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Ga-actions on Complex Affine Threefolds
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This  thesis  consists  of two papers  and  a summary.  The  papers  both  deal with  affine algebraic complex  varieties,  and  in particular such  varieties  in dimension  three  that have a non-trivial action  of one of the  one-dimensional  algebraic  groups  Ga   :=  (C, +) and  Gm  :=  (C*, ·).  The methods  used  involve  blowing up  of subvarieties, the correspondances between  Ga - and  Gm - actions  on an affine variety  X with locally nilpotent derivations  and Z-gradings  respectively  on O(X) and passing from a filtered algebra  A to its associated graded  algebra  gr(A).

In Paper  I, we study  Russell’s hypersurface  X , i.e. the affine variety  in the affine space A4 given by the equation  x + x2y + z3 + t2 = 0. We reprove by geometric means Makar-Limanov’s result which states  that X is not isomorphic to A3 – a result which was crucial to Koras-Russell’s proof of the linearization conjecture  for Gm -actions on A3. Our method consist in realizing X as an open part  of a blowup M  −→ A3 and to show that each Ga -action on X descends to A3 . This follows from considerations of the graded  algebra  associated to O(X ) with respect  to a certain filtration.

In Paper  II, we study  Ga-threefolds X  which have  as their  algebraic  quotient  the  affine plane  A2  = Sp(C[x, y]) and  are a principal  bundle  above the  punctured plane  A2  :=  A2 \ {0}. Equivalently, we study  affine Ga -varieties  Pˆ  that extend  a principal  bundle  P over A2, being P together  with an extra  fiber over the origin in A2. First  the trivial  bundle  is studied,  and some examples of extensions  are given (including  smooth  ones which are not isomorphic  to A2 × A). The  most  basic among  the  non-trivial  principal  bundles  over A2 is SL2 (C)  −→ A2, A  1→  Ae1 where e1  denotes  the first unit  vector,  and we show that any non-trivial  bundle  can be realized as a pullback  of this  bundle  with  respect  to  a morphism  A2  A2. Therefore  the  attention is then  restricted to extensions  of SL2(C)  and  find two families of such extensions  via a study of the  graded  algebras  associated  with  the  coordinate  rings  O(Pˆ)  '→ O(P ) with  respect  to  a filtration  which is defined in terms  of the Ga -actions  on P and Pˆ  respectively.

Place, publisher, year, edition, pages
Uppsala: Uppsala universitet, Matematiska institutionen , 2013. , 32 p.
Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 81
Keyword [en]
Complex affine varieties, algebraic group actions of C^+ and C^*, locally nilpotent derivations, graded algebras, principal bundles, Russell's hypersurface
National Category
Geometry
Identifiers
URN: urn:nbn:se:uu:diva-203708ISBN: 978-91-506-2357-4 (print)OAI: oai:DiVA.org:uu-203708DiVA: diva2:637411
Public defence
2013-09-06, Häggsalen, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2013-08-16 Created: 2013-07-17 Last updated: 2013-08-16Bibliographically approved
List of papers
1. Russell's hypersurface from a geometric point of view
Open this publication in new window or tab >>Russell's hypersurface from a geometric point of view
(English)Manuscript (preprint) (Other academic)
National Category
Geometry
Identifiers
urn:nbn:se:uu:diva-203706 (URN)
Available from: 2013-07-17 Created: 2013-07-17 Last updated: 2013-07-17
2. Extensions of principal G_a-bundles over A^2_*
Open this publication in new window or tab >>Extensions of principal G_a-bundles over A^2_*
(English)Manuscript (preprint) (Other academic)
National Category
Geometry
Identifiers
urn:nbn:se:uu:diva-203707 (URN)
Available from: 2013-07-17 Created: 2013-07-17 Last updated: 2013-07-17

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