Adjoint divisors on algebraic curves

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that if is an ample invertible sheaf satisfying some technical numerical hypotheses then is normally generated. As a corollary we show that the sheaf ω_C^⊗2 on a numerically connected curve C of arithmetic genus p_a?3 is normally generated if ω_C is ample and does not exist a subcurve B⊂C such that p_a(B)=1=B(C−B).     

Poisson pencils, algebraic integrability, and separation of variables

In this paper we review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the bihamiltonian structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.     

The algebraic independence of the sum of divisors functions

Let σj(n)=_∑d|ndj be the sum of divisors function, and let I be the identity function. When considering only one input variable n, we show that the set of functions is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n=rσ(r)/(2r−σ(r)), with some restrictions on r. This generalizes the known forms for both even and odd perfect numbers.     

On algebraic integrability of Gelfand-Zeitlin fields

We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in \mathfrakg\mathfrakl \mathfrak{g}\mathfrak{l} (n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties X_D {X_\mathcal{D}} . We construct an étale cover [^(\mathfrakg)]_D {\hat{\mathfrak{g}}}_\mathcal{D} of X_D {X_\mathcal{D}} and show that X_D {X_\mathcal{D}} and [^(\mathfrakg)]_D {\hat{\mathfrak{g}}}_\mathcal{D} are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on X_D {X_\mathcal{D}} to Hamiltonian vector fields on [^(\mathfrakg)]_D {\hat{\mathfrak{g}}}_\mathcal{D} and integrate these vector fields to an action of a connected, commutative algebraic group.     

Dicritical divisors and Jacobian problem

The dicritical divisor coming out of integrable holonomic systems gets miraculously married to the jacobian problem of algebraic geometry.     

A general class of centers for the Poincaré problem

We consider the classical Poincaré problem of a linear center perturbed by homogeneous polynomials of degree n?2. Using certain parity properties of a related differential equation, we develop a technique for obtaining center conditions for an arbitrary value of n and use it to exhibit explicitly new center conditions for n=4,…,8.     

Poincaré invariants

We construct an obstruction theory for relative Hilbert schemes in the sense of [K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1) (1997) 45-88] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V, our obstruction theory determines a virtual fundamental class , which we use to define Poincaré invariants

     

Poincaré-Perron's method for the Dirichlet problem on stratified sets

We extend standard Poincaré-Perron's method to the Dirichlet problem on a class of multistructures. This method is based on the spherical mean theorem, the construction of fundamental solutions and on Harnack's inequality on such domains, that we first establish.     

Linear algebra algorithms for divisors on an algebraic curve

We use an embedding of the symmetric th power of any algebraic curve of genus into a Grassmannian space to give algorithms for working with divisors on , using only linear algebra in vector spaces of dimension , and matrices of size . When the base field is finite, or if has a rational point over , these give algorithms for working on the Jacobian of that require field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess' 1999 Ph.D. thesis, which works with function fields as extensions of . However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of .

     

Equivariant completions of homogenous algebraic varieties by homogenous divisors

Complete smooth complex algebraic varieties with an almost transitive action of a linear algebraic group are studied. They are classified in the case, when the complement of the open orbit is a homogeneous hypersurface. If the group and the isotropy subgroup at a generic point are both reductive, then there exists a natural one-to-one correspondence between these two-orbit varieties and compact riemannian symmetric spaces of rank one.     

The three-magnon problem and integrability of rung-dimerized spin ladders

We study the integrability problem for rung-dimerized spin ladder by the Bethe ansatz in three-magnon sector. It is shown that solvability of the three-magnon problem takes place for the same values of coupling constants in the Hamiltonian which guarantee solvability of the Yang–Baxter equation for the corresponding R-matrix. Bibliography: 15 titles.     

Hamiltonicity and integrability of the Suslov problem

The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This subject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed.     

The Cauchy problem and integrability of a modified Euler-Poisson equation

We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in when . We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space as a Hamiltonian equation, we concentrate on one space dimension () and show that the equation is bihamiltonian.

     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003