Cohomological support loci for Abel-Prym curves
Sebastian Casalaina-Martin, Martí Lahoz, Filippo Viviani
对于包含在Prym簇中的Abel-Prym曲线，我们确定了它的扭曲理想层的上同调支撑轨迹和它的θ-对偶的维数.
For an Abel-Prym curve contained in a Prym variety, we determine the cohomological support loci of its twisted ideal sheaves and the dimension of its theta-dual.
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Two-pointed Prym-Brill-Noether Loci and coupled Prym-Petri theorem
本文建立了具有特殊消没的双点Prym-Brill-Noether轨迹，利用D型vexillary退化轨迹的K-理论公式确定其K-理论类，并借助耦合Prym-Petri映射证明了双点Prym-Petri定理，推广了前人的无点和带点情形。
Given the Prym variety of an étale double cover one can define analogues of the classical Brill-Noether loci on Jacobians of curves. Recent work by Lahoz and Naranjo shows that the Brill-Noether locus $V^2$ completely determines the covering. In this paper we describe the singular locus and the irreducible components of $V^2$.
Prym loci of branched double coverings and generalized Andreotti-Mayer loci
本文将安德烈奥蒂-迈耶轨迹推广至非主极化阿贝尔簇模空间，证明分支二重覆盖的普里姆轨迹是广义安德烈奥蒂-迈耶轨迹的不可约分支。
In this paper we give a bound on the dimension of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. This improves the already known bounds. The idea is to adapt the techniques introduced by the authors in collaboration with A. Ghigi and G. P. Pirola for the Torelli map to the case of the Prym maps of (ramified) double covers.
We study Shimura curves of PEL type in the space of polarised abelian varieties $A^δ_p$ generically contained in the ramified Prym locus. We generalise to ramified double covers, the construction done in [10] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is ${\mathbb P}^1$. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Pry…
In this paper we construct Shimura subvarieties of dimension bigger than one of the moduli space of polarised abelian varieties of a given dimension, which are generically contained in the Pym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely to use families of Galois covers of the projective line. The case of abelian covers is treated in detail, since in this case it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield…
Secant loci on moduli of Prym varieties
本文将 Lazarsfeld 关于一般极化 K3 曲面曲线无意外割线的结果推广到 Prym 情形，证明标准与非标准一般 Nikulin 曲面上曲线的 Prym-典范嵌入无意外割线，并据此确定奇数亏格稳定 Prym 曲线模空间中泛差除子的类。
Prym-Brill-Noether theory for ramified double covers
该研究开展分歧双复叠的Prym-Brill-Noether理论研究，改进并拓展了多位学者的相关经典结果，还证明一般Du Val曲线是Prym-Brill-Noether一般的。
Euler Characteristics of Brill-Noether Loci on Prym Varieties
本文提出 Prym 簇上带点 Brill-Noether 轨迹的连通 K-理论类公式，推广 Concini 与 Pragacz 的结果，并据此计算其全纯欧拉示性数。
We use Young tableaux to compute the dimension of $V^r$, the Prym-Brill-Noether locus of a folded chain of loops of any gonality. This tropical result yields a new upper bound on the dimensions of algebraic Prym-Brill-Noether loci. Moreover, we prove that $V^r$ is pure-dimensional and connected in codimension $1$ when $\dim V^r \geq 1$. We then compute the first Betti number of this locus for even gonality when the dimension is exactly $1$, and compute the cardinality when the locus is finite and the edge lengths are generic.
