For every integer $g \geq 1$ we define a universal Mumford curve of genus $g$ in the framework of Berkovich spaces over $\mathbb{Z}$. This is achieved in two steps: first, we build an analytic space $\mathcal{S}_g$ that parametrizes marked Schottky groups over all valued fields. We show that $\mathcal{S}_g$ is an open, connected analytic space over $\mathbb{Z}$. Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of $\mathcal{S}_g$, both locally and globally. As a result, we can define the universal Mumford curve $\mathcal{C}_g$ as a relative curve over $\mathcal{S}_g$ such that every Schottky uniformized curve can be described as a fiber of a point in $\mathcal{S}_g$. We prove that the curve $\mathcal{C}_g$ is itself uniformized by a universal Schottky group acting on the relative projective line $\mathbb{P}^1_{\mathcal{S}_g}$. Finally, we study the action of the group $Out(F_g)$ of outer automorphisms of the free group with $g$ generators on $\mathcal{S}_g$, describing the quotient $Out(F_g) \backslash \mathcal{S}_g$ in the archimedean and non-archimedean cases. We apply this result to compare the non-archimedean Schottky space with constructions arising from geometric group theory and the theory of moduli spaces of tropical curves.
This text is an exposition of non-Archimedean curves and Schottky uniformization from the point of view of Berkovich geometry. It consists of two parts, the first one of an introductory nature, and the second one more advanced. The first part is meant to be an introduction to the theory of Berkovich spaces focused on the case of the affine line. We define the Berkovich affine line and present its main properties, with many details: classification of points, path-connectedness, metric structure, variation of rational functions, etc. Contrary to many other introductory texts, we do not assume th…
In this article, we prove that there do not exist stable Schottky-Jacobi forms for the universal Jacobian locus and also prove that there exist non-trivial stable Schottky-Jacobi forms for the universal hyperelliptic locus.
Given a non-trivial complete valued field $K$ with value group $Λ$, we construct a $Λ$-tree space associated to $K$ analog of the Bruhat-Tits tree, and locally finite trees associated to compact subsets of the projective line. We propose a definition of hyperbolic matrix and Schottky group over such field $K$. To any such Schottky group $Γ$, we associate a compact set with an action of $Γ$, such that the quotient graph of the associated tree is a finite graph, and $Γ$ is identified with its fundamental group. Finally explain a method to construct such groups. This results extend the classical …
A Mumford curve of genus g (>1) over a non-archimedean valued field k of positive characteristic has at most max{12(g-1), 2 g^(1/2) (g^(1/2)+1)^2} automorphisms. This bound is sharp in the sense that there exist Mumford curves of arbitrary high genus that attain it (they are fibre products of suitable Artin-Schreier curves). The proof provides (via its action on the Bruhat-Tits tree) a classification of discontinuous subgroups of PGL(2,k) that are normalizers of Schottky groups of Mumford curves with more than 12(g-1) automorphisms. As an application, it is shown that all automorphisms of the …
We construct the universal Mumford curve of given genus as a family of Mumford curves over the deformation space of degenerate curves in the category of arithmetic formal geometry. Furthermore, we give explicit formulas of abelian differentials and their periods of the universal Mumford curve.
A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing moduli and deformation parameters. By considering each generalized Tate curve as a family of degenerating Riemann surfaces, we give explicit formulas of the period isomorphism between its de Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. A remarkable fact is that similar formulas hold also for families of Mumfor…
In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space $\overline{\mathcal{T}}_g$ whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the trop…
The Equivalence of Hilbert and Mumford Stability for Vector Bundles
本文证明了对于平滑曲线上任意秩的向量丛，Hilbert稳定性和Mumford稳定性的概念是一致的. Hilbert稳定性的概念是由Gieseker和Morrison在1984年引入的，他们证明了对于平滑曲线和秩为2的向量丛，它与Mumford稳定性一致。M. Teixidor i Bigas.我们的证明使用了一种新的方法，避免了复杂的计算。我们的结果可以作为Teixidor建议的构造平滑曲线模空间上稳定向量丛的泛模空间的Hilbert稳定紧化的第一步。
In this article we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne-Mumford stable curves $\overline{M}_{_{g, n}}$. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ i.e., over singular curves will remain vector bundles. The total space and the fibers over $\overline{M}_{_{g, n}}$ will have good singularities.
Following Mumford and Chiodo, we compute the Chern character of the derived pushforward $\textrm{ch} (R^\bulletπ_\ast\mathscr{O}(\mathsf{D}))$, for $\mathsf D$ an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.
O'Grady's generalized Franchetta conjecture (GFC) is concerned with codimension 2 algebraic cycles on universal polarized K3 surfaces. In \cite{BL17}, this conjecture has been studied in the Betti cohomology groups. Following a suggestion of Voisin, we investigate this problem in the Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we compute the 4-th DB-cohomology group of universal oriented polarized K3 surface…
Using abelian differentials and periods of the universal Mumford curve, we study the universal expression and asymptotic behavior of tau functions defined for stably degenerating families of algebraic curves with additional data. Furthermore, we apply this study to constructing solutions to the KP hierarchy which are expressed by nonarchimedean theta functions and by mixtures of quasi-periodic solutions with solitons containing real solutions defined on families of M-curves.
We construct modular compactifications of the universal Jacobian stack over the moduli stack of reduced curves with marked points depending on stability parameters obtained out of fixing a vector bundle on the universal curve. When restricted to the locus of stable marked curves, our compactifications are Deligne-Mumford irreducible smooth stacks endowed with projective moduli spaces and, following Esteves approach to the construction of fine compactifications of Jacobians, they parametrize torsion-free rank-1 simple sheaves satisfying a stability condition with respect to the fixed vector bun…
We construct an infinite collection of universal -- independent of $(g,n)$ -- polynomials in the Miller-Morita-Mumford classes $κ_m\in H^{2m}( \overline{\cal M}_{g,n},\bq)$, defined over the moduli space of genus $g$ stable curves with $n$ labeled points. We conjecture vanishing of these polynomials in a range depending on $g$ and $n$.
In this paper we describe compactified universal Jacobians, i.e. compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to Oda-Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne-Mumford stack that compactifies the moduli space of line bundles. We apply this descriptio…
We describe a higher dimensional generalization of Ramanujan's differential equations satisfied by the Eisenstein series $E_2$, $E_4$, and $E_6$. This will be obtained geometrically as follows. For every integer $g\ge 1$, we construct a moduli stack $\mathcal{B}_g$ over $\mathbf{Z}$ classifying principally polarized abelian varieties of dimension $g$ equipped with a suitable additional structure: a symplectic-Hodge basis of its first algebraic de Rham cohomology. We prove that $\mathcal{B}_g$ is a smooth Deligne-Mumford stack over $\mathbf{Z}$ of relative dimension $2g^2 + g$ and that $\mathca…
The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has proved that the Deligne-Mumford compactification $\overline{\mathcal{M}}(0,n)$ of $n$-pointed curves of genus zero can be realized as the Chow quotient $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$. In our recent papers, the constructive description of the orbit space $G_{n,2}/T^n$ has been obtained. In getting this result our notions of the CW-complex of the admissib…
