A Note on the k-very Ampleness of Semi-stable Bundles on an Algebraic Curve

We give a criterion for k-very ampleness of semi-stable bundles on an algebraic curve.     

Generalized Adjunction of k‐Very Ample Vector Bundles on Algebraic Surfaces II

We study the k‐very ampleness of the adjoint bundle K_S + det E associated to a (k — 1)‐very ample vector bundle E with degree greater than or equal to 4k + 5 on an algebraic surface S. We classify polarized surfaces (S, E) which the k‐very ampleness of K_S + det E fails.     

On the adjunction mapping for surfaces of Kodaira dimension ≤0 in char. p

Here we prove in positive characteristic the spannedness or very ampleness or k-ampleness of the adjuntion bundle K_S⊕L with S surface of Kodaira dimension ≤0 L∈Pic(S), L ample, under numerical conditions on L which are similar to the ones required in characteristic 0 by Reider method.     

Mumford's example and a general construction

We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability.     

On 2-very ample divisors of genus ⩽8 on algebraic surfaces

In this paper we study the k-very ampleness of certain classes of divisors of genus 8 on rational and ruled surfaces appearing in the papers [La] of A. Lenteri and [An] of M. Andreatta.Research carried out under the EC HCM project AGE (Algebraic Geometry in Europe), contract number ERBCHRXCT 940557.     

On the 2 very ampleness of the adjoint bundle

Let S be a smooth projective surface over C polarized by a 2-very ample line bundle L=O _S(L), i.e. for any 0-dimensional subscheme (Z,O _Z ) of length 3 the restriction map Γ(L)→Γ(L⊗O _Z) is a surjection. This generalization of very ampleness was recently introduced by M. Beltrametti and A.J. Sommese. The authors prove that, if L·L≥13, the adjoint line bundleK _S⊗L is 2-very ample apart from a list of well understood exceptions and up to contracting down the smooth rational curves E such that E·E=−1, L·E=2. The appendix contains an inductive argument in order to extend the result in higher dimension.     

A Frobenius variant of Seshadri constants

We define and study a version of Seshadri constant for ample line bundles in positive characteristic. We prove that lower bounds for this constant imply the global generation or very ampleness of the corresponding adjoint line bundle. As a consequence, we deduce that the criterion for global generation and very ampleness of adjoint line bundles in terms of usual Seshadri constants holds also in positive characteristic.     

An ampleness criterion with the extendability of singular positive metrics

Coman, Guedj and Zeriahi proved that, for an ample line bundle L on a projective manifold X, any singular positive metric on the line bundle L| _V along a subvariety ${V \subset X}$ can be extended to a global singular positive metric on L. In this paper, we prove that the extendability of singular positive metrics on a line bundle along a subvariety implies the ampleness of the line bundle.     

Slopes of vector bundles on projective curves and applications to tight closure problems

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous -primary ideal in a two-dimensional normal standard-graded algebra in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.

     

非常极凸空间的推广及其对偶概念

本文研究了k非常极凸和k非常极光滑空间的问题.利用Banach空间理论的方法,证明了k非常极凸空间和k非常极光滑空间是一对对偶概念,并且k非常极凸空间(k非常极光滑空间)是严格介于k一致极凸空间和k非常凸空间(k一致极光滑空间和k非常光滑空间)之间的一类新的Banach空间,得到了k非常极凸空间和k非常极光滑空间的若干等价刻画以及k非常极凸(k非常极光滑性)与其它凸性(光滑性)之间的蕴涵关系,推广了非常极凸空间和非常极光滑空间,完善了k非常极凸空间及其对偶空间的研究.     

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     

On Higher Order Embeddings and n - Connectedness

Let L be a very ample line bundle on a smooth complex projective surface. The relations between the higher order embedding properties of L, like k - spannedness, k - very ampleness, k - jet ampleness, and the higher numerical connectedness properties of divisors D ε |L| are Investigated. Some applications to adjoint bundles as well as pluricanonical bundles are discussed.     

Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.

     

Primitive higher order embeddings of abelian surfaces

In recent years several concepts of higher order embeddings have been studied: -spannedness, -very ampleness and -jet ampleness. In the present note we consider primitive line bundles on abelian surfaces and give numerical criteria which allow to check whether a given ample line bundle satisfies these properties.

     

A generalization of Le Potier’s vanishing theorem

The main result is a general vanishing theorem for the cohomology of the ample vector bundles obtained as Schur functors of some vectors bundle which is not assumed to be ample itself. This is a generalization of Le Potiers vanishing theorem. It is also proven that for two partitions I and J such that IJ, the ampleness of S_IE implies that of S_JE.Mathematics Subject Classification (2000):14F17     

On the jumping conics of a semistable rank two vector bundle on P2

Stromme (see [S], (1.7)) introduced the notion of jumping conic of a normalized semistable rank two vector bundle E on ℙ² and he remarked that the locus of jumping conics of E has codimension ≤3+2c₁(E), with equality for general E. Here we introduce a concept of jumping conic of a semistable rank two vector bundle on ℙ² (see (I.1)) by generalizing the notion of jumping line of the second kind introduced by Hulek in [H]. Our definition agrees with Stromme's for c₁=−1, but not for c₁=0. In contrast with the case of jumping lines, where we have a different behaviour in the case c₁ even or c₁ odd, the set of jumping conics according to our definition is always a divisor (possibly empty) in the ℙ⁵ of all conics of ℙ² (see th. (I.8)), whose degree depends on c₂(E).     

The k - Very Ampleness and k - Spannedness on Polarized Abelian Surfaces

We give a numerical criterion for an ample line bundle on an abelian surface to be it-very ample for