Hodge integrals on admissible covers and tautological projections of Prym classes
本文提出一种算法，利用可容许覆盖模空间上三个Hodge丛的比较公式，计算Prym轨迹类的tautological投影，并应用于计算Prym映射次数及证明低亏格下雅可比类的不可整除性。
Intersection theory and Siegel-Veech constants for Prym eigenform loci in $Ω\mathcal{M}_3(2,2)^{\rm odd}$
本文研究了Prym特征形式在特定模空间中的鞍连接的Siegel-Veech常数，涉及交集理论和实二次域的非平方判别式特征。
We study the rational Picard group of the projectivized moduli space of holomorphic n-differentials on complex genus g stable curves. We define (n - 1) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the univer…
On algebraic equivalences among the 27 Abel-Prym curves on a generic abelian 5-fold
本文证明了在一个5维的广义主极化阿贝尔簇上，27条Abel-Prym曲线的代数等价的$\mathbb{Q}$ -向量空间的维数为20。
This paper is devoted to the classification of connected components of Prym eigenform loci in the strata H(2,2)^odd and H(1,1,2) in the Abelian differentials bundle in genus 3. These loci, discovered by McMullen are GL^+(2,R)-invariant submanifolds (of complex dimension 3) that project to the locus of Riemann surfaces whose Jacobian variety has a factor admitting real multiplication by some quadratic order Ord_D. It turns out that these subvarieties can be classified by the discriminant D of the corresponding quadratic orders. However there algebraic varieties are not necessarily irreducible. …
Tropical covers, tropical abelian varieties and Prym varieties
本文研究热带普里姆簇，基于无分支伽罗华循环覆盖的热带曲线，通过循环伽罗华群在热带阿贝尔簇上的作用进行分析，并扩展了双覆盖的阿贝尔-普里姆映射。
We construct pointed Prym-Brill-Noether varieties parametrizing line bundles assigned to an irreducible étale double covering of a curve with a prescribed minimal vanishing at a fixed point. We realize them as degeneracy loci in type D and deduce their classes in case of expected dimension. Thus, we determine a pointed Prym-Petri map and prove a pointed version of the Prym-Petri theorem implying that the expected dimension holds in the general case. These results build on work of Welters and De Concini-Pragacz on the unpointed case. Finally, we show that Prym varieties are Prym-Tyurin varietie…
Let $C$ be a smooth non rational projective curve over the complex field $\mathbb{C}$. If $A$ is an abelian subvariety of the Jacobian $J(C)$, we consider the Abel-Prym map $φ_A : C \rightarrow A$ defined as the composition of the Abel map of $C$ with the norm map of $A$. The goal of this work is to investigate the degree of the map $φ_A$ in the case where $A$ is one of the components of an isotypical decomposition of $J(C)$. In this case we obtained a lower bound for $\mathrm{deg}(φ_A)$ and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of A…
The tropical Abel--Prym map
本文证明双曲度量图自由二重覆盖的热带Abel--Prym映射是度数2的调和映射，给出部分逆命题，分析源图双曲性对映射单射性的影响，刻画Abel--Prym图性质并统计相关覆盖数量。
Polarizations of Prym varieties for Weyl groups via abelianization
设$π: Z \ra X$是光滑射影曲线的Galois覆盖，其Galois群是单连通李群$G$的Weyl群.对于任何主导权重$λ$，考虑曲线$Y = Z/\Stab(λ)$。Kanev对应定义了$Y$的雅可比矩阵的一个阿贝尔子簇$P_λ$。在某些情况下，我们计算了$\Jac(Y)$的正则主极化限制到$P_λ$的极化类型.特别地，在组$E_8$的情况下，我们获得了Prym-Tyurin变种的家族。主要思想是利用Donagi-Prym簇的阿贝尔化映射到曲线$X$上的主$G$ -丛的模栈。
On Picard bundles over Prym varieties
我们研究的双理性（到它的形象）的阿贝尔-Prym态射与Prym-Tuyrin品种。我们利用这个结果证明了当Abel-Prym态射不是双有理时，Prym簇上的Picard丛是简单的，并且是稳定的.由此得到行列式不动的稳定向量丛的模空间上的Picard丛是单的.证明了亏格为2的曲线上的秩为2的向量丛模空间上的Picard丛是稳定的.
