Robayo, Maria Fernanda. Birational diffeomorphisms of the sphere of finite order. 2015, Doctoral Thesis, University of Basel, Faculty of Science.

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Abstract
In this thesis in real algebraic geometry, we obtain the classification of conjugacy classes of elements of prime order in the group of birational diffeomorphisms of the twodimensional sphere.
More precisely, the twodimensional sphere S(\R) is the surface defined by the equation x^2+y^2+z^2=1 in \R^3. It corresponds to the real part of the smooth complex projective surface S_\C defined by the equation x^2+y^2+z^2=w^2 in the threedimensional complex projective space \P^3, that is to say, to the part which is contained in the chart w=1. We can study the algebraic diffeomorphisms of S(\R): birational maps from the complex surface S_\C to itself such that the coefficients that define the map are real, and the restriction to the real part S(\R) is a diffeomorphism. Those maps are called birational diffeomorphisms of the sphere and they form a group denoted by Aut(S(\R)), the subject of this thesis.
The interest in this group stems from many reasons. For instance, J. Kollár and F. Mangolte showed that Aut(X(\R)) is big  that is, dense in the group of all diffeomorphisms Diff(X(\R))  if X is a smooth real compact rational surface, as in the case of the sphere. This is in stark contrast to the group of automorphisms of the complex surface, which is finite dimensional.
One motivation to classify the conjugacy classes of the group Aut(S(\R)) is that, by definition, it is contained in the group of all birational maps of the complex surface S_\C, and this latter group is isomorphic to the Cremona group, i.e. the group of birational maps of the complex projective plane. The problem of classification of conjugacy classes of elements of finite order in the Cremona group has been of interest for a lot of mathematicians. L. Bayle and A. Beauville classified the elements of order two using the tools of the minimal model program developed in dimension two by Yu. Manin and V.A. Iskovskikh. T. de Fernex generalised the classification for elements of prime order (except for one case, done by A. Beauville and J. Blanc). Because of the relation of those two groups we can relate the conjugacy classes of birational diffeomorphisms of the sphere to the conjugacy classes of elements in the Cremona group. For example, we find that the sphere features an example of a large familiy of conjugacy classes of involutions that all correspond to one single class for the complex projective plane, to name just one of the differences between the two.
More precisely, the twodimensional sphere S(\R) is the surface defined by the equation x^2+y^2+z^2=1 in \R^3. It corresponds to the real part of the smooth complex projective surface S_\C defined by the equation x^2+y^2+z^2=w^2 in the threedimensional complex projective space \P^3, that is to say, to the part which is contained in the chart w=1. We can study the algebraic diffeomorphisms of S(\R): birational maps from the complex surface S_\C to itself such that the coefficients that define the map are real, and the restriction to the real part S(\R) is a diffeomorphism. Those maps are called birational diffeomorphisms of the sphere and they form a group denoted by Aut(S(\R)), the subject of this thesis.
The interest in this group stems from many reasons. For instance, J. Kollár and F. Mangolte showed that Aut(X(\R)) is big  that is, dense in the group of all diffeomorphisms Diff(X(\R))  if X is a smooth real compact rational surface, as in the case of the sphere. This is in stark contrast to the group of automorphisms of the complex surface, which is finite dimensional.
One motivation to classify the conjugacy classes of the group Aut(S(\R)) is that, by definition, it is contained in the group of all birational maps of the complex surface S_\C, and this latter group is isomorphic to the Cremona group, i.e. the group of birational maps of the complex projective plane. The problem of classification of conjugacy classes of elements of finite order in the Cremona group has been of interest for a lot of mathematicians. L. Bayle and A. Beauville classified the elements of order two using the tools of the minimal model program developed in dimension two by Yu. Manin and V.A. Iskovskikh. T. de Fernex generalised the classification for elements of prime order (except for one case, done by A. Beauville and J. Blanc). Because of the relation of those two groups we can relate the conjugacy classes of birational diffeomorphisms of the sphere to the conjugacy classes of elements in the Cremona group. For example, we find that the sphere features an example of a large familiy of conjugacy classes of involutions that all correspond to one single class for the complex projective plane, to name just one of the differences between the two.
Advisors:  Blanc, Jérémy 

Committee Members:  Mangolte, Frédéric 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Algebra (Blanc) 
UniBasel Contributors:  Robayo, Maria Fernanda and Blanc, Jérémy 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  11285 
Thesis status:  Complete 
Bibsysno:  Link to catalogue 
Number of Pages:  70 S. 
Language:  English 
Identification Number: 

Last Modified:  22 Apr 2018 04:32 
Deposited On:  22 Jul 2015 13:24 
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