A characterization of bielliptic curves and applications to their moduli spaces

We deal with the covers of degree 4 naturally associated to a bielliptic curve of genus g≥6, giving a proof of the unirationality of the moduli space ? _g ^be of such curves, of the rationality of the Hurwitz scheme ℌ ^be _{4, g} of bielliptic curves of even genus g, whereas, when g is odd, we construct a finite map ℂ^{2 g -2}→? _g ^be and compute its degree. Received: March 25, 2000; in final form: March 10, 2001?Published online: May 29, 2002     

Remarks on normal generation of line bundles on algebraic curves

We determine necessary and sufficient conditions for nonspecial line bundles of degree 2% - 4 and 2g - 5 being not normally generated. Furthermore, we also determine necessary and suffcient conditions for speciality 1 line bundles of degree 2g -7,2% - 8, and 2g - 9 being not normally generated.     

Some presentations for the special linear groups on finite fields

We simplify the Steinberg presentation of SL _n (F _d ), where n≥1 and F _d is any finite field with d elements. That presentation has the elementary matrices e _ij (r), with i,j∈{1,...,n}, i≠=j and r∈F _d , as generators, and (E1)–(E3), described at the opening of this work, as relations. The presentation that we shall obtain reduces the number of generators e _ij (r) and relations (E1)–(E3). In particular, relations (E3) are considerably reduced. Received: March 16, 1998; in final form: November 3, 2000?Published online: October 2, 2001     

Mirror fibrations and root stacks of weighted projective spaces

We show that the orbifold Chow ring of a root stack over a well-formed weighted projective space can be naturally seen as the Jacobian algebra of a function on a singular variety.     

Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme. Received 6 July 2000; in revised form 16 June 2001     

Vector fields on smooth threefolds vanishing on complete intersections

A result of J. Wahl shows that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducible and reduced curve which is the scheme theoretic intersection of two ample divisors, X is isomorphic to the 3-dimensional projective space or the 3-dimensional quadric. Received: 24 April 2001     

Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

A note on families of hyperelliptic curves

We give a stack-theoretic proof for some results on families of hyperelliptic curves. Received: 5 February 2008     

Recurrent random walks on homogeneous spaces of p-adic algebraic groups of polynomial growth

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two. Received: 23 November 2007     

Complemented ideals in A({\mathbb{R}}^{d}) of algebraic curves

It is shown that for an algebraic curve the ideal of real analytic functions vanishing on X is complemented in if and only if in every a ∈ X every irreducible component of the germ X_a is either regular or a point. Received: 5 January 2009     

Representation,interpolation, and reiteration theorems for generalized approximation spaces

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l ^q (ℬ))={f∈X:{E _n (f)}∈l ^q (ℬ)} in which the weighted l ^q -space l ^q (ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces. Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003     

Oscillation and asymptotic behaviour of a class of higher-order non-linear difference equations

The dynamical characteristics of scalar difference equations of the form?x _{n +1}=f ₁(x _{n −τ1})+f ₂(x _{n −τ2}), n=0,1,2,...,?are investigated. A necessary and sufficient condition is obtained for all positive solutions to be oscillatory about a unique positive equilibrium point and sufficient criteria for the global attractivity of the equilibrium are established. Also, the stability and periodicity of more general equations are studied via comparison with the corresponding properties of an associated first-order non-linear equation. Received: September 5, 2001; in final form: March 7, 2002?Published online: April 14, 2003     

The base components of the dualizing sheaf of a curve on a surface

This note studies the structure of the divisorial fixed part of for a 1-connected curve D on a smooth surface S. It is shown that if the divisorial fixed part F of is non empty then it has arithmetic genus ≤ 0 and each component of F is a smooth rational curve. The structure of curves D, with non empty divisorial fixed part F for , is also described. Received: 16 August 2007     

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers and the relative Jacobian of a family of curves . As one of the applications, we show that in the case of a single curve C this action induces a -form of a Lefschetz sl₂- action on the Chow groups of C ^[N]. Another application gives a new grading on the ring CH ₀(J) of 0-cycles on the Jacobian J of C (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in CH and in CH and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in [5]. We show that our algebras of operators preserve the subrings of tautological cycles and act on them as some explicit differential operators. This work was partially supported by the NSF grant DMS-0601034.     

Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves

In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems introduced by Hernández Ruipérez and Tejero Prieto.     

The mean value property for harmonic functions on graphs and trees

Following the Euclidean example, we introduce the strong and weak mean value property for finite variation measures on graphs. We completely characterize finite variation measures with bounded support on radial trees which have the strong mean value property. We show that for counting measures on bounded subsets of a tree with root o, the strong mean value property is equivalent to the invariance of the subset under the action of the stabilizer of o in the automorphism group. We finally characterize, using the discrete Laplacian, the finite variation measures on a generic graph which have the weak mean value property and we give a non-trivial example. Received: July 21, 2000; in final form: March 13, 2001?Published online: March 19, 2002     

A trust region method based on interior point techniques for nonlinear programming

An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Received: May 1996 / Accepted: August 18, 2000?Published online October 18, 2000     

Topology on Asymptotic Algebras of Generalized Functions and Applications

 Starting from a locally convex metrisable topological space and from any asymptotic scale, we construct a generalized extension of this space. To those extensions, we associate Hausdorff topologies. We introduce the notion of a temperate map, with respect to a given asymptotic scale, between two locally convex metrisable semi-normed spaces. We show that such mappings extend in a canonical way to mappings between the respective generalized extensions. We give an application to nonlinear Dirichlet boundary value problems with singular data in the framework of generalized extensions. (Received 2 November 1998; in revised form 25 February 1999)