
ramification locus of some ramified cover. The corresponding
fundamental group is the free group $F_n$, which always has lots of
representations into the symmetric group $Sym_n$. In particular, the
construction gives lots of complex curves. There were many studies of the
The case of dimension two is very different. Only very special plane
curves are ramification divisors, and only very special groups are
fundamental groups. Moreover, there are restrictions on representations.
We argue that the groups $\Gamma$ appearing here are (a very
broad) generalizations of the Artin's braid group, and define the
corresponding class of discrete groups. We call them twisted
Artin-Brieskorn groups. Then we discuss combinatorics of their
representations into symmetric groups $Sym_n$, and discuss the related
Chisini conjecture of algebraic geometry.
Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel <firstname.lastname@example.org>
Announcement from: Yaeli Malka <email@example.com>