On the IRS compactification of moduli space

带边黎曼曲面模空间的组合模型及其紧化

A combinatorial model for the moduli of bordered Riemann surfaces and a compactification

本文利用对称度量带状图及不可收缩子图序列，构造带边标记黎曼曲面模空间的组合紧化，并在零亏格下得到解析模空间的轨形胞腔分解。

模空间组合模型及其紧化的比较

Comparing combinatorial models of moduli space and their compactifications

本文比较了二维配边模空间的两种组合模型（径向狭缝配置和容许胖图），通过“临界图”映射构建了显式同伦等价，并证明该等价在它们的自然紧化（单水平调和紧化与Sullivan图）之间诱导了胞腔同胚。

六次曲线具有简单奇点的模空间及其紧化

Moduli spaces of sextic curves with simple singularities and their compactifications

本文研究六次曲线具有简单奇点的模空间，通过周期映射证明其可嵌入算术商空间，并展示GIT与Looijenga紧化的等价性，以及皮卡晶格的显式描述。

The Compactification of the Moduli Space of Convex RP(2) Surfaces, I

There is a canonical identification, due to the author, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne-Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to t…

The curious moduli spaces of unmarked Kleinian surface groups

Fixing a closed hyperbolic surface S, we define a moduli space AI(S) of unmarked hyperbolic 3-manifolds homotopy equivalent to S. This 3-dimensional analogue of the moduli space M(S) of unmarked hyperbolic surfaces homeomorphic to S has bizarre local topology, possessing many points that are not closed. There is, however, a natural embedding of M(S) into AI(S) and a compactification of AI(S) such that this embedding extends to an embedding of the Deligne-Mumford compactification of M(S) into the compactification of AI(S).

Teichmuller geometry of moduli space, I: Distance minimizing rays and the Deligne-Mumford compactification

Let $S$ be a closed, oriented surface with a finite (possibly empty) set of points removed. In this paper we relate two important but disparate topics in the study of the moduli space $\M(S)$ of Riemann surfaces: Teichmüller geometry and the Deligne-Mumford compactification. We reconstruct the Deligne-Mumford compactification (as a metric stratified space) purely from the intrinsic metric geometry of $\M(S)$ endowed with the Teichmüller metric. We do this by first classifying (globally) geodesic rays in $\M(S)$ and determining precisely how pairs of rays asymptote. We construct an "iterated ED…

多尺度微分的模空间

The moduli space of multi-scale differentials

本文构造了具有指定零点和极点的阿贝尔微分模空间的紧化（多尺度微分模空间），它是带正规交叉边界的复轨形，局部为关联簇紧化显式爆破的正规化，还定义了射影化族得到适当Deligne-Mumford栈，并通过实定向爆破延拓了$\textrm{GL}_2(\textrm{R})$作用。

Connecting $p$-gonal loci in the compactification of moduli space

Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus $g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. We are interested in the branch locus ${\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of $\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in $\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for $g\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\bar{\mathcal{M}_{g}}$. To do so we exhi…