Stampfli, Immanuel. Contributions to automorphisms of affine spaces. 2013, PhD Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_10504
Abstract
We study aspects of the group G_n of polynomial automorphisms of the affine space A^n, the socalled affine Cremona group. Shafarevich introduced on G_n the structure of an indvariety, an infinitedimensional analogon to a (classical) variety. The aim of this thesis is to study G_n within the framework of indvarieties. The thesis consists of five articles. In the following we summarize them.
1. On the Topologies on indVarieties and related Irreducibility Questions.
In the literature there are two ways of endowing an affine indvariety with a topology. One possibility is due to Shafarevich and the other due to Kambayashi. We specify a large class of affine indvarieties where these two topologies differ. We give an example of an affine indvariety that is reducible with respect to Shafarevich’s topology, but irreducible with respect to Kambayashi’s topology. Moreover, we give a counterexample of a supposed irreducibility criterion given by Shafarevich which is different from a counterexample given by Homma. We finish the article with an irreducibility criterion similar to the one given by Shafarevich.
2. On Automorphisms of the Affine Cremona Group (joint with Hanspeter Kraft)
We show that every automorphism of the group G_n is inner up to field automorphisms when restricted to the subgroup TG_n of tame automorphisms. This generalizes a result of Julie Déserti who proved this in dimension n = 2 where all automorphisms are tame, i.e. TG_2 = G_2. The methods are different, based on arguments from algebraic group actions.
3. A Note on Automorphisms of the Affine Cremona Group
Let G be an indgroup and let U be a unipotent indsubgroup. We prove that an abstract automorphism f: G > G maps U isomorphically onto a unipotent indsubgroup of G, provided that f fixes a closed torus T in G that normalizes U and the action of T on U by conjugation fixes only the neutral element. As an application we generalize the main result of the article "On Automorphisms of the Affine Cremona Group" as follows: If an abstract automorphism of G_3 fixes the subgroup of tame automorphisms TG_3, then it also fixes a whole family of nontame automorphisms (including the Nagata automorphism).
4. Automorphisms of the Plane Preserving a Curve (joint with Jérémy Blanc)
We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.
5. Centralizer of a Unipotent Automorphism in the Affine Cremona Group
Let g be a unipotent element of G_3. We describe the centralizer Cent(g) inside G_3. First, we treat the case when g is a modified translation. In the other case, we describe the subset Cent(g)_u of unipotent elements of Cent(g) and prove that it is a closed normal subgroup of Cent(g). Moreover, we show that Cent(g) is the semidirect product of Cent(g)_u with a closed algebraic subgroup R of Cent(g). Finally, we prove that the subgroup of Cent(g) consisting of those elements that induce the identity on the algebraic quotient Spec O(A^3)^g form a characteristic subgroup of Cent(g).
1. On the Topologies on indVarieties and related Irreducibility Questions.
In the literature there are two ways of endowing an affine indvariety with a topology. One possibility is due to Shafarevich and the other due to Kambayashi. We specify a large class of affine indvarieties where these two topologies differ. We give an example of an affine indvariety that is reducible with respect to Shafarevich’s topology, but irreducible with respect to Kambayashi’s topology. Moreover, we give a counterexample of a supposed irreducibility criterion given by Shafarevich which is different from a counterexample given by Homma. We finish the article with an irreducibility criterion similar to the one given by Shafarevich.
2. On Automorphisms of the Affine Cremona Group (joint with Hanspeter Kraft)
We show that every automorphism of the group G_n is inner up to field automorphisms when restricted to the subgroup TG_n of tame automorphisms. This generalizes a result of Julie Déserti who proved this in dimension n = 2 where all automorphisms are tame, i.e. TG_2 = G_2. The methods are different, based on arguments from algebraic group actions.
3. A Note on Automorphisms of the Affine Cremona Group
Let G be an indgroup and let U be a unipotent indsubgroup. We prove that an abstract automorphism f: G > G maps U isomorphically onto a unipotent indsubgroup of G, provided that f fixes a closed torus T in G that normalizes U and the action of T on U by conjugation fixes only the neutral element. As an application we generalize the main result of the article "On Automorphisms of the Affine Cremona Group" as follows: If an abstract automorphism of G_3 fixes the subgroup of tame automorphisms TG_3, then it also fixes a whole family of nontame automorphisms (including the Nagata automorphism).
4. Automorphisms of the Plane Preserving a Curve (joint with Jérémy Blanc)
We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.
5. Centralizer of a Unipotent Automorphism in the Affine Cremona Group
Let g be a unipotent element of G_3. We describe the centralizer Cent(g) inside G_3. First, we treat the case when g is a modified translation. In the other case, we describe the subset Cent(g)_u of unipotent elements of Cent(g) and prove that it is a closed normal subgroup of Cent(g). Moreover, we show that Cent(g) is the semidirect product of Cent(g)_u with a closed algebraic subgroup R of Cent(g). Finally, we prove that the subgroup of Cent(g) consisting of those elements that induce the identity on the algebraic quotient Spec O(A^3)^g form a characteristic subgroup of Cent(g).
Advisors:  Kraft, Hanspeter 

Committee Members:  Dubouloz, A. 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Algebra (Kraft) 
Item Type:  Thesis 
Thesis no:  10504 
Bibsysno:  Link to catalogue 
Number of Pages:  79 S. 
Language:  English 
Identification Number: 

Last Modified:  30 Jun 2016 10:53 
Deposited On:  22 Oct 2013 12:49 
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