论文标题
二维球的分支覆盖空间的压实
Compactification of the space of branched coverings of the two-dimensional sphere
论文作者
论文摘要
对于封闭式的表面$σ$，我们将其变性定义为奇数表面，这些表面是局部同构对圆盘楔形的同型。令$ x_ {σ，n} $是保存$ n $ fold的分支覆盖物$σ\ rightArrow s^2 $的定向的同构类别的集合。我们将$ x_ {σ，n} $与映射的同构类别类别一起完成，这些类别通过$σ$的退化覆盖了球体。在$σ= s^2 $的情况下，我们在获得的完成$ \ bar {x} _ {x} _ {σ，n} $上与$ x_ {s^2，n} $重合的拓扑与拓扑构成的拓扑是由$ p/q $ p/q $ n $ nmogene promone $ nomials primation unomials pynomials pregation y n of polational uncoration from from of pogipation o。 \ Mathbb {C} \ Mathrm {p}^1 \ cong s^2 $。 我们证明，$ \ bar {x} _ {σ，n} $与hurwitz space $ h（σ，n）\ subset x_ {σ，n} $的diaz-edidin-natanzon-turaev compactification compactification y hurwitz space $ h（σ，n）\ subset x_ {σ，n）\ subset x_ {σ，n} $由等异形的覆盖物组成的所有关键值都简单。
For a closed oriented surface $ Σ$ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let $X_{Σ,n}$ be the set of isomorphism classes of orientation preserving $n$-fold branched coverings $ Σ\rightarrow S^2 $ of the two-dimensional sphere. We complete $X_{Σ,n}$ with the isomorphism classes of mappings that cover the sphere by the degenerations of $ Σ$. In case $ Σ=S^2$, the topology that we define on the obtained completion $\bar{X}_{Σ,n}$ coincides on $X_{S^2,n}$ with the topology induced by the space of coefficients of rational functions $ P/Q $, where $ P,Q $ are homogeneous polynomials of degree $ n $ on $ \mathbb{C}\mathrm{P}^1\cong S^2$. We prove that $\bar{X}_{Σ,n}$ coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space $H(Σ,n)\subset X_{Σ,n}$ consisting of isomorphism classes of branched coverings with all critical values being simple.
