I. Krichever
We prove that Prym varieties of algebraic curves with two smooth fixed points of involution are exactly the indecomposable principally polarized abelian varieties whose theta-functions provide explicit formulae for integrable 2D Schrödinger equation.
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The classical definition of Prym varieties deals with the unramified covers of curves. The aim of the present paper is to give explicit algebraic descriptions of the Prym varieties associated to ramified double covers of algebraic curves. We make a careful study of the connection with the concept of algebraic completely integrable systems and we apply the methods to some problems of Mathematical Physics.
We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the generic fiber of the momentum map of the periodic Volterra lattice $$\dot a_i=a_i(a_{i-1}-a_{i+1}), \qquad i=1,...,n,\quad a_{n+1}=a_1,$$ is an affine part of a hyperelliptic Prym variety, obtained by removing $n$ translates of the theta divisor, and we conclude that this integrable system is algebraic completely integrable.
We present a Prym construction which associates abelian varieties to vertex-transitive strongly regular graphs. As an application we construct Prym-Tyurin varieties of arbitrary exponent $\geq 3$, generalizing a result by Lange, Recillas and Rochas.
Our main theorem is an improvement of the Criterion of Kanev about Prym-Tyurin varieties induced by correspondences, which includes correspondences with fixed points. We give some examples of Prym--Tyurin varieties using this criterion.
We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.
The space of Hodge cycles of the general Prym variety is proved to be generated by its Neron-Severi group.
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 investigate a family of correspondences associated to étale coverings of degree 3 of hyperelliptic curves. They lead to Prym-Tyurin varieties of exponent 3. We identify these varieties and derive some consequences.
We prove the generic injectivity of the Prym map, sending a double covering of an elliptic curve ramified at r>4 points to its polarized Prym variety. For r=6 the map is birational.
Recovering curves from isogeny classes of Prym varieties
本文提出从Prym簇同源类恢复代数曲线的猜想，并据此分析同源Jacobian及多个同源Prym的曲线对构造。
In this paper, using a generalization of the notion of Prym variety for covers of quasi-projective varieties, we prove a structure theorem for the Mordell-Weil group of the abelian varieties over function fields that are twists of Abelian varieties by Galois covers of irreducible quasi-projective varieties. In particular, the resutls we obtain contribute in the construction of Jacobians (of covers of the projective line) of high rank.
We establish some upper and lower bounds for the number of rational points of Prym varieties over finite fields.
Algebraicity of Hodge classes on some generalized Prym Varieties
本文通过重新审视Chad Schoen在特定Prym簇上构造的代数环，利用无分歧几何类域论证明某些广义Prym簇上Hodge类的代数性。
We show that the Prym variety associated to a triple covering f: Y --> X of curves is principally polarized of dimension > 1, if and only if f is non-cyclic, etale and X is of genus 2. We investigate some properties of these Prym varieties and their moduli.
We prove an extension of the Babbage-Enriques-Petri theorem for semi-canonical curves. We apply this to show that the Prym variety of a generic element of a codimension $k$ subvariety of $\mathcal{R}_{g}$ is not isogenous to another distinct Prym variety, under some mild assumption on $k$.
This article proposes a generalization of tautological rings introduced by Beauville and Moonen for Jacobians. The main result is that, under certain hypotheses, the special subvarieties of Prym varieties are algebraically equivalent and their classes belong to the tautological ring.
The Prym map of type (g,n,r) associates to every cyclic covering of degree n of a curve of genus g, ramified at a reduced divisor of degree r, the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.
We study the birationality (onto its image) of the Abel-Prym morphism associated with a Prym-Tuyrin variety. We use such result to prove that Picard bundles over Prym varieties are simple and moreover they are stable when the Abel-Prym morphism is not birational. As a consequence we obtain that Picard bundles over moduli spaces of stable vector bundles with fixed determinant are simple. We prove that Picard bundles over moduli spaces of rank 2 vector bundles on curves of genus 2 bundles are stable.