On the Existence of Shimura curves in the Prym locus of abelian covers of projective line
该研究借助希格斯丛理论与Viehweg-Zuo刻画，证明了g≥8时射影直线全分歧Z_{2p}等类型阿贝尔覆盖的Prym轨迹中不存在非紧Shimura曲线，p为不小于5的素数。
The rationality of ineffective spin genus-4 thetanull loci
本文证明了由亏格4且thetanull消失的曲线C与无效thetacharacteristic θ构成的对[C,θ]所给出的除子是有理簇，并得出Prym模空间中的类似除子也是有理的。
On the Prym map for cyclic covers of genus two curves
本研究针对亏格二曲线的非分歧循环覆盖，证明Prym映射恰在双椭圆覆盖轨迹上分歧，通过关联带循环极化的阿贝尔曲面解释映射余微分，并给出Ramanan关于高次亏格二循环覆盖非超椭圆结论的新证明。
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
Variétés de Prym associées aux revêtements n-cycliques d'une courbe hyperelliptique
设H是一条超椭圆曲线，f：C --> H是一个n次循环根覆盖，对应于Pic^0（H）中的一个n阶线丛.证明了当n不能被4整除时，Prym簇P=Prym（C / H）作为阿贝尔簇同构于Jacobian的乘积.
We study the Prym varieties arising from étale cyclic coverings of degree 7 over a curve of genus 2. These Prym varieties are products of Jacobians JY x JY of genus 3 curves Y with polarization type D=(1,1,1,1,1,7). We describe the fibers of the Prym map between the moduli space of such coverings and the moduli space of abelian sixfolds with polarization type D, admitting an automorphism of order 7.
Polarizations of Prym Varieties of pairs of coverings
对于光滑射影曲线的任何一对覆盖$f_i: X \ra X_i, i = 1,2$，人们可以将雅可比$J_X$的交换子簇，对$(f_1,f_2)$的Prym簇$P(f_1,f_2)$联系起来。在某些情况下，我们可以计算$JX$的正则主极化的限制类型。得到了2个指数为6的Prym-Tyurin簇。
Branching of Hitchin's Prym cover for SL(2)
证明了从谱曲线的雅可比矩阵到秩为2的稳定丛的模的映射是沿一个不可约因子的一般简单分支。这一观察结果证伪了最近一篇论文中提出的“SU（2）WZW连接的阿贝尔化”的关键步骤[Yoshida，Annals 2006]
We prove that a very general hypersurface of bidegree (2, n) in P^2 x P^2 for n bigger than or equal to 2 is not stably rational, using Voisin's method of integral Chow-theoretic decompositions of the diagonal and their preservation under mild degenerations. At the same time, we also analyse possible ways to degenerate Prym curves, and the way how various loci inside the moduli space of stable Prym curves are nested. No deformation theory of stacks or sheaves of Azumaya algebras like in recent work of Hasset-Kresch-Tschinkel is used, rather we employ a more elementary and explicit approach via…
Hyperelliptic genus 3 curves with involutions and a Prym map
本文刻画了含两个额外对合的亏格3超椭圆曲线，给出显式方程，并证明在固定椭圆商曲线时Prym映射为2:1，同时构造了不与对偶同构的极化阿贝尔曲面族。
Pryms of $\mathbb{Z}_3\times\mathbb{Z}_3$ coverings of genus 2 curves
本文研究了Z_3×Z_3覆盖的二阶曲线的Prym变种和映射，证明了Prym-Teorelli定理成立，并探讨了任意交换群G的覆盖情况。
We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.
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。
Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space A_6 of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general ppav of dimension 6 is a Prym-Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E_6 lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E_6-covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge-Hurwitz ei…
Products of Jacobians as Prym-Tyurin varieties
设$X_1, ..., X_m$表示特征为0的代数闭域上亏格为$g_i \geq 2$的光滑射影曲线，$n$表示至少等于$1+\max_{i=1}^m g_i$的任意整数.我们证明了相应雅可比簇的乘积$JX_1 \times ... \times JX_m$具有指数为$n^{m-1}$的Prym-Tyurin簇的结构.这个指数是相当小的指数的结构的Prym-Tyurin品种已知存在的任意主要极化的阿贝尔簇。此外，它是由明确的对应。
The Equations of Singular Loci of Ample Divisors on (Subvarieties of) Abelian Varieties
本文考虑复阿贝尔簇上的一个样本线丛的线性系$|L|$中因子的奇异轨迹所对应的理想层。我们证明了一个有效的结果，其（连续）的整体生成后，适当的扭曲权力的$L$。此外，我们表明，类似的结果举行的复阿贝尔簇的子簇。
We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.
