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UID:news739@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190211T212734
DTSTART;TZID=Europe/Zurich:20130503T103000
SUMMARY:Seminar Algebra and Geometry: Sergei Kovalenko (Ruhr-Universität B
 ochum)
DESCRIPTION:Gizatullin surfaces are normal affine surfaces completable by a
   zigzag\, i. e. by a linear chain of smooth rational curves. An equivalen
 t  characterization of such surfaces V\, except for the surface C∗ × C
 ∗\, is that the automorphism group acts with a big orbit O\, i. e. V \\ 
 O is finite. Considering some examples of Gizatullin surfaces like the aff
 ine plane A2 or the Danielewski surfaces V = {xy − P(z) = 0} ⊆ A3 it f
 ollows that the big orbit O coincides with the smooth locus Vreg . Gizatul
 lin formulated in his pioneer works the following conjecture: \\r\\nConjec
 ture (Gizatullin): The big orbit of a Gizatullin surface V coincides with 
 its smooth locus\, i. e. O = Vreg . \\r\\nWe show that the action of the a
 utomorphism group of a smooth  Gizatullin surface with a distinguished and
  rigid extended divisor is  not transitive in general. Thus such surfaces 
 represent counterexamples  to Gizatullin’s conjecture. For such surfaces
  we give an explicit orbit  decomposition of the natural action of the aut
 omorphism group. Moreover\,  the automorphism group of such smooth Gizatul
 lin surfaces can be  represented as an amalgamated product of two automorp
 hism subgroups.
X-ALT-DESC:Gizatullin surfaces are normal affine surfaces completable by a 
  zigzag\, i. e. by a linear chain of smooth rational curves. An equivalent
   characterization of such surfaces <i>V</i>\, except for the surface <b>C
 </b><sup>∗</sup> × <b>C</b><sup>∗</sup>\, is that the automorphism gr
 oup acts with a big orbit <i>O</i>\, i. e.<i> V </i>\\ <i>O</i> is finite.
  Considering some examples of Gizatullin surfaces like the affine plane <b
 >A</b><sup>2</sup> or the Danielewski surfaces V = {xy − P(z) = 0} ⊆ <
 b>A</b><sup>3</sup> it follows that the big orbit <i>O</i> coincides with 
 the smooth locus <i>V</i><sub>reg</sub> . Gizatullin formulated in his pio
 neer works the following conjecture: \n<b>Conjecture (Gizatullin):</b> The
  big orbit of a Gizatullin surface <i>V</i> coincides with its smooth locu
 s\, i. e. O = <i>V</i><sub>reg</sub> . \nWe show that the action of the au
 tomorphism group of a smooth  Gizatullin surface with a distinguished and 
 rigid extended divisor is  not transitive in general. Thus such surfaces r
 epresent counterexamples  to Gizatullin’s conjecture. For such surfaces 
 we give an explicit orbit  decomposition of the natural action of the auto
 morphism group. Moreover\,  the automorphism group of such smooth Gizatull
 in surfaces can be  represented as an amalgamated product of two automorph
 ism subgroups.
DTEND;TZID=Europe/Zurich:20130503T120000
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