The Gauss map for bielliptic Prym varieties
本文研究双椭圆 Prym 簇 theta 除子的 Gauss 映射，完整刻画其次数、识别与 Jacobian 次数相同的情形，并构造低次数例子及 5 维 Andreotti-Mayer 轨迹的完整结果。
We discuss the geometry of the moduli space of Prym varieties. The article is based on series of lectures given in Bedlewo and Luminy. The first section of the paper contains a detailed historical account of the lives of Friedrich Prym and Friedrich Schottky.
We survey Lagrangian fibrations of holomorphic symplectic varieties, both compact and non-compact, whose fibres are Jacobians and Prym varieties.
We prove that Prym varieties are characterized geometrically by the existence of a symmetric pair of quadrisecant planes of the associated Kummer variety. We also show that Prym varieties are characterized by certain (new) theta-functional equations. For this purpose we construct and study a difference-differential analog of the Novikov-Veselov hierarchy.
To any pair of coverings $f_i: X \ra X_i, i = 1,2$ of smooth projective curves one can associate an abelian subvariety of the Jacobian $J_X$, the Prym variety $P(f_1,f_2)$ of the pair $(f_1,f_2)$. In some cases we can compute the type of the restriction of the canonical principal polarization of $JX$. We obtain 2 families of Prym-Tyurin varieties of exponent 6.
Prym varieties that are not isomorphic to Jacobian
通过研究具有素数阶自同构的曲线上的分歧双覆盖的Prym簇，利用伽罗瓦理论构造了非同构于雅可比簇的Prym簇。
Irreducible symplectic varieties via relative Prym varieties
本文利用带反辛对合的 K3 曲面及商曲面线性系构造相对 Prym 簇，给出其为本原或不可约辛簇的判据，并在无穷多维度中产生不可约辛簇序列。
We describe the isotypical decomposition of the Jacobian variety JW of the Galois extension W-->T of any fourfold cover of smooth connected irreducible projective complex curves X-->T, in terms of Prym's of intermediate covers. We also compute the degree of the isogenies involved and as a result we obtain new proofs of the bigonal and trigonal constructions. Furthermore, we give examples of families of Jacobians isogenous to a product of Jacobians and of families of Prym varieties isogenous to the product of elliptic curves.
In this paper we study Prym varieties and their moduli space using the well known techniques of the infinite Grassmannian. There are three main results of this paper: a new definition of the BKP hierarchy over an arbitrary base field (that generalizes the classical one over the complex numbers; a characterization of Prym varieties in terms of dynamical systems, and explicit equations for the moduli space of (certain) Prym varieties. For all of these problems the language of the infinite Grassmannian, in its algebro-geometric version, allows us to deal with these problems from the same point of…
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 $V^2$.
We construct a relative Chow-Kunneth decomposition for a conic bundle over a surface such that the middle projector gives the Prym variety of the associated double covering of the discriminant of the conic bundle. This gives a refinement (up to an isogeny) of Beauville's theorem on the relation between the intermediate Jacobian of the conic bundle and the Prym variety of the double covering.
Chip-firing games, Jacobians, and Prym varieties
该研究对度量图芯片射击游戏理论与热带Prym簇理论做自成体系的介绍，探讨二者和代数对应概念的关联，提出未来研究方向。
Prym--Tyurin varieties coming from intermediate coverings of curves
该研究构造了奇素数指数$p$的普里姆-秋林簇，明确了其生成条件，分析了对应簇的几何并证明了相关映射的一般性有限性。
Pseudoreflections on Prym Varieties
研究g≥5时Prym变种在模空间中的定位，发现存在三个非空不可约族，与Jacobian变种不同，后者在 genus>3 时为空。
Under the condition that the Prym map is injective in characteristic $p$, we prove that the special subvarieties in the moduli space of abelian varieties of dimension $l$ and polarization type $D$, $A_{l,D}$, arising from families of abelian covers of $¶^1$ are of a very restrictive nature. In other words, if the family is one-dimensional or if it contains an eigenspace of certain type for the group action on the cohomology of fibers, then the Shimura varieties arising from such families can only be constructed by the group action of the family.
