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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-203708 |
| Date | January 2013 |
| Creators | Hedén, Isac |
| Publisher | Uppsala universitet, Matematiska institutionen, Uppsala : Uppsala universitet, Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 81 |
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