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We derive a local formula for the parity of the 2 ∞ -Selmer rank of Jacobians of curves of genus 2 or 3 which admit an unramified double cover. We give an explicit example to show how this local formula gives rank parity predictions against which the 2-parity conjecture may be tested. Our results yield applications to the parity conjecture for semistable curves of genus 3. [ABSTRACT FROM AUTHOR]

We undertake a study of conic bundle threefolds p: X → W over geometrically rational surfaces whose associated discriminant covers Δ → Δ ⊂ W are smooth and geometrically irreducible. We first show that the structure of the Galois module CH² Xk of rational equivalence classes of curves is captured by a group scheme that is a generalization of the Prym variety of Δ → Δ. This generalizes Beauville's result that the algebraically trivial curve classes on Xk are parametrized by the Prym variety. We apply our structural result on curve classes to study the refined intermediate Jacobian torsor (IJT) obstruction to rationality introduced by Hassett-Tschinkel and Benoist-Wittenberg. The first case of interest is where W = P² and Δ is a smooth plane quartic. In this case, we show that the IJT obstruction characterizes rationality when the ground field has less arithmetic complexity (precisely, when the 2-torsion in the Brauer group of the ground field is trivial). We also show that a hypothesis of this form is necessary by constructing, over any k ⊂ R, a conic bundle threefold with Δ a smooth quartic where the IJT obstruction vanishes, yet X is irrational over k. [ABSTRACT FROM AUTHOR]

In the moduli space Rg$\mathcal {R}_g$ of double étale covers of curves of a fixed genus g, the locus formed by covers of curves with a semicanonical pencil consists of two irreducible divisors Tge$\mathcal {T}^e_g$ and Tgo$\mathcal {T}^o_g$. We study the Prym map on these divisors, which shows significant differences between them and has a rich geometry in the cases of low genus. In particular, the analysis of T5o$\mathcal {T}^o_5$ has enumerative consequences for lines on cubic threefolds. [ABSTRACT FROM AUTHOR]

An algebraic subvariety Z of A g is totally geodesic if it is the image via the natural projection map of some totally geodesic submanifold X of the Siegel space. We say that X is the symmetric space uniformizingZ. In this paper we determine which symmetric space uniformizes each of the low genus counterexamples to the Coleman-Oort conjecture obtained studying Galois covers of curves. It is known that the counterexamples obtained via Galois covers of elliptic curves admit two fibrations in totally geodesic subvarieties. The second result of the paper studies the relationship between these fibrations and the uniformizing symmetric space of the examples. [ABSTRACT FROM AUTHOR]

We show that the moduli space R ‾ 9 of Prym curves of genus 9 is uniruled. This is the largest genus for which such a result is known to hold. [ABSTRACT FROM AUTHOR]

We study Shimura curves of PEL type in the space of polarized abelian varieties A p δ generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math.21(2) (2019) 1850009] 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 ℙ 1 . Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci. [ABSTRACT FROM AUTHOR]

The aim of this paper is to demonstrate the rich interaction between the Kowalewski-Painlev\'{e} analysis, the properties of algebraic completely integrable (a.c.i.) systems, the geometry of its Laurent series solutions, and the theory of Abelian varieties. We study the classification of metrics for which geodesic flow on the group $SO(n)$ is a.c.i. For $n=3$, the geodesic flow on $SO(3)$ is always a.c.i., and can be regarded as the Euler rigid body motion. For $n=4$, in the Adler-van Moerbeke's classification of metrics for which geodesic flow on $SO(4)$ is a.c.i., three cases come up; two are linearly equivalent to the Clebsch and Lyapunov-Steklov cases of rigid body motion in a perfect fluid, and there is a third new case namely the Kostant-Kirillov Hamiltonian flow on the dual of $so(4)$. Finally, as was shown by Haine, for $n\geq 5$ Manakov's metrics are the only left invariant diagonal metrics on $SO(n)$ for which the geodesic flow is a.c.i.