We describe certain Lagrangian fibrations by Prym varieties on holomorphic symplectic varieties that deform to compactifications of Sp-Hitchin systems.
Prym varieties and cubic threefolds over $\mathbb{Z}$
本文发展了特征2域上的Prym簇与三次三维簇理论，证明了光滑三次三维簇在任意域上的非有理性及Torelli定理，并解决了Deligne关于算术Torelli映射的猜想。
The aim of the paper is twofold. First, some results of Shiota and Plaza-Martin on Prym varieties of curves with an involution are generalized to the general case of an arbitrary automorphism of prime order. Second, the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of the Sato Grassmannian are given.
Let $(C,ι)$ be a stable curve with an involution. Following a classical construction one can define its Prym variety $P$, which in this case turns out to be a semiabelian group variety and usually not complete. In this paper we study the question whether there are ``good'' compactifications of $P$ in analogy to compactified Jacobians. The answer to this question depends on whether we consider degenerations of principally polarized Prym varieties or degenerations with the induced (non-principal) polarization. We describe degeneration data of such degenerations. The main application of our theor…
In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and g>5. We also show that a very generic Prym variety of dimension at least 4 is not isogenous to a Jacobian.
Given Prym-Tyurin varieties of exponent $q$ with respect to a finite group $G$, a subgroup $H$ and a set of rational irreducible representations of $G$ satisfying some additional properties, we construct a Prym-Tyurin variety of exponent $[G:H]q$ in a natural way. We study an example of this result, starting from the dihedral group $\mathbf{D}_p$ for any odd prime $p$. This generalizes the construction of arXiv:math/0412103v2[math.AG] for $p=3$. Finally, we compute the isogeny decomposition of the Jacobian of the curve underlying the above mentioned example.
For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally non-equivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated to such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separation of variables in a series of classical and new algebraic integrable systems linearized on such varieties. We also consider some special cases of the covering C -> E, in particular, when the correspondi…
Let $X_1, ..., X_m$ denote smooth projective curves of genus $g_i \geq 2$ over an algebraically closed field of characteristic 0 and let $n$ denote any integer at least equal to $1+\max_{i=1}^m g_i$. We show that the product $JX_1 \times ... \times JX_m$ of the corresponding Jacobian varieties admits the structure of a Prym-Tyurin variety of exponent $n^{m-1}$. This exponent is considerably smaller than the exponent of the structure of a Prym-Tyurin variety known to exist for an arbitrary principally polarized abelian variety. Moreover it is given by explicit correspondences.
It is well known that the Prym variety of an étale cyclic covering of a hyperelliptic curve is isogenous to the product of two Jacobians. Moreover, if the degree of the covering is odd or congruent to 2 mod 4, then the canonical isogeny is an isomorphism. We compute the degree of this isogeny in the remaining cases and show that only in the case of coverings of degree 4 it is an isomorphism.
We prove that the Jacobian of a general curve C of genus g=2a+1, with g>4, can be realized as a Prym-Tyurin variety for the Brill-Noether curve W^1_{a+2}(C). As a consequence of this result we are able to compute the class of the sum of the secant divisors for the curve C, embedded with a complete linear series g^{a-1}_{3a-2}
Let p:C' -> C be an unramified double covering of irreducible smooth curves and let P be the attached Prym variety. We prove the schematic theta-dual equalities in the Prym variety T(C')=V^2 and T(V^2)=C', where V^2 is the Brill-Noether locus of P associated to p considered by Welters. As an application we prove a Torelli Theorem analogous to the fact that the g-th symmetric product of a curve D of genus g determines the curve.
