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Register a new user$mathbb{A}^2$ -Fibrations between affine spaces are trivial $mathbb{A}^2$-bundles
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Authors
Adrien Dubouloz
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We give a criterion for a flat fibration with affine plane fibers over a smooth scheme defined over a field of characteristic zero to be a Zariski locally trivial $mathbb{A}^2$-bundle. An application is a positive answer to a version of the Dolgachev-Weisfeiler Conjecture for such fibrations: a flat fibration $mathbb{A}^m$ $rightarrow$ $mathbb{A}^n$ with all fibers isomorphic to $mathbb{A}^2$ is the trivial $mathbb{A}^2$-bundle.
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In this paper we characterize the rank two vector bundles on $mathbb{P}^2$ which are invariant under the actions of the parabolic subgroups $G_p:=mathrm{Stab}_p(mathrm{PGL}(3))$ fixing a point in the projective plane, $G_L:=mathrm{Stab}_L(mathrm{PGL}(3))$ fixing a line, and when $pin L$, the Borel subgroup $mathbf{B} = G_p cap G_L$ of $mathrm{PGL}(3)$. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.
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