Inversion of the Abel--Prym map in presence of an additional involution
本文研究了存在额外对合时Abel--Prym映射的反演问题，提出了第一型对合对的条件，并利用Prym theta函数定义了Jacobi反演的类比，给出了Hitchin和Kovalewski系统等曲线实例。
In a previous paper, the authors proved that the Prym variety of any non-cyclic etale triple cover of a smooth curve of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map of a certain moduli space of admissible $S_3$-covers of genus 7 to the moduli space of principally polarized abelian surfaces. The main result is that this map is finite surjective of degree 10.
This is the second part of a paper describing a new concept of separation of variables applied to the classical Clebsch integrable case. The quadratures obtained in Part I (also uploaded in arXiv.org) lead to a new type of the Abel map which contains Abelian integrals on two different algebraic curves. Here we interprete it as from the product of the two curves to the Prym variety of one of them, show that the map is well defined although not a bijection. We analyse its properties and formulate a new extention of the Riemann vanishing theorem, which allows to invert the map in terms of theta-f…
Prym-Tyurin varieties via Hecke algebras
令$G$表示有限群，$π: Z \to Y$表示具有伽罗瓦群的光滑射影曲线的伽罗瓦覆盖$G$。对于$G$的每个子群$H$，存在相应的Hecke代数$\mathbb{Q}[H \backslash G/H]$对曲线$X = Z/H$的雅可比矩阵的标准作用。对于$G$的每一个有理不可约表示$\mathcal{W}$，我们在Hecke代数中关联一个幂等元，它导致曲线$X$的对应，从而导致雅可比矩阵$JX$的交换子簇$P$。给出了关于$\mathcal{W}$，$H$的充分条件，以及$G$对$Z$的作用，从而暗示$P$是Prym-Tyurin簇.用这种方法得到了许多新的任意指数的Prym-Tyurin簇族。
In the paper we consider the Jacobian locus $\overline{J_g}$ and the Prym locus $\overline{P_{g+1}}$, in the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$, for $g\geq 7$, and we study the extrinsic geometry of $\overline{J_g}\subset \overline{P_{g+1}}$, under the inclusion provided by the theory of generalized Prym varieties as introduced by Beauville. More precisely, we study certain geodesic curves with respect to the Siegel metric of $A_g$, starting at a Jacobian variety $[JC]\in A_g$ of a curve $[C]\in M_g$ and with direction $ζ\in T_{[JC]}J_g$. We prove th…
We prove an analogue of Kirchhoff's matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebast…
Abelian varieties in Brill--Noether loci
本文改进了Abramovich-Harris和Debarre-Fahlaoui的结果，给出了亏格为g的曲线C的完全分类，使得严格包含在C的Jacobian J（C）中的Brill-Noether轨迹W^ s_d（C）包含一个簇Z，该簇Z在由包含在J（C）中的正维交换子簇A的元素平移下稳定，且使得dim（Z）=d-dim（A）-2s，即，最大的可能是这样的Z。
A characterization of Prym varieties
证明了具有两个光滑对合不动点的代数曲线的Prym簇是不可分解的主极化阿贝尔簇，其θ函数为可积的二维薛定谔方程提供了显式公式.
