Seminar Algebra and Geometry: Bruno Laurent (Grenoble)

Almost homogeneous curves and surfaces

The varieties which are homogeneous under the action of an algebraic group are very symmetric objects. More generally, we get a much wider class of objects, having a very rich geometry, by allowing the varieties to have not a unique orbit, but a dense orbit. Such varieties are said to be almost homogeneous; this includes the case of toric varities, when the group is an algebraic torus.

In this talk, I will explain how to classify the pairs (X,G) where X is a curve or a surface and G is a smooth and connected algebraic group acting on X with a dense orbit.

For curves, I will mainly focus on the regular ones, defined over an arbitrary field. Over an algebraically closed field, the "natural" notion of non-singularity is "smoothness". However, over an arbitrary field, the weaker notion of "regularity" is more suitable. I will recall the difference between those two notions and show that there exist regular homogeneous curves which are not smooth.

For surfaces, I will restrict to the smooth ones, defined over an algebraically closed field. The situation is more complicated than for curves. Moreover, new phenomena and several difficulties appear in positive characteristic, and I will highlight them.

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