On a conjecture of W. Veys

In this work we show a singular K3 surface S of such that producing a counter-example to a conjecture of Veys. Received: 22 October 1999     

The generalized Hodge conjecture for the quadratic complex of lines in projective four–space

We verify the generalized Hodge conjecture GHC(X,5,2) for the quadratic complex of lines in projective four–space. Received: 27 February 1998 / Revised version: 13 May 1998     

Sur l'effectivité numérique des images inverses de fibrés en droites

Sans résumé

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Re?u le 7 Octobre 1996     

Cycles algébriques sur les surfaces K3 réelles

Sans résumé

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Received 14 June 1995; in final form 30 October 1995     

The Albanese map of a 3-fold of general type whose canonical map is composed with a pencil

Received: 15 October 1999; in final form: 13 June 2000 / Published online: 29 April 2002     

Seshadri constants in a family of surfaces

The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. Received: 16 April 2001 / Published online: 26 April 2002     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

On non-projective normal surfaces

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron–Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus. Received: 29 January 1999     

On the conormal bundle of a G-stable subvariety

Let G be a reductive algebraic group and X a smooth G-variety. For a smooth locally closed G-stable subvariety M⊂X, we prove that the G-complexity of the (co)normal bundle of M is equal to the G-complexity of X. In particular, if X is spherical, then all (co)normal bundles are again spherical G-varieties. If X is a G-module with finitely many orbits, the closures of the conormal bundles of the orbits coincide with the irreducible components of the commuting variety. We describe properties of these closures for the representations associated with short gradings of simple Lie algebras. Received: 22 April 1998     

Rational curves and ampleness properties¶of the tangent bundle of algebraic varieties

Let X be a projective manifold, a locally free ample subsheaf of the tangent bundle T _X . If and or n, we prove that . Furthermore we investigate ampleness properties of T _X on large families of curves and the relation to rational connectedness. Received: 2 July 1996     

Concavity estimates for a class of nonlinear elliptic equations in two dimensions

We study solutions of the nonlinear elliptic equation on a bounded domain in . It is shown that the set of points where the graph of the solution has negative Gauss curvature always extends to the boundary, unless it is empty. The meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the Hessian determinant and a suitable power of the solutions. As a consequence of the result, we give a new proof for power concavity of solutions to certain semilinear boundary value problems in convex domains. Received: 12 January 2000; in final form: 15 March 2001 / Published online: 4 April 2002     

Kauffman bracket skein module of a connected sum of 3-manifolds

We show that for the Kauffman bracket skein module over the field of rational functions in variable A, the module of a connected sum of 3-manifolds is the tensor product of modules of the individual manifolds. Received: 12 January 1998 / Revised version: 15 September 1999     

On a question of Miles Reid

A conjecture of Miles Reid states that the relative canonical algebra for a pencil of curves of genus greater than one is always generated in degrees at most three (1-2-3 conjecture). We give some explicit counter-examples to the conjecture. Received: 10 December 1998/ Revised version: 15 April 1999