On the Torelli problem and Jacobian Nullwerte in genus three
Jordi Guardia
给出了由周期矩阵恢复非超椭圆亏格为3的曲线的闭公式，并导出了3维Jacobian Nullwerte之间的一些恒等式。
We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Jacobian Nullwerte, Periods and Symmetric Equations for Hyperelliptic Curves
我们提出了一个解决方案的超椭圆肖特基问题，基于使用的雅可比零和对称模型的超椭圆曲线。这两种成分本身都很有趣，因为第一种提供了可以几何描述的周期矩阵，第二种具有显着的算术性质。
A Torelli theorem for curves over finite fields
我们在阿那贝尔几何的背景下研究了有限域上的双曲曲线及其雅可比行列式。
We prove a generic Torelli theorem for Jacobian elliptic surfaces, provided that the geometric genus is large compared to the irregularity. The result is effective to the extent that defining equations for the base curve are recovered from the period data.
The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism type. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. In Voisin's original proof of the global Torelli theorem for smooth cubic fourfolds the Jacobian ring does not intervene. In this paper we present a proof of the global Torelli theorem for cubic fourfolds that relies on the Jacobian ring and the (derived) global Torelli theorem for K3 surfaces. It emphasizes, once again, the relation between K3 surfaces and smooth cubic fourfolds. More generally, fo…
Mumford and Newstead generalized the classical Torelli theorem to higher rank i.e., a smooth, projective curve $X$ is uniquely determined by the second intermediate Jacobian of the moduli space of stable rank $2$ bundles on $X$, with fixed odd degree determinant. In this article we prove the analogous result in the case $X$ is an irreducible nodal curve with one node. As a byproduct, we obtain the degeneration of the second intermediate Jacobians and the associated Néron model of a family of such moduli spaces.
The Torelli locus and Newton polygons
本文围绕阿贝尔簇模空间中的 Torelli 轨迹展开，综述其几何性质及其在正特征下光滑曲线 Jacobian 的算术不变量与 Newton 多边形研究中的应用。
Torelli groups and Jacobian varieties of non-orientable compact Klein surfaces
紧不可定向Klein曲面的Torelli群是模群的子群，模群由平凡地作用在曲面的第一同调群上的映射类组成。证明了若曲面的亏格至少为$3$，则Torelli群在曲面的Teichmüller空间上自由作用于不动点.这给出了克莱因曲面的Torelli空间在其复二重曲面的Torelli空间中的嵌入。我们还构建真正的环面克莱因曲面，我们称之为雅可比矩阵的表面。我们证明了这个雅可比矩阵同构于复二重矩阵雅可比矩阵实部的一个分量。
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。
Jacobian rings and the infinitesimal Torelli Theorem
本文研究Jacobian环，并识别非退化超曲面在环面中的混合Hodge组件，通过Laurent多项式计算微分的核，从而完成微分Torelli定理的证明，特别是在n≥4的情况下。
Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities
研究了具有孤立奇点的射影超曲面$V$的定义方程的Jacobian理想的合矢与$V$的Torelli性质（Dolgachev-Kapranov意义下）之间的关系.我们特别表明，超曲面与一个小的Tjurina数是Torelli在这个意义上。当$V$是平面曲线，或者更有趣的是，是$P^3$中的曲面时，我们讨论了沿$V$的对数向量场的自反层的稳定性.本文还给出了与1维完全交相关的合合的最小度的一个新的下界。
Torelli theorem via Fourier-Mukai transform
我们表明，傅里叶变换的雅可比曲线互换“$δ$功能”的曲线和θ因子。Torelli定理是一个直接的结果。
Supersingular elliptic surfaces and Infinitesimal Torelli
本文通过产品-商结构证明Katsura的例子是Artin奇点椭圆面，并给出新的证明表明自身例子不满足微分Torelli条件。
In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic subvarieties and the Jacobian locus.
Let $g \geq 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We verify the well known fact that the map induced on tangent spaces by the Torelli map is dual to the multiplication map $\mathrm{Sym}^2 H^0(C, ω_C) \rightarrow H^0(C, ω_C^{\otimes 2})$.
Generic Torelli with denominators for elliptic surfaces
本文证明了在非正则性与几何亏格的一定约束下，非常一般的雅可比椭圆曲面可由其极化有理霍奇结构唯一确定，从而建立了带分母的泛型 Torelli 型结果。
A Generalized Torelli Theorem
给定复数上亏格为$g$的光滑射影曲线$C$，Torelli定理断言对$(J(C),W^{g-1})$确定$C$，其中$W^{g-1}$是$C$在Abel-Jacobi映射下的Jacobian内的$g-1$的第一次对称幂的像.我们证明了在$1\le d\le g-1$的范围内用整数$d$代替$g-1$的定理成立。
Let $g \geq 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We show that the restriction of the Torelli map to the hyperelliptic locus is an immersion in characteristic not $2$. In characteristic $2$, we show the Torelli map restricted to the hyperelliptic locus fails to be an immersion because it is generically inseparable; moreover, the induced map on tangent spaces has kernel of dimension $g-2$ at every point.
