On spherical ideals of Borel subalgebras

The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of These are ideals of such that their G-saturation is a spherical G-variety. We classify all maximal spherical ideals of for all simple G.Received: 25 March 2004     

Enumeration of one-nodal rational curves in projective spaces

We give a formula computing the number of one-nodal rational curves that pass through an appropriate collection of constraints in a complex projective space. The formula involves intersections of tautological classes on moduli spaces of stable rational maps. We combine the methods and results from three different papers.     

On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.     

On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Finite orbit modules for parabolic subgroups of exceptional groups

Let G be a reductive linear algebraic group, P a parabolic subgroup of G and P_u its unipotent radical. We consider the adjoint action of P on the Lie algebra _u of P_u. Each higher term _u^(l) of the descending central series of _u is stable under this action. For classical G all instances when P acts on _u^(l) with a finite number of orbits were determined in [9], [10], [3] and [4]. In this note we extend these results to groups of type F₄ and E₆. Moreover, when P acts on _u^(l) with an infinite number of orbits, we determine whether P still acts with a dense orbit. For G of type E₇ and E₈ we investigate only the case of a Borel subgroup.We present a complete classification of all instances when _u^(l) is a prehomogeneous space for a Borel subgroup B of a reductive algebraic group for any l ≥ 0.     

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermat's quartic.     

Connections for weighted projective lines

We introduce a notion of a connection on a coherent sheaf on a weighted projective line (in the sense of Geigle and Lenzing). Using a theorem of Hübner and Lenzing we show, under a mild hypothesis, that if one considers coherent sheaves equipped with such a connection, and one passes to the perpendicular category to a nonzero vector bundle without self-extensions, then the resulting category is equivalent to the category of representations of a deformed preprojective algebra.     

Einige Existenz- und Einschließungssätze für Fixpunkte expandierender Integraloperatoren

Some existence theorems for fixed points of nonlinear integral operators expanding a cone in a Banach space in the sense of Krasnoselskii [5] are given. Using special cones we find pointwise inclusions for the fixed points. The question of finding the best bounds for the fixed points leads to a nonlinear optimization problem with infinitely many restrictions.

     

Conservation laws with singular nonlocal sources

Several physical phenomena are modeled by conservation laws with fluxes or sources that are singular in the origin. Here an integro-differential regularization of those problems is proposed. The existence of positive solutions with finite total variation is proved.     

On distance preserving transformations of Euclidean-like planes over the rational field

Aequationes mathematicae -     

Singular rational curves with points of nearly-maximal weight

In this article we study rational curves with a unique unibranch genus-g singularity, which is of κ-hyperelliptic type in the sense of [27]; we focus on the cases $\kappa =0$ and $\kappa =1$, in which the semigroup associated to the singularity is of (sub)maximal weight. We obtain a partial classification of these curves according to the linear series they support, the scrolls on which they lie, and their gonality.     

Castelnuovo-Mumford regularity for nonhyperelliptic curves

The purpose here is to show that an irreducible, reduced, projective, nonhyperelliptic curve of degree d and genus g is n-regular for if Received: 10 July 2003     

A System of Differential-Operator Equations of Different Orders in Hilbert Spaces

We consider an abstract Cauchy problem for a system of nonhomogeneous abstract differential equations in Hilbert spaces. The “main” equation is of the second order and “boundary” equations are of the first order. Existence of a solution is proved. Application to mixed (initial boundary-value) problems for one-dimensional second order hyperbolic equations and for fourth order PDEs with the time derivative in boundary conditions has been shown. The first author was partially supported by 60% funds of the University of Bologna and G.N.A.M.P.A. of INdAM; the second author was supported by the Israel Ministry of Absorption.     

On certain finite-dimensional algebras generated by two idempotents

This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)^m=(QP)^m and (PQ)^m-1≠(QP)^m-1. The main result is the classification of all these algebras, implying that for each m?2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given.     

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit of N is contained in the orbit closure and has codimension two. We show that the pointed variety is smoothly equivalent to a simple surface singularity of type A _n , or to the cone over a rational normal curve.     

Weierstrass semigroups whose minimum positive integers are even

We consider three subsets of the set of 2n-semigroups, where for a positive integer n a 2n-semigroup means a numerical semigroup whose minimum positive integer is 2n. These three subsets are obtained by the Weierstrass semigroups of total ramification points on a cyclic covering of the projective line, the Weierstrass semigroups of ramification points on a double covering of a non-singular curve and the Weierstrass semigroups of points on a non-singular curve. We show that the three subsets are different for n ≧ 3. Partially supported by Grant-in-Aid for Scientific Research (17540046), Japan Society for the Promotion of Science. Received: 19 June 2006     

On the stochastic quasi-linear symmetric hyperbolic system

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly.     

Deformations of algebraic curves and integrable nonlinear evolution equations

A brief exposition of applications of the methods of algebraic geometry to systems integrable by the IST method with variable spectral parameters is presented. Usually, theta-functional solutions for these systems are generated by some deformations of algebraic curves. The deformations of algebraic curves are also related with theta-functional solutions of Yang-Mills self-duality equations which contain special systems with a variable spectral parameter as a special reduction. Another important situation in which the deformations of algebraic curves naturally occur is the KdV equation with string-like boundary conditions. Most important concrete examples of systems integrable by the IST method with variable spectral parameter having different properties within a framework of the behavior of moduli of underlying curves, analytic properties of the Baker-Akhiezer functions, and the qualitative behavior of the solutions, are vacuum axially symmetric Einstein equations, the Heisenberg cylindrical magnet equation, the deformed Maxwell-Bloch system, and the cylindrical KP equation.Dedicated to the memory of J.-L. Verdier     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.