Seminar Algebra and Geometry: Isac Hedén (Universität Basel)

Extensions of principal G_a-bundles over the punctured plane

We study complex affine G_a-threefolds X with the affine plane A² as their algebraic quotient, and which are a principal bundle over
the punctured affine plane A²_*. Changing the point of view, we look for affine extensions of G_a-principal bundles over A²_* that are obtained by adding an extra fiber to the bundle projection over the origin. It turns out that SL₂ plays a special role, since every G_a-principal bundle can be obtained as a pullback of SL₂. We construct two series of such extensions which generalize the basic extension obtained by adding a zero section to SL₂ viewed as a G_m-principal bundle over P¹ x P¹ \ Δ.

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