Problems in affine algebraic geometry

在“代数”几何学中，我们研究由代数(而不是更简单的，如线性的，或更复杂的，如可微的)方程定义的对象。在这种情况下，“仿射空间”主要是作为我们真正感兴趣的物体的友好遮蔽物而熟悉的，例如三维空间中的球体。然而，当为它们本身而研究时，它们提出了令人惊讶的棘手问题，尽管付出了大量努力，但基本上仍未解决问题：它们具有哪些特征？它们的对称性是什么？有什么方法可以将一个嵌入到另一个中？这些都是“仿射代数几何”的基本问题。它们听起来往往很简单，但通常是用经典(射影)代数几何中的高度复杂的技术来解决的。这一建议中要研究的一些相关问题是：仿射空间上环面作用的线性化，“奇异仿射空间”的结构(代数簇在拓扑上与仿射空间是不可区分的)，具有“简单拓扑”的仿射曲面的研究(例如可压缩的仿射曲面)，非常“齐次”仿射曲面的结构(具有大的，非常传递的自同构群)，平面有理曲线的研究。这项研究是先前工作的延续，是这一领域大型、活跃的国际活动的一部分。

In "algebraic" geometry we study objects defined by algebraic (as opposed to simpler, e.g. linear, or more complicated, e.g. differentiable) equations. "Affine spaces" are familiar in this context mainly as friendly shelters for objects that really interest us, a sphere in 3-space for instance. Yet, when studied for their own sake, they pose surprisingly thorny and, despite a lot of effort, largely unresolved problems: What properties characterise them? What are their symmetries? What are the ways to embed one into another? These are the basic questions of "affine algebraic geometry". They often sound simple, but are addressed typically with highly sophisticated techniques from classical (projective) algebraic geometry.   Some of the interrelated problems to be studied in this proposal are: the linearization of torus actions on affine space, the structure of "exotic affine spaces" (algebraic varieties topologically indistinguishable from affine spaces), the study of affine surfaces with "simple topology" (e.g contractible ones), the structure of very "homogeneous" affine surfaces (having a large, very transitive automorphism group), the study of plane rational curves.   This research is a continuation of previous work and forms part of a large, lively international activity in this area.