论文标题
飞机上两个点的高度是多少？
What is the height of two points in the plane?
论文作者
论文摘要
在这里，我们描述了在投影平面中两个点的希尔伯特方案上的理性点的分布。 更具体地说，我们明确描述了一个高度功能的两参数家族$ h_ {s，t} $，使得与任何投影嵌入的高度函数相当于某些$ h_ {s，t} $，以通过有限函数乘以乘法。对于一定范围的参数$（s，t）$，我们证明了一个有界高度的理性点的渐近公式，对于其他$（s，t）$，我们获得了上限。证明确定了与晶格计数问题的等效性，我们使用数字的几何形状来解决该问题。
Here we describe the distribution of rational points on the Hilbert scheme of two points in the projective plane. More specifically, we explicitly describe a two-parameter family of height functions $H_{s, t}$, such that the height function associated to any projective embedding is equivalent to some $H_{s, t}$, up to multiplication by a bounded function. For a certain range of the parameters $(s, t)$, we prove an asymptotic formula for the number of rational points of bounded height, and for other $(s, t)$ we obtain an upper bound. The proof establishes an equivalence to a lattice point counting problem, which we solve using the geometry of numbers.
