Completely decomposable Jacobian varieties in new genera

Decomposable theta divisors and generic vanishing

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding addition map, and show that the minimum can only be achieved if X is a theta divisor. Conjecturally, the latter happens only on Jacobians of curves and intermediate Jacobians of cubic threefolds. As an application, we prove that nondegenerate generic vanishing subschemes of indecomposable principally polarized abelian varieties are automatically reduced and irreduc…

关于(2,2)-可分解亏格4的Jacobian簇

On (2,2)-decomposable genus 4 Jacobians

研究亏格4 Jacobian簇的(2,2)-可分解性，利用Prym簇和Igusa四次三维体几何刻画该曲线轨迹，并分析了允许2-同源到亏格2 Jacobian簇平方的情形。

可分解阿贝尔 $G$-曲线与特殊子簇

Decomposable abelian $G$-curves and special subvarieties

研究直线上的阿贝尔伽罗瓦覆盖族，证明当一般元素的雅可比簇完全可分解时，其产生特殊子簇的充要条件是一个数值条件。

The arithmetic of genus two curves

Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, moduli spaces…

具有完全可分解雅可比簇的4、5、6次广义Howe曲线

Generalized Howe curves of genus 4, 5, and 6 with completely decomposable Jacobians

本文通过限制在雅可比簇分解为四个椭圆曲线乘积的Howe曲线上，将超特殊曲线的存在性问题简化为椭圆曲线的超奇异性问题，从而更高效地构造了4次超特殊曲线，并推广到5次和6次曲线。

Homotopy theory of simplicial presheaves in completely decomposable topologies

There are two approaches to the homotopy theory of simplicial (pre-)sheaves. One developed by Joyal and Jardine works for all sites but produces a model structure which is not finitely generated even in the case of sheaves on a Noetherian topological space. The other one developed by Brown and Gersten gives a nice model structure for sheaves on a Noetherian space of finite dimension but does not extend to all sites. In this paper we define a class of sites for which a generalized version of the Brown-Gersten approach works.

Generators of Jacobians of Hyperelliptic Curves

This paper provides a probabilistic algorithm to determine generators of the m-torsion subgroup of the Jacobian of a hyperelliptic curve of genus two.

A generalisation of Kani-Rosen decomposition theorem for Jacobian varieties

In this short paper we generalise a theorem due to Kani and Rosen on decomposition of Jacobian varieties of Riemann surfaces with group action. This generalisation extends the set of Jacobians for which it is possible to obtain an isogeny decomposition where all the factors are Jacobians.

广义费马曲线雅可比簇的分解

Decomposition of jacobians of generalized Fermat curves

本文研究了广义费马曲线雅可比簇的分解，扩展了Humbert-Edge曲线的相关研究成果，并增加了因子数量的计数。

关于亏格二曲线雅可比簇的广义Kummer四重簇

On the generalised Kummer fourfold of the Jacobian of a genus two curve

本文构造了亏格二曲线雅可比簇的广义Kummer四重簇的双有理模型，利用三次曲线线性系统解释加法律，证明模型有轻微奇点并存在到四维射影空间的有限分歧覆盖。

Triangle varieties and surface decomposition of hyper-Kähler manifolds

We introduce and study the notion of "surface decomposable" variety, and discuss the possibility that any projective hyper-Kähler manifold is surface decomposable, which would produce new evidence for Beauville's weak splitting conjecture. We show that surface decomposability relates to the Beauville-Fujiki relation, a constraint on the cohomology ring of the variety, and that general varieties with $h^{2,0}\not=0$ are not surface decomposable. We also formalize the notion of triangle variety that is useful to produce surface decomposition. We show the existence of these geometric structures o…

Generators of Jacobians of Genus Two Curves

We prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve.

Fourier-Mukai and autoduality for compactified Jacobians. I

To every singular reduced projective curve X one can associate, following E. Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier-Mukai) transform with kernel the Poincare' sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of D. Arinkin …

Decomposable Subspaces, Linear Sections of Grassmann Varieties, and Higher Weights of Grassmann Codes

Given a homogeneous component of an exterior algebra, we characterize those subspaces in which every nonzero element is decomposable. In geometric terms, this corresponds to characterizing the projective linear subvarieties of the Grassmann variety with its Plucker embedding. When the base field is finite, we consider the more general question of determining the maximum number of points on sections of Grassmannians by linear subvarieties of a fixed (co)dimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to…

关于 $\mathbb{P}^n$ 中一般位置点的完全可分解定义方程

On completely decomposable defining equations of points in general position in $\mathbb{P}^n$

本文从新视角研究线性一般位置有限点集的定义方程，证明理想可由低次分量及指定次数的完全可分解形式生成，并推出秩2二次方程情形，重新证明并改进 Treger 与 Saint-Donat 的结果。

Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus

We consider the generalized Jacobian $\widetilde{J}_0(N)$ of a modular curve $X_0(N)$ with respect to a reduced divisor given by the sum of all cusps on it. When $N$ is a power of a prime $\geq 5$, we exhibit that the group of rational torsion points $\widetilde{J}_0(N)(\mathbb{Q})_{\mathrm{Tor}}$ tends to be much smaller than the classical Jacobian.

雅可比图

Jacobian graphs

本文提出雅可比图，一种谱上与随机图不可区分但局部结构差异显著的正则图家族，基于曲线广义雅可比的几何性质和有限域上的字符和等分布定理。