Counting curves on orbifolds

相同类型弧的计数

Counting arcs of the same type

该研究在数学几何拓扑领域，证明了紧曲面中同型弧的通用计数结果，拓展至带尖点曲面无限弧和轨形弧的计数，核心基于测地流空间上特定测度收敛性的结论。

The Bolza curve and some orbifold ball quotient surfaces

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror $M$ of the orbifold ball quotient $(X,M)$ and by taking the quotient by an involution, we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees $1,2$ and $3$. We also exhibit an arrangement of f…

带有任意数域系数的负曲率轨道空间的贝蒂数

Betti Numbers of Negatively Curved Orbifolds with Coefficients in Arbitrary Fields

该研究证明了负曲率轨道空间的贝蒂数随体积线性增长，并扩展了格罗莫夫的定理至轨道空间，解决了萨梅特关于任意特征的问题。

Pseudoholomorphic curves in four-orbifolds and some applications

The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on symplectic 4-manifolds concerning pseudoholomorphic curves and Seiberg-Witten theory. They form the technical backbone of the proof that a symplectic s-cobordism of elliptic 3-manifolds (with a canonical contact structure on the boundary) is smoothly a product. One interesting feature of the theory is that existence of pseudoholomorphic curves gives certain restricti…

Orbifold points on Prym-Teichmüller curves in genus three

Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for the number of cusps and the Euler characteristic, complete the topological characterisation of Prym-Teichmüller curves in genus 3. Crucial for the determination of the orbifold points is the analysis of families of genus 3 cyclic covers of degree $4$ and $6$, branched over four poi…

Orbifold points on Prym-Teichmüller curves in genus four

For each discriminant $D>1$, McMullen constructed the Prym-Teichmüller curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_D(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau-Nguyen and Möller, complete the topological characterisation of all Prym-Teichmüller curves and determine their genus. The study of orbif…

Menger曲线与一个闭双曲3-轨道流形的球面CR一致化

Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold

本文证明双曲群$G_{6,3}$在$\mathbf{PU}(2,1)$中的离散凸余紧表示的无穷远3-轨道流形是底空间为3-球面、奇异轨迹为$\mathbb{Z}_3$-锥形链环$C(6,-2)$的闭双曲3-轨道流形，回答了Kapovich猜想。

Counting minimal surfaces in negatively curved 3-manifolds

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.

Depth of cohomology support loci for quasi-projective varieties via orbifold pencils

The present paper describes a relation between the quotient of the fundamental group of a smooth quasi-projective variety by its second commutator and the existence of maps to orbifold curves. It extends previously studied cases when the target was a smooth curve. In the case when the quasi-projective variety is a complement to a plane algebraic curve this provides new relations between the fundamental group, the equation of the curve, and the existence of polynomial solutions to certain equations generalizing Pell's equation. These relations are formulated in terms of the depth which is an in…

Counting lattice points in compactified moduli spaces of curves

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli space and constant term the orbifold Euler characteristic of the compactified moduli space. We also prove a recursive formula which can be used to effectively calculate these polynomials.

Counting curve types

Let $S$ be a closed orientable hyperbolic surface, and let $\mathcal{O}(K,S)$ denote the number of mapping class group orbits of curves on $S$ with at most $K$ self-intersections. Building on work of Sapir [16], we give upper and lower bounds for $\mathcal{O}(K,S)$ which are both exponential in $\sqrt{K}$.

Counting intersections of normal curves

A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication. An efficient solution of the word problem for mapping class groups of punctured surfaces is proposed, with efficiency understood in a more restrictive way than the most common one.

Mirzakhani's Curve Counting: From Simple to All

Mirzakhani obtained the asymptotic growth, when $L\to\infty$, of the number of curves in the mapping class group orbit of some given simple curve and with length at most $L$. Years later she extended this result from simple to arbitrary curves. Here we give a short and relative low-tech argument showing how to derive the general result from the one for simple curves.

Counting lattice points in the moduli space of curves

We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.

The Goldman bracket determines intersection numbers for surfaces and orbifolds

In the mid eighties Goldman proved an embedded curve could be isotoped to not intersect a closed geodesic if and only if their Lie bracket (as defined in that work) vanished. Goldman asked for a topological proof and about extensions of the conclusion to curves with self-intersection. Turaev, in the late eighties, asked about characterizing simple closed curves algebraically, in terms of the same Lie structure. We show how the Goldman bracket answers these questions for all finite type surfaces. In fact we count self-intersection numbers and mutual intersection numbers for all finite type orie…

带一个孔的环面上给定字长的精确曲线计数

Exact curve counting of given word length on the once-punctured torus

针对带一个孔的环面，研究人员得出映射类群轨道内给定字长曲线数的精确公式，解决了Cha在2016年提出的相关猜想。

Counting curves on surfaces

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasi-polynomial behaviour. This "elementary curve-counting…