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     

Seshadri constants on surfaces of general type

We study Seshadri constants of the canonical bundle on minimal surfaces of general type. First, we prove that if the Seshadri constant ε(K _X , x) is between 0 and 1, then it is of the form (m − 1)/m for some integer m ≥ 2. Secondly, we study values of ε(K _X , x) for a very general point x and show that small values of the Seshadri constant are accounted for by the geometry of X.     

Algebraic loop groups and moduli spaces of bundles

We study algebraic loop groups and affine Grassmannians in positive characteristic. The main results are normality of Schubert-varieties, the construction of line-bundles on the affine Grassmannian, and the proof that they induce line-bundles on the moduli-stack of torsors. Received March 15, 2001 / final version received April 11, 2002?Published online November 19, 2002     

Normal del Pezzo surfaces containing a nonrational singularity

Working over perfect ground fields of arbitrary characteristic, I classify minimal normal del Pezzo surfaces containing a nonrational singularity. As an application, I determine the structure of 2-dimensional anticanonical models for proper normal algebraic surfaces. The anticanonical ring may be non-finitely generated. However, the anticanonical model is either a proper surface, or a proper surface minus a point. Received: 5 June 2000 / Revised version: 10 November 2000     

Embeddings of general blowups of abelian surfaces

Blowups of algebraic surfaces polarized by tensor powers of ample line bundles were studied by Coppens [8]. In this note we complete the picture in the case of abelian surfaces studying blowups of surfaces polarized by primitive line bundles. Received: 16 November 1999 / Revised version: 13 June 2000     

Pluricanonical maps of varieties of maximal Albanese dimension

Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of in terms of the set . An immediate consequence of this is that the Kodaira dimension is invariant under smooth deformations. We then study the pluricanonical maps . We prove that if X is of general type, is generically finite for and birational for . More generally, we show that for the image of is of dimension equal to and for , is the stable canonical map. Received July 7, 2000 / Published online April 12, 2001     

On a conjecture of Cox and Katz

This note shows that a certain toric quotient of the quintic Calabi-Yau threefold in provides a counterexample to a recent conjecture of Cox and Katz concerning nef cones of toric hypersurfaces. Received: 8 February 2001; in final form: 17 September 2001 / Published online: 1 February 2002     

Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type A _{2 n} , this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence. Received: Received: 25 October 2001 / Revised version: 19 November 2001     

Heights for line bundles on arithmetic varieties

Let X be an arithmetic variety and L be an element of the Néron-Severi group of its generic fiber X _K . Then there are only finitely many line bundles on X, generically belonging to L, such that the degrees of on the irreducible components of the special fibers of X and the height of are bounded. The concept of a height used here is recalled. Several elementary properties of this height are proven. Received: 9 March 1996     

Very ample line bundles on quasi-abelian varieties

Received January 27, 1999; in final form May 31, 1999 / Published online October 30, 2000     

Very Ampleness of <Emphasis Type="Italic">d</Emphasis>-standard Classes on Rational Surfaces

We consider a rational surface X_r, obtained by blowing up ² along a curvilinear zero-dimensional subscheme of length r of the regular locus of a reduced irreducible plane curve of degree d, with d 4; and we give sufficient conditions for d-standard classes to be very ample (resp. base point free or non special) on such a rational surface X_r.Postdoctoral Fellow of the Fund for Scientific Research-Flanders (Belgium).     

Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold

Generalizing the Harder–Narasimhan filtration of a vector bundle it is shown that a principal G-bundle over a compact K?hler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where , this reduction is the Harder–Narasimhan filtration of the vector bundle associated to by the standard representation of . The reduction of in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and the canonical reduction, then the first condition says that the principal L-bundle obtained by extending the structure group of the P-bundle using the natural projection of P to L is semistable. Denoting by the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in , the associated vector bundle is of positive degree; here is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to for is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here. Received: 10 November 1999 / Revised version: 31 October 2001 / Published online: 26 April 2002