We study geometric and algebraic geometric properties of the continuous and discrete Neumann systems on cotangent bundles of Stiefel varieties Vn,r. The systems are integrable in the non-commutative sense, and by applying a 2r × 2r–Lax representation, we show that generic complex invariant manifolds are open subsets of affine Prym varieties on which the complex flow is linear. The characteristics of the varieties and the direction of the flow are calculated explicitly. Next, we construct a family of multi-valued integrable discretizations of the Neumann systems and describe them as translations on the Prym varieties, which are written explicitly in terms of divisors of points on the spectral curve. [ABSTRACT FROM AUTHOR]

Given a finite group G and an abelian variety A acted on by G , to any subgroup H of G , we associate an abelian subvariety A H on which the associated Hecke algebra H H for H in G acts. Any irreducible rational representation W ˜ of H H induces an abelian subvariety of A H in a natural way. In this paper we give equations for this abelian subvariety. In a special case these equations become much easier. We work out some examples. [ABSTRACT FROM AUTHOR]

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical divisors. [ABSTRACT FROM AUTHOR]

An old conjecture of Voisin describes how 0-cycles on a surface S should behave when pulled-back to the self-product S m for m > p g (S) . We show that Voisin's conjecture is true for a 3-dimensional family of surfaces of general type with p g = q = 2 and K 2 = 7 constructed by Cancian and Frapporti, and revisited by Pignatelli–Polizzi. [ABSTRACT FROM AUTHOR]

We study Shimura curves of PEL type in A g generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases, we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is one-dimensional and the quotient of the base curve by the group is ℙ 1 . We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples obtained in this way up to genus 28. We obtain 43 Shimura curves contained in the unramified Prym locus and 9 families contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus. [ABSTRACT FROM AUTHOR]

The aim of this survey paper is to investigate the algebraic complete integrability of Euler-Arnold's body description of the four dimensional rigid body, or equivalently of geodesics in SO(4) using left-invariant metrics that arise from inertia tensors, namely non-degenerate maps Λ : so(4)→ so(4)* ≡ so(4) together with the canonical inner product associated to the Killing form. Algebraic complete integrability is motivated by Arnold-Liouville's classical notion of complete integrability : one extends the value of space and time coordinates from ℝ to ℂ, and then the regular invariant manifolds are complex instead of real tori; in addition one demands such complex tori to be projective. Using different methods, as systematized by Adler-Haine-van Moerbeke-Mumford, to study the integrability of the geodesic flow on the rotation group, we will see that the linearization is carried on an abelian surface and each time a Prym variety appears related to this problem.

In this paper I present the basic ideas and properties of the complex algebraic completely integrable dynamical systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several examples of algebraic completely integrable systems : Kowalewski’s top, geodesic flow on SO(4), Hénon-Heiles system, Garnier potential, two coupled nonlinear Schrödinger equations and Yang-Mills system.

In this paper, I present an overview of the active area of interactions between algebraic geometry and algebraic completely integrable systems. These are integrable systems whose trajectories are straight line motions on complex algebraic tori (abelian varieties). We make, via the Kowalewski-Painlevé analysis, a detailed study of the level manifolds of the system. These manifolds are described explicitly as being affine part of complex algebraic tori and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-van Moerbeke method’s which will be used is primarily analytical but heavily inspired by algebraic geometrical methods. We will also discuss several interesting examples of algebraic completely integrable systems.

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 Rg is not isogenous to another distinct Prym variety, under some mild assumption on k. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present a conceptual approach to understanding the extension of the Prym map from the space of admissible double covers of stable curves to different toroidal compactifications of the moduli space of principally polarized abelian varieties. By separating the combinatorial problems from the geometric aspects we can reduce this to the computation of certain monodromy cones. In this way we not only shed new light on the extension results of Alexeev, Birkenhake, Hulek, and Vologodsky for the second Voronoi toroidal compactification, but we also apply this to other toroidal compactifications, in particular the perfect cone compactification, for which we obtain a combinatorial characterization of the indeterminacy locus, as well as a geometric description up to codimension six, and an explicit toroidal resolution of the Prym map up to codimension four. [ABSTRACT FROM AUTHOR]

We present a Lie algebra theoretical schema leading to integrable systems, based on the Kostant-Kirillov coadjoint action. Many problems on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the Adler-Kostant-Symes theorem and the van Moerbeke-Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety (or some subabelian variety of it) of the spectral curve. The complex flows generated by the constants of the motion are straight line motions on these varieties. We study the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view and their relation with the Kac-Moody extension of some algebras. We will present in detail the Griffith's aproach and his cohomological interpretation of linearization test for solving integrable systems without reference to Kac-Moody algebras. We will discuss several examples of integrable systems of relevance in mathematical physics.

