Problems in affine algebraic geometry

在“代数”几何学中，我们研究由代数方程定义的对象（相对于简单的，例如线性，或更复杂的，例如可微）。“仿射空间”在这方面是熟悉的，主要是作为我们真正感兴趣的对象的友好庇护所，例如三维空间中的球体。然而，当研究它们本身时，它们提出了令人惊讶的棘手问题，尽管付出了很多努力，但在很大程度上没有解决的问题：是什么属性困扰着它们？它们的对称性是什么？有哪些方法可以将一个嵌入另一个？这些都是“仿射代数几何”的基本问题。它们通常听起来很简单，但通常是用经典（射影）代数几何中高度复杂的技术来解决的。 本建议中要研究的一些相互关联的问题是：仿射空间上环面作用的线性化，“奇异仿射空间”的结构（代数簇拓扑上不可区分的仿射空间），研究仿射曲面与“简单拓扑”（例如可收缩的），非常“齐次”仿射曲面的结构（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，是这一领域一项大型、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.