We study the enumerative geometry of the moduli space R_g of Prym varieties of dimension g-1 (also known as the space of admissible double covers). Our main result is that the compactification of R_g is of general type as soon as g>13. We achieve this by computing the class of two types of cycles on R_g: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym-Green conjecture on syzygies of Prym-canonical curves. In the appendix we show that even though R_g has non-canonic…
It is known that the Jacobian of an algebraic curve which is a 2-fold covering of a hyperelliptic curve ramified at two points contains a hyperelliptic Prym variety. Its explicit algebraic description is applied to some of the integrable Henon-Heiles systems with a non-polynomial potential. Namely, we identify the generic complex invariant manifolds of the systems as a hyperelliptic Prym subvariety of the Jacobian of the spectral curve of the corresponding Lax representation. The exact discretization of the system is described as a translation on the Prym variety.
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 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…
Jacobian varieties with group algebra decomposition not affordable by Prym varieties
本文研究有限群作用诱导的雅可比簇的群代数分解，发现仿射群$\operatorname{Aff}(\mathbb{F}_q)$（除少数例外）不能由Prym簇实现该分解，并在例外情形给出具体分解。
Let $π: Z \ra X$ be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply-connected Lie group $G$. For any dominant weight $λ$ consider the curve $Y = Z/\Stab(λ)$. The Kanev correspondence defines an abelian subvariety $P_λ$ of the Jacobian of $Y$. We compute the type of the polarization of the restriction of the canonical principal polarization of $\Jac(Y)$ to $P_λ$ in some cases. In particular, in the case of the group $E_8$ we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety…
A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GL(n) defined on certain infinite-dimensional Grassmannians is studied. It is proved that every finite-dimensional orbit of the action of traceless elements of these commutative Lie algebras is isomorphic to the Prym variety associated with a morphism of algebraic curves. Conversely, it is shown that every Prym variety can be realized as a finite-dimensional orbit of the action of traceless diagonal el…
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…
We study the $p$-rank stratification of the moduli space of Prym varieties in characteristic $p > 0$. For arbitrary primes $p$ and $\ell$ with $\ell \not = p$ and integers $g \geq 3$ and $0 \leq f \leq g$, the first theorem generalizes a result of Nakajima by proving that the Prym varieties of all the unramified ${\mathbb Z}/\ell$-covers of a generic curve $X$ of genus $g$ and $p$-rank $f$ are ordinary. Furthermore, when $p \geq 5$ and $\ell = 2$, the second theorem implies that there exists a curve of genus $g$ and $p$-rank $f$ having an unramified double cover whose Prym has $p$-rank $f'$ fo…
Let $P \cup P'$ be the two component Prym variety associated to an étale double cover $\tilde{C} \to C$ of a non-hyperelliptic curve of genus $g \geq 6$ and let $|2Ξ_0|$ and $|2Ξ_0'|$ be the linear systems of second order theta divisors on $P$ and $P'$ respectively. The component $P'$ contains canonically the Prym curve $\tilde{C}$. We show that the base locus of the subseries of divisors containing $\tilde{C} \subset P'$ is scheme-theoretically the curve $\tilde{C}$. We also prove canonical isomorphisms between some subseries of $|2Ξ_0|$ and $|2Ξ_0'|$ and some subseries of second order theta …
We study the behavior of theta characteristics on an algebraic curve under the specialization map to a tropical curve. We show that each effective theta characteristic on the tropical curve is the specialization of $2^{g-1}$ even theta characteristics and $2^{g-1}$ odd theta characteristics. We then study the relationship between unramified double covers of a tropical curve and its theta characteristics, and use this to define the tropical Prym variety.
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。
A note on smooth quotients of Prym varieties
该文研究平展二重覆盖的Prym簇上几何起源伪反射，证明g≥4时此类伪反射阶为2，推出g≥5时对应非平凡有限群仅为ℤ/2ℤ或(ℤ/2ℤ)²（后者仅g≤7时存在），改进了已有结果。
Given a tame Galois branched cover of curves pi: X -> Y with any finite Galois group G whose representations are rational, we compute the dimension of the (generalized) Prym variety corresponding to any irreducible representation ρof G. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic.
Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym va…
We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.
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