Gunther Cornelissen, Fumiharu Kato
A Mumford curve of genus g=5,6,7 or 8 over a non-archimedean field of characteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g-1) automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lame type) are explicitly determined.
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In the present paper, we investigate the question if the skeleton of a Mumford curve of genus two can be tropicalized faithfully in dimension three, i.e. if there exists an embedding of the curve in projective three space such that the tropicalization maps the skeleton of the curve isometrically to its image. Baker, Payne and Rabinoff showed that the skeleton of every analytic curve can be tropicalized faithfully. However the dimension of the ambient space in their proof can be quite large. We will define a map from the skeleton to the tropicalization of the Jacobian, which is an isometry on t…
Non-existence of Shimura curves of Mumford type generically in the non-hyperelliptic locus
研究证明在非双曲曲线的Teichmüller族中，不存在具有严格最大Higgs场的Shimura曲线，特别是Mumford型Shimura曲线不在该族中。
We define Gromov-Witten classes and invariants of smooth proper tame Deligne-Mumford stacks of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth proper tame Deligne-Mumford stack over a Dedekind domain, we prove that the invariants of fibers in different characteristics are the same. We show that genus zero Gromov-Witten invariants define a potential which satisfies the WDVV equation and we deduce from this a reconstruction theorem for genus zero Gromov-Witten invariants in arbitrary characteristic.
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.
The maximal number of automorphisms of a Mumford curve over a non-archimedean valued field of positive characteristic is known. In this note, the unique family of curves that attains this bound in genus not equal to 5,6,7 or 8 is explicitly determined, as is its automorphism group.
Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms
本文在p-adic Mumford曲线的K-有理点上构造自伴算子，生成Feller半群，通过热方程谱的极限点对应微分形式零点，从而从谱恢复亏格，超椭圆情形可显式提取。
Let $X \subseteq \mathbb{P}^r$ be a scroll of codimension $e$ and degree $d$ over a smooth projective curve of genus $g$. The purpose of this paper is to prove a linear Castelnuovo-Mumford regularity bound that reg$(X) \leq d-e+1+g(e-1)$. This bound works over an algebraically closed field of arbitrary characteristic.
Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain …
In the moduli space M_g of genus g Riemann surfaces, consider the locus RM_O of Riemann surfaces whose Jacobians have real multiplication by the order O in a totally real number field F of degree g. If g = 2 or 3, we compute the closure of RM_O in the Deligne-Mumford compactification of M_g and the closure of the locus of eigenforms over RM_O in the Deligne-Mumford compactification of the moduli space of holomorphic one-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of RM_O Boundary strata of RM_O are parameterized by configurations of el…
Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$. The purpose of this note is to show that if $k$ has characteristic zero, then $S$ is finite -- this is in contrast to the situation where $k$ is the algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surfac…
The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications of this space. By allowing to parameterize as well curves with controlled singularities (the so called stable curves), Deligne and Mumford constructed a projective compactification. The properness of this compactification translates into the stable reduction theorem that they prove, its projectivity is a later theorem of Knudsen and Mumford. Th…
Division polynomials in Mumford coordinates
本文提出了一种基于Mumford坐标的有效方法计算除法多项式，通过显式求解3-4阶 torsion 分子在双曲曲线上的多项式，展示了如何直接从除法多项式计算n阶 torsion 分子，或通过求解Jacobi反问题获得。
Oort has conjectured that there do not exist Shimura curves lying generically in the Torelli locus of curves of genus $g \geq 8$. We show that there do not exist one-dimensional Shimura families of semi-stable curves of genus $g\geq 5$ of Mumford type. We also show that there do not exist Shimura curves lying generically in the Torelli locus of hyperelliptic curves of genus $g\geq 8$. The first result proves a slightly weaker form of the conjecture for the case of Shimura curves of Mumford type. The second result proves the conjecture for the Torelli locus of hyperelliptic curves. We also pres…
We use rigid analytic uniformization by Schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a Mumford curve in terms of its genus.
Given a canonical genus three curve $X=\{F=0\}$, we construct, emulating Mumford discussion for hyperelliptic curves, a set of equations for an affine open subset of the jacobian $JX$. We give explicit algorithms describing the law group in $JX$. Finally we introduce a related construction by means of an imbedding of the open set previously described in a Grassmanian variety.
We study the singular iso-level manifold $M_g(0)$ of the genus $g$ Mumford system associated to the spectral curve $y^2=x^{2g+1}$. We show that $M_g(0)$ is stratified by $g+1$ open subvarieties of additive algebraic groups of dimension $0,1,...,g$ and we give an explicit description of $M_g(0)$ in terms of the compactification of the generalized Jacobian. As a consequence, we obtain an effective algorithm to compute rational solutions to the genus $g$ Mumford system, which is closely related to rational solutions of the KdV hierarchy.