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.

The aim of this survey paper is to investigate the algebraic complete integrability of Euler-Arnold's body description of the four dimensional rigid body, or equivalently of geodesics in SO(4) using left-invariant metrics that arise from inertia tensors, namely non-degenerate maps Λ : so(4) → so(4)* ≡ so(4) together with the canonical inner product associated to the Killing form. Algebraic complete integrability is motivated by Arnold-Liouville's classical notion of complete integrability : one extends the value of space and time coordinates from ℝ to &8450;, and then the regular invariant manifolds are complex instead of real tori; in addition one demands such complex tori to be projective. Using different methods, as systematized by Adler-Haine-van Moerbeke-Mumford, to study the integrability of the geodesic flow on the rotation group, we will see that the linearization is carried on an abelian surface and each time a Prym variety appears related to this problem. [ABSTRACT FROM AUTHOR]

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 Hénon–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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, improving on results of Abramovich, Harris, Debarre and Fahlaoui [1,8], we give the full classification of curves C of genus g such that a Brill–Noether locus , strictly contained in the jacobian of C, contains a variety Z stable under translations by the elements of a positive dimensional abelian subvariety and such that , i.e., the maximum possible for such a Z. [Copyright &y& Elsevier]

Abstract: We show that the second fundamental form of the Prym map lifts the second Gaussian map of the Prym-canonical bundle. We prove, by degeneration to binary curves, that is surjective for the general point of for . [Copyright &y& Elsevier]

We consider 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. We prove that 풫:ℛg,r→풜δg−1+r/2 is generically injective if \[ r>6\text { and }g\geq 2,\quad r=6\text { and }g\geq 3,\quad r=4 \text { and }g\geq 5\quad \mbox {or}\quad r=2\text { and }g\geq 6. \] We also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian. [ABSTRACT FROM AUTHOR]

We study the geometry and the period map of nodal complex prime Fano threefolds with index 1 and degree 10. We show that these threefolds are birationally isomorphic to Verra threefolds, i.e., hypersurfaces of bidegree (2, 2) in P × P. Using Verra's results on the period map for these threefolds and on the Prym map for double étale covers of plane sextic curves, we prove that the fiber of the period map for our nodal threefolds is the union of two disjoint surfaces, for which we give several descriptions. This result is the analog in the nodal case of a result of Debarre O, Iliev A, Manivel L (arXiv: 0812.3670) in the smooth case. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion. These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In Section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems. In Section 3, we discuss the concept of the algebraic complete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how to decide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in Section 4 we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifold invariant by the complex flow is covering of abelian variety. [ABSTRACT FROM AUTHOR]

Abstract: The classical definition of Prym varieties deals with the unramified covers of curves. The aim of this article is to give explicit algebraic descriptions of the Prym varieties associated with 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 such as the Hénon–Heiles system, the Kowalewski rigid body motion and Kirchhoff’s equations of motion of a solid in an ideal fluid. [Copyright &y& Elsevier]

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's flop. They have the same singularities as two of Fujiki's examples, namely, 28 isolated singular points analytically equivalent to the Veronese cone of degree 8, but a different Euler number. The family of curves used in this construction forms a linear system on a K3 surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type (1,2). No example of such fibration is known on nonsingular irreducible symplectic varieties. [ABSTRACT FROM AUTHOR]

Abstract: The B-quadrilateral lattice (BQL) provides geometric interpretation of Miwa’s discrete BKP equation within the quadrialteral lattice (QL) theory. After discussing the projective-geometric properties of the lattice we give the algebro-geometric construction of the BQL emphasizing the role of Prym varieties and the corresponding theta functions. We also present the reduction of the vectorial fundamental transformation of the QL to the BQL case. [Copyright &y& Elsevier]

Let W be a Weyl group and P ⊂ W, a parabolic subgroup. In this paper, we give the decomposition of the permutation representation Ind 1 into irreducibles for each exceptional W and maximal parabolic P. We find that there is an 'extra' common irreducible component which appears for exceptional groups and not for classical groups. This work is motivated by the study of Prym varieties and integrable systems. [ABSTRACT FROM AUTHOR]