Courbes et surfaces presque homogènes

Abstract : The varieties having a dense orbit under the action of a group are said to be almost homogeneous. Those are objects with a very rich geometry and have been extensively studied for the last 50 years ; this includes toric varieties. The purpose of this thesis is to classify the pairs (X,G) where X is an algebraic curve or surface, defined over an arbitrary field, and G is a smooth connected algebraic group, acting faithfully on X with a dense orbit. The classification relies on the study of the equivariant regular completions of X.The study of almost homogeneous curves highlights the class of seminormal curves. We get a full classification of the pairs (X,G) when X is a seminormal curve. We also describe all almost homogeneous curves (over an arbitrary field), thus generalizing a result of Vladimir Popov. Finally, we determine the linearized line bundles over seminormal almost homogeneous curves.The last chapter deals with the case of surfaces. Again, we get a classification of the pairs (X,G) when X is a surface and G is not affine. When G is affine, the surface is rational. We then describe, over an algebraically closed field, the homogeneous surfaces and their relatively minimal equivariant regular completions. In characteristic zero, we also determine the acting groups. Many new phenomena occur in positive characteristic, and some of our results are incomplete in this setting.
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Bruno Laurent. Courbes et surfaces presque homogènes. Géométrie algébrique [math.AG]. Université Grenoble Alpes, 2018. Français. ⟨NNT : 2018GREAM046⟩. ⟨tel-01999851⟩

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