Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Tilting generators via ample line bundles

It is known that a tilting generator on an algebraic variety X gives a derived equivalence between X and a certain non-commutative algebra. In this paper, we present a method to construct a tilting generator from an ample line bundle, and construct it in several examples.     

Very ample line bundles on quasi-abelian varieties

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

Effective bounds for very ample line bundles

Let be an ample line bundle on a non singular projective -fold . It is first shown that is very ample for . The proof develops an original idea of Y.T. Siu and is based on a combination of the Riemann-Roch theorem together with an improved Noetherian induction technique for the Nadel multiplier ideal sheaves. In the second part, an effective version of the big Matsusaka theorem is obtained, refining an earlier version of Y.T. Siu: there is an explicit polynomial bound of degree in the arguments, such that is very ample for . The refinement is obtained through a new sharp upper bound for the dualizing sheaves of algebraic varieties embedded in projective space. Oblatum 30-I-1995 & 18-V-1995     

Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(K _X L))/2=3 are classified by means of the main techniques of adjunction theory.     

Some remarks on ample line bundles on abelian varieties

Let L be an ample line bundle on an abelian variety A. We show that L² is very ample if (A,L) is not isomorphic to (A₁×A₂,o(D₁×A₂+A₁×D₂)) where A_i is an abelian variety (i=1,2), D_i is an ample divisor on A_i (i=1,2) and (A₁,o(D₁))=1, and if (A,L)2. As an application we show that L² is base point free if L is an ample line bundle on bielliptic surface.In conclusion, the author would like to thank the referee for very helpful advice.     

A note on tensor products of ample line bundles on abelian varieties

Let be a polarized abelian variety defined over the complex number field. Then we classify with such that is not -jet ample nor -very ample.

     

Parabolic ample bundles III: Numerically effective vector bundles

In this continuation of [Bi2] and [BN], we define numerically effective vector bundles in the parabolic category. Some properties of the usual numerically effective vector bundles are shown to be valid in the more general context of numerically effective parabolic vector bundles.