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# Cremona groups and their subgroups.

27.10.2011 | 17:00 c.t.

Abstract:   The Cremona group of degree $n$ is the group of rational substitutions of  $n$ variables. More precisely, it is the group  $Cr_F(n) = Aut_FF(x_1, ..., x_n)$  of automorphisms of the field of rational functions  $F(x_1, ..., x_n)$  in  $n$  variables with coefficients in a field  $F$  which are identical on constant functions. In the case  $n = 1$,  the group  $Cr_F(1)$  consists of fractional-linear transformations  $x\mapsto {ax+b \over cx+d}$  with  $ad-bc\ne 0$.  It is isomorphic to the group  $PGL(2,F)$   of  $F$-rational points of the algebraic group  $PGL_F(2)$.  In the case  $n > 1$,  the group is very big, it is not isomorphic to any algebraic group and contains the group  $PGL_F(n+1,F)$  as its proper subgroup. In geometric language, the Cremona group  $Cr_F(n)$  is equal to the group of birational automorphisms of the projective space  $P_F^n$  or of any algebraic variety birationally isomorphic to the projective space. The  Cremona groups were studied  for over 150 years, however the main problems are still unsolved. In my talk I will review some old and new results on this subject with emphasis on the classification of algebraic subgroups of  $Cr_F(n)$.

Tee/Kaffee/Gebäck
ab  16:45 Uhr,
Arnimallee 3,  Raum 006

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Koordinator:  Prof. Dr. Alexander Schmitt

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### Zeit & Ort

27.10.2011 | 17:00 c.t.

Institut für Mathematik<br> Arnimallee 3, Hörsaal 001