On the chow group and the intermediate jacobian of a conic bundle

Sunto In questo lavoro si studiano fibrati in coniche X su una superficie non singolare S. I fibrati in coniche si presentano nella classificazione delle varietà algebriche a tre dimensioni aventi dimensione di Kodaria negativa. Si descrivono qui, a meno di isogenie, il gruppo A²(X) dei cicli di codimensione due algebrieamente equivalenti a zero e la iacobiana intermedia J(X). Quando S è una superficie razionale, A²(X) e J(X) vengono descritti a meno di isomorfismi ottenendo risultati già provati da Beauville nel caso S=P ².     

A conic bundle description of Moishezon twistor spaces without effective divisors of degree one

In [K2] Moishezon twistor spaces over the connected sum (), which do not contain effective divisors of degree one, were constructed as deformations of the twistor spaces introduced in [LeB]. We study their structure for by constructing a modification which is a conic bundle over . We show that they are rational. In case n = 4 we give explicit equations for such conic bundles and use them to construct explicit birational maps between these conic bundles and . Received October 8, 1996; in final form May 16, 1997     

Symplectic automorphisms on Kummer surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface X by a finite abelian group G of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of H²(X,\mathbbZ){H^2(X,\mathbb{Z})} which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.     

A sylvester theorem for conic sections

If S is a finite set of points in the plane and no conic contains all points of S, then S determines a conic which contains exactly five points ofS.     

A conic manifold perspective of elliptic operators on graphs

We give a simple, explicit, sufficient condition for the existence of a sector of minimal growth for second-order regular singular differential operators on graphs. We specifically consider operators with a singular potential of Coulomb type and base our analysis on the theory of elliptic cone operators.     

A note on Turán type and mean inequalities for the Kummer function

Turán-type inequalities for combinations of Kummer functions involving Φ(a±ν,c±ν,x) and Φ(a,c±ν,x) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279-293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1-9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ(a±ν,c,x). The application to modeling credit risk is also summarized.     

A conic scalarization method in multi-objective optimization

This paper presents the conic scalarization method for scalarization of nonlinear multi-objective optimization problems. We introduce a special class of monotonically increasing sublinear scalarizing functions and show that the zero sublevel set of every function from this class is a convex closed and pointed cone which contains the negative ordering cone. We introduce the notion of a separable cone and show that two closed cones (one of them is separable) having only the vertex in common can be separated by a zero sublevel set of some function from this class. It is shown that the scalar optimization problem constructed by using these functions, enables to characterize the complete set of efficient and properly efficient solutions of multi-objective problems without convexity and boundedness conditions. By choosing a suitable scalarizing parameter set consisting of a weighting vector, an augmentation parameter, and a reference point, decision maker may guarantee a most preferred efficient or properly efficient solution.     

A construction of a class of maximal Kummer curves

In this note, we present a simple method for constructing maximal curves defined over by the equation yⁿ=T(x), where T(x) is a q-polynomial over .     

Three problems related to the Kummer problem

We give a brief survey of the principal results concerning the individual Kummer problem and present new results of the author concerning this problem. Institute of Cybernetics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 8, pp. 1061–1068, August, 1997.     

A bundle view of boundary-value problems: generalizing the Gardner-Jones bundle

Holomorphic families of linear ordinary differential equations on a finite interval with prescribed parameter-dependent boundary conditions are considered from a geometrical viewpoint. The Gardner-Jones bundle, which was introduced for linearized reaction-diffusion equations, is generalized and applied to this abstract class of λ-dependent boundary-value problems, where λ is a complex eigenvalue parameter. The fundamental analytical object of such boundary-value problems (BVPs) is the characteristic determinant, and it is proved that any characteristic determinant on a Jordan curve can be characterized geometrically as the determinant of a transition function associated with the Gardner-Jones bundle. The topology of the bundle, represented by the Chern number, then yields precise information about the number of eigenvalues in a prescribed subset of the complex λ-plane. This result shows that the Gardner-Jones bundle is an intrinsic geometric property of such λ-dependent BVPs. The bundle framework is applied to examples from hydrodynamic stability theory and the linearized complex Ginzburg-Landau equation.     

The projectivised cotangent bundle of an algebraic surface

Partially supported by project PS 90-0069 (Ministerio de Educacion y Ciencia, Spain) and by project Geometry of algebraic varieties, European Science Programme, SC 1-0398-C(A).