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

                                    at 15:00
 
                        Note the unusual time and place
 
                                 Maxim Leyenson
 
                                 will speak on
 
          On ramified covers  of algebraic surfaces, and combinatorics
                       of some infinite discrete groups.
 
Abstract:
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
corresponding representations.
 
  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   <techm@math.technion.ac.il> 
Announcement from: Yaeli Malka   <yaeli.malka@weizmann.ac.il>