A Mumford group is a discontinuous subgroup $Γ$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $Γ$ is a curve of genus 0. As abstract group $Γ$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphi…
We give a sufficient condition for a first order infinitesimal deformation of a curve on a 3-fold to be obstructed. As application we construct generically non-reduced components of the Hilbert schemes of uniruled 3-folds and the Hom scheme from a general curve of genus five to a general cubic 3-fold.
We use the mirror theorem for toric Deligne-Mumford stacks, proved recently by the authors and by Cheong-Ciocan-Fontanine-Kim, to compute genus-zero Gromov-Witten invariants of a number of toric orbifolds and gerbes. We prove a mirror theorem for a class of complete intersections in toric Deligne-Mumford stacks, and use this to compute genus-zero Gromov-Witten invariants of an orbifold hypersurface.
The $\mathbb Z/p$-equivariant cohomology of genus zero Deligne-Mumford space with $1+p$ marked points
本文证明了带$1+p$个标记点的亏格零Deligne-Mumford空间相关纤维化的Serre谱序列在$E_2$页坍塌，并据此得出其$\mathbb Z/p$-等变上同调中的挠元均为非等变元，从而确定唯一的有趣等变运算是量子Steenrod幂运算。
An origami (also known as square-tiled surface) is a Riemann surface covering a torus with at most one branch point. Lifting two generators of the fundamental group of the punctured torus decomposes the surface into finitely many unit squares. By varying the complex structure of the torus one obtains easily accessible examples of Teichmüller curves in the moduli space of Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves. A p-adic origami is defined as a covering of Mumford curves with at most one branch point, where the bottom curve has genus one. A classification o…
A conjecture of Mumford predicts a complete set of relations between the generators of the cohomology ring of the moduli space of rank 2 semi-stable sheaves with fixed odd degree determinant on a smooth, projective curve of genus at least 2. The conjecture was proven by Kirwan. In this article, we generalize the conjecture to the case when the underlying curve is irreducible, nodal. In fact, we show that these relations (in the nodal curve case) arise naturally as degeneration of the Mumford relations shown by Kirwan in the smooth curve case. As a byproduct, we compute the Hodge-Poincare polyn…
There is a well-known stratification of the moduli space $M_g$ of Deligne-Mumford stable curves of genus $g$ by the loci of stable curves with a fixed number $i$ of nodes, where $i \le 3g-3$. The associated moduli stack ${\cal M}_g$ admits an analogous stratification. Our main objects of study are those one-dimensional substacks of the moduli stack, which are the irreducible components of the stratum corresponding to stable curves with exactly $3g-4$ nodes. We describe these substacks explicitely as quotient stacks, and relate them to other, and simpler, moduli stacks of (permutation classes o…
Maximal Mumford Curves from Planar Graphs
本文研究极大Mumford曲线的构造，通过在典范曲线的Hilbert概型中利用平面图曲线的一阶形变，构建了具有全维数的MM曲线族。
We study the Newton polygon jumping locus of a Mumford family in char $p$. Our main result says that, under a mild assumption on $p$, the jumping locus consists of only supersingular points and its cardinality is equal to $(p^r-1)(g-1)$, where $r$ is the degree of the defining field of the base curve of a Mumford family in char $p$ and $g$ is the genus of the curve. The underlying technique is the $p$-adic Hodge theory.
We determine, for all genus $g\geq2$ the Riemann surfaces of genus $g$ with $4g$ automorphisms. For $g\neq$ $3,6,12,15$ or $30$, this surfaces form a real Riemann surface $\mathcal{F}_{g}$ in the moduli space $\mathcal{M}_{g}$: the Riemann sphere with three punctures. The set of real Riemann surfaces in $\mathcal{F}_{g}$ consists of three intervals its closure in the Deligne-Mumford compactification of $\mathcal{M}_{g}$ is a closed Jordan curve.
Mumford-type Shimura curves contained in the Torelli locus
该研究构造了两类亏格4的Shimura曲线族的显式方程，其雅可比簇为定义在有理数域上、自同态代数为整数环且带额外Hodge张量的Mumford型阿贝尔四维簇，是该类阿贝尔簇的首批显式实例。
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 …
A general smooth curve of genus six lies on a quintic del Pezzo surface. In \cite{AK11}, Artebani and Kondō construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne-Mumford stack $\mathfrak{P}_0$ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in $\mathfrak{P}_0$ containing special curves.
There exists on each Teichmüller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmüller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q…
It was conjectured in \cite{Namikawa_ExtendedTorelli} that the Torelli map $M_g\to A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}_g^{\rm cent}$. A counterexample in genus $g=9$ was found in \cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $g\le8$, thus completely solving the problem in every genus.