A Torelli theorem for graphs via quasistable divisors
本文证明图的拟稳定除子偏序集可完全决定图的双连通分支（至多收缩分离边），从而给出 Torelli 定理的组合版本，并将其推广到热带曲线。
Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $β_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( β_{C'}) = β_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are i…
We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We verify this methods via numerical experiments with curves up to genus 7.
Lazzeri's Jacobian of oriented compact riemannian manifolds
本文的研究对象是F. Lazzeri，（未发表），与每一个紧定向的黎曼流形的尺寸两倍的奇数。我们开始调查的Torelli型问题和Schottky型问题的Lazzeri的雅可比矩阵，特别是我们研究的情况下，环面平坦的度量。此外，我们还研究了Kahler流形的Lazzeri Jacobian及其与其它Jacobian的关系。最后，我们研究了一个丛的Lazzeri的Jacobian。
A Torelli Theorem for Graph Isomorphisms
该研究在图论与代数几何交叉领域，利用易计算除子刻画了图雅可比同构诱导的循环双射可强化为图同构的条件，完善了紧化Torelli映射纤维研究的工具。
Extended Torelli map to the Igusa blowup in genus 6, 7, and 8
在\cite{Namikawa_ExtendedTorelli}中证明了与曲线的雅可比矩阵相关联的Torelli映射$M_g\to A_g$扩展到从稳定曲线的Deligne-Mumford模空间$\bar{M}_g$到Igusa blowup $\bar{A}_g^{\rm cent}$（的归一化）的正则映射。在\cite{AlexeevBrunyate}中发现了一个$g=9$属的反例。在这里，我们证明了扩展映射是正则的所有$g\le8$，从而完全解决了在每个属的问题。
The Oort conjecture on Shimura curves in the Torelli locus of curves
Oort证明了当$g$足够大时，不存在类属$g$曲线的Torelli轨迹中包含的Shimura曲线.本文证明了Mumford型Shimura曲线的Oort猜想，以及参数化主极化$g$ -维阿贝尔簇的Shimura曲线与$g>11$的椭圆曲线的$g$ -重自积同构.证明了亏格为$g>7$的超椭圆曲线的Torelli轨迹中不存在通有包含的Shimura曲线.因此，我们得到了关于光滑亏格- $g$曲线的有限性结果，该曲线具有完全可分解的Jacobian，这与Ekedahl和Serre的一个问题有关。
Generic Torelli theorem for Prym varieties of ramified coverings
本文证明了从亏格为g的具有r个分支点的曲线的双覆盖空间到阿贝尔簇的模空间的Prym映射是泛内射的，若r>6且g>1，r=6且g>2，r=4且g>4，r=2且g>5.我们还表明，一个非常通用的Prym各种尺寸至少为4是不同构的雅可比矩阵。
Residues and Infinitesimal Torelli for Equisingular Curves
本文基于余数理论，为等奇异平面曲线提供微分Torelli定理，并扩展Green--Voisin哲学到奇异和等奇异情况，构建统一的Torelli型问题框架。
Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.
Syzygies and logarithmic vector fields along plane curves
研究了平面曲线$C$定义方程的雅可比理想的合矢与沿$C$的对数向量场层的稳定性、因子$C$的自由度以及$C$的Torelli性质（Dolgachev-Kapranov意义下）之间的关系.我们特别表明，曲线与少量的节点和尖点是Torelli在这个意义上。
Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components
本文针对非埃卡特型三次三重体，通过非交换霍奇理论、布里奇兰模空间理论和周环理论三种途径，证明了库兹涅佐夫分量的等变范畴可决定原三次三重体的同构类，并进一步建立了等变无穷小范畴托雷利定理。
The moduli space of cubic threefolds with a non-Eckardt type involution via intermediate Jacobians
本文研究带非 Eckardt 型对合的三次三维簇，将其周期映射到中间 Jacobian 不变部分，并利用 Prym 簇与 bigonal construction 证明全局 Torelli 定理。
Orbifold points on Prym-Teichmüller curves in genus three
Prym-Teichmüller曲线$W_D(4)$构成了模空间$\mathcal{M}_3$中已知的本原Teichmüller曲线的主要例子。我们确定，对于每个非平方判别$D>1$，在$W_D(4)$的orbifold点的数量和类型。这些结果与Lanneau-Nguyen和Möller关于尖点数和欧拉示性数的公式一起，完成了亏格为3的Prym-Teichmüller曲线的拓扑刻画.确定轨道点的关键是分析$4$和$6$度的属3循环覆盖族，分支于$\mathbb{P}^1$的四个点。作为我们的研究的副产品，我们提供了一个明确的描述这两个家庭的雅可比和Prym-Torelli图像，以及相应的平面的描述。
A non-linear differential equation for the periods of elliptic surfaces
本文推导了椭圆面周期的非线性偏微分方程，并证明了其周期映射的通用微分Torelli定理。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。