Inversion of the Abel--Prym map for real curves with involutions
本文研究具有自映射的实曲线，详细阐述了普里姆θ函数的对称性，填补了现有文献中的空白。
Lagrangian loci in moduli of abelian surfaces
本文利用 Donagi-Markman 三次条件、二元三次形式分析及 Cauchy-Kowalevski 定理，刻画阿贝尔曲面模空间中可由拉格朗日纤维化产生的曲面芽与曲线芽。
Flat Vector Bundles on Very General Curves and Codimension of Non-Abelian Hodge Loci
本文基于复Hodge结构的变体秩和水平，界定了非阿贝尔Hodge领域在相对de Rham模空间上的码维，并通过Landesman和Litt的成果，研究了等monodromy叶状结构及代数monodromy群的李代数秩。
Abelian Varieties with Quaternion Multiplication
本文利用Prym构造研究了具有四元数群作用的低维阿贝尔簇。在特殊情况下，我们描述的志村品种参数化等阿贝尔簇，以及映射到这个志村品种从自然参数空间的四元数阿贝尔簇。我们的描述是基于模的立方三倍与九个节点，我们研究的一些细节。
We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym-Brill-Noether locus for generic unramified double covers of curves with fixed even gonality on the base. Our methods also give a new proof of the classical Prym-Brill-Noether Theorem for generic unramified double…
In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In the first case the proof is constructive and can be extended to …
Cyclic Coverings of genus 2 curves of Sophie Germain type
本文研究亏格 2 曲线循环非分歧覆盖能否由其 Prym 簇重构，并结合 GL_2 型 Abel 簇算术结果与 theta 对偶证明 Sophie Germain 素数情形下一般 Prym Torelli 定理成立。
We provide a bound on the $Θ$-regularity of an arbitrary reduced and irreducible curve embedded in a polarized abelian variety in terms of its degree and codimension. This is an "abelian" version of Gruson-Lazarsfeld-Peskine's bound on the Castelnuovo--Mumford regularity of a non-degenerate curve embedded in a projective space. As an application, we provide a Castelnuovo type bound for the genus of a curve in a (non necessarily principally) polarized abelian variety. Finally, we bound the $Θ$-regularity of a class of higher dimensional subvarieties in Jacobian varieties, i.e. the Brill-Noether…
The trigonal construction and the second moment of the tropical Prym variety
本文利用热带三角构造计算亏格不超过4的热带曲线双重覆盖的Prym簇第二矩，结果由多项式与分段多项式项组成，并将后者与Prym--Torelli映射的扩展问题相关联。
We construct a three-parameter family of non-hyperelliptic and bielliptic plane genus-three curves whose associated Prym variety is two-isogenous to the Jacobian variety of a general hyperelliptic genus-two curve. Our construction is based on the existence of special elliptic fibrations with the section on the associated Kummer surfaces that provide a simple geometric interpretation for the rational double cover induced by the two-isogeny between the abelian surfaces.
Cohomology of local systems on loci of d-elliptic abelian surfaces
我们考虑了$M_2$中d-椭圆曲线的轨迹和$A_2$中d-椭圆曲面的轨迹。我们展示了如何描述这些轨迹的一个产品的模曲线的concurents可以用来计算上同调的自然局部系统，无论是混合霍奇结构和$\ell$ -进伽罗瓦表示。特别研究了d=2的情形，计算了Hodge结构的Grothendieck群中亏格为2的n点双椭圆曲线的模空间的欧拉示性数.
Putting the p back in Prym
本文填补了特征2下Prym簇缺乏系统性论述的空白，主要贡献是分类了特征2中产生Prym簇的曲线分歧覆盖，并发展了相对设定下的相关理论及扩展了Welters准则。
On Brill-Noether loci over Quot schemes and a Torelli theorem
通过在相关极化Quot格式的导出范畴中工作，并在其上定义Brill-Noether轨迹和Abel-Jacobi映射，证明了光滑射影曲线的一个非交换Torelli型结果.
In this text we prove that if an abelian variety $A$ admits an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $Θ_A$ to be the divisor $Θ_C\cap A$, where $Θ_C$ denotes the theta divisor of $J(C)$, then the embedding of $Θ_A$ into $A$ induces an injective push-forward homomorphism (under certain conditions) at the level of Chow groups. We show that this is the case for every Prym varietiy arising from an unramified double cover of smooth projective curves. As a consequence we prove that there does not exist a universal codimension two cycle on the product of a v…
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