In the paper, we study the GIT construction of the moduli space of Chow semistable curves of genus 4 in P^3. By using the GIT method developed by Mumford and a deformation theoretic argument, we give a modular description of this moduli space. We classify Chow stable or Chow semistable curves when they are irreducible or nonreduced. Then we work out the case when a curve has two components. Our classification provides some clues to understand the birational map from the moduli space of stable curves of genus 4 to the moduli space of Chow semistable curves of genus 4 in P^3.
We compute the rational Chow class of the locus of genus 2 curves admitting a d-to-1 map to a genus 1 curve, recovering a result of Faber-Pagani when d=2. The answer exhibits quasi-modularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve. Along the way, we give a classification of Harris-Mumford admissible covers of a genus 1 curve by a genus 2 curve, which may be of independent interest. As an application of the main calculation, we compute the number of d-elliptic curves in a very general family of genus 2 curves obtained by fixing five branch points of the…
Let $\mathcal{M}$ be the Artin-Mumford curve over the finite prime field $\mathbb{F}_p$ with $p>2$. By a result of Valentini and Madan, $\mbox{Aut}_{\mathbb{F}_p}(\mathcal{M})\cong H$ with $H=(C_p\times C_p)\rtimes D_{p-1}$. We prove that if $\mathcal{X}$ is an algebraic curve of genus $g=(p-1)^2$ such that $\mbox{Aut}_{\mathbb{F}_p}(\mathcal{X})$ contains a subgroup isomorphic to $H$ then $\mathcal{X}$ is birationally equivalent over $\mathbb{F}_p$ to the Artin-Mumford curve $\mathcal{M}$.
Let $\mathbb{F}_q$ be the finite field of order $q=p^h$ with $p>2$ prime and $h>1$, and let $\mathbb{F}_{\bar{q}}$ be a subfield of $\mathbb{F}_q$. From any two $\bar{q}$-linearized polynomials $L_1,L_2 \in \overline{\mathbb{F}}_q[T]$ of degree $q$, we construct an ordinary curve $\mathcal{X}_{(L_1,L_2)}$ of genus $(q-1)^2$ which is a generalized Artin-Schreier cover of the projective line $\mathbb{P}^1$. The automorphism group of $\mathcal{X}_{(L_1,L_2)}$ over the algebraic closure $\overline{\mathbb{F}}_q$ of $\mathbb{F}_q$ contains a semidirect product $Σ\rtimes Γ$ of an elementary abelian …
In this paper we compute the generating function for the Euler characteristic of the Deligne-Mumford compactification of the moduli space of smooth n-pointed genus 2 curves.
We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a degeneration of a curve of class $(3,3)$. We prove that the compactified moduli space is a smooth Deligne--Mumford stack with 4 boundary components. We relate it to the moduli space of genus 4 curves; we show that it compactifies the blow-up of the hyperelliptic locus. We also relate it to a compactification of the Hurwitz space of triple coverings of $\math…
The automorphism group of $\overline{M}_{g,n}$
本文证明在 $2g-2+n\geq 3$ 时，稳定曲线模叠及其粗模空间的自同构群同构于对称群 $S_n$，并计算了其余情形。
Mumford representation and Riemann Roch space of a divisor on a hyperelliptic curve
本文研究带 Weierstrass 无穷远点的超椭圆曲线，证明可由零次除子的 Mumford 表示直接构造相应 Riemann–Roch 空间的基，并用于得到 Goppa 码生成矩阵。
For a power $q$ of a prime $p$, the Artin-Schreier-Mumford curve $ASM(q)$ of genus $g=(q-1)^2$ is the nonsingular model $\mathcal{X}$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c,\, c\neq 0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin-Schreier-Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|<1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{g}(\sqrt{g}+1)^2$, far away from the Hurwitz bound $84(g-1)$ valid in zero characteristic. In this…
In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g=3. Furthermore, we determine the part of the orbifold cohomology of the Deligne-Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of M_3.
We first give an exposition of how the Polyakov path integral for the bosonic string produces a natural mapping class group invariant measure, $d(Poly)$, on the Teichmüller space of Riemann surfaces of each fixed genus. The description of $d(Poly)$ via the Mumford isomorphisms for determinant bundles over Teichmüller space is also explained. We then report on our recent results with I.Biswas and D.Sullivan, where we succeed in coherently fitting together this Polyakov-Mumford construction over the universal direct limit, $T_\infty$, of all the finite genus Teichmüller spaces. The virtual autom…
We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two recursions, namely the string equation and the dilaton equation. We establish, for each integer m>0, an analogous pair of recursions which determine these intersection numbers on the moduli spaces of m-stable pointed curves of genus one.
