Problems in affine algebraic geometry

在“代数”几何中，我们研究由代数方程（相对于更简单的，如线性，或更复杂的，如可微）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庇护所，例如三维空间中的球体。然而，当研究它们本身的时候，它们带来了令人惊讶的棘手问题，尽管付出了很多努力，但在很大程度上仍未解决：它们的特征是什么？它们的对称性是什么？将一个嵌入到另一个的方法是什么？这些是“仿射代数几何”的基本问题。它们通常听起来很简单，但通常使用来自经典（射影）代数几何的高度复杂的技术来解决。本建议研究的一些相关问题是：环面作用在仿射空间上的线性化，“外来仿射空间”（拓扑上与仿射空间难以区分的代数变异）的结构，具有“简单拓扑”的仿射曲面（例如可收缩曲面）的研究，非常“齐次”仿射曲面的结构（具有一个大的、非常传递的自同构群），平面有理曲线的研究。这项研究是以前工作的延续，是这一领域一个大型的、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.