One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only im…
The supermoduli space of genus zero SUSY curves with Ramond punctures
该研究构造了带Ramond穿孔的零亏格超黎曼曲面超模空间及无SUSY结构的零亏格超曲线模空间的显式商，分别证明二者是Deligne-Mumford超栈与代数超栈。
We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne-Mumford stability. For every pair of integers 0<m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne-Mumford stack. We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks. In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulti…
Oort has conjectured that there do not exist Shimura curves contained generically in the Torelli locus of genus-$g$ curves when $g$ is large enough. In this paper we prove the Oort conjecture for Shimura curves of Mumford type and Shimura curves parameterizing principally polarized $g$-dimensional abelian varieties isogenous to $g$-fold self-products of elliptic curves for $g>11$. We also prove that there do not exist Shimura curves contained generically in the Torelli locus of hyperelliptic curves of genus $g>7$. As a consequence, we obtain a finiteness result regarding smooth genus-$g$ curve…
The cohomology ring of the moduli space of stable holomorphic vector bundles of rank n and degree d over a Riemann surface of genus g>1 has a standard set of generators when n and d are coprime. When n=2 the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when n>2 and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.
We study algebraic (Artin) stacks over $\bar{\mathcal M}_g$ giving a functorial way of compactifying the relative degree $d$ Picard variety for families of stable curves. We also describe for every $d$ the locus of genus $g$ stable curves over which we get Deligne-Mumford stacks strongly representable over $\bar{\mathcal M}_g$.
A well-known principle of Mumford asserts that all moduli spaces of curves are varieties of general type, except a finite number of cases that occur for relatively small genus, when these varieties tend to be uniruled. In all known cases, the transition from uniruledness to being of general type is quite sudden. For the first time, we show that naturally defined moduli spaces of curves can have intermediate birational type. We show that the moduli space of even spin curves of genus 8 is of Calabi-Yau type. Then we prove that the 41-dimensional moduli space M_{11, 11} of 11-pointed curves of ge…
The Gromov-Witten theory of Deligne-Mumford stacks is a recent development, and hardly any computations have been done beyond 3-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed by hand, determine all genus 0 invariants of the stack P^2_{D,2}. Here D is a smooth plane curve and P^2_{D,2} is locally isomorphic to the stack quotient [U/(Z/(2))], where U -> V \subset P^2 is a double cover branched along D \cap V. The introduction discusses an enumerative application of these invariants.
A uniform quantitative Manin-Mumford theorem for curves over function fields
本文提出一个关于函数域上曲线的统一定量Manin-Mumford定理，利用Zhang配对、算术Hodge指数定理和Elkies下界，证明了曲线在雅可比嵌入中的torsion点数量上限及Bogomolov型结果。
Let M_g be the moduli space of smooth curves of genus g >= 3, and \bar{M}_g the Deligne-Mumford compactification in terms of stable curves. Let \bar{M}_g^{[1]} be an open set of \bar{M}_g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a Q-divisor D on \bar{M}_g is nef over \bar{M}_g^{[1]}, that is, (D . C) >= 0 for all irreducible curves C on \bar{M}_g with C \cap \bar{M}_g^{[1]} \not= \emptyset.
We investigate the moduli of genus 10 curves that are endowed with a faithful action of the icosahedral group $\mathcal{A}_5$. We show among other things that this has the structure of a Deligne-Mumford stack whose underlying coarse moduli space essentially consists of two copies of the pencil of plane sextics that was introduced by Winger in 1924, but with the unique unstable member (a triple conic) replaced by a smooth non-planar curve. The orbifold defined by any member has genus zero and comes with 4 orbifold points. We show that by numbering the points, we get a fine moduli space whose ba…
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 relat…
After a short exposition of the basic properties of the tautological ring of the moduli space of genus g Deligne-Mumford stable curves with n markings, we explain three methods of detecting non-tautological classes in cohomology. The first is via curve counting over finite fields. The second is by obtaining length bounds on the action of the symmetric group S_n on tautological classes. The third is via classical boundary geometry. Several new non-tautological classes are found.
This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of $μ_{r}$-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual fundamental classes. Our result verifies the genus 0 part of the so-called decomposition conjecture which compares Gromov-Witten theory of étale gerbes with that of the bases. We also verify this conjecture in all genera for toric gerbes over toric Deligne-Mumford stacks.
Belyi pair for the orientation cover of $ \overline{{\mathcal M}_{0,5}^{\mathbb R}}$
本文针对具有5个标记点的零亏格实曲线模空间Deligne-Mumford紧化的定向覆盖，计算其亏格4儿童画所对应的Belyi对，并发现该曲线为著名的Bring曲线。
We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular Kähler-Einstein metrics when ${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsen compactification $\overline{\mathcal{M}}_{0,n}$. As a consequence, we obtain a formula computing the volumes of ${\mathcal{M}}_{0,n}$ with respect to these metrics using intersection of boundary divisors of $\overline{\mathcal{M}}_{0,n}$. In the case of rational weights, following an idea of Y. Kawamata, we show that these metrics actually represent the first Chern class of some lin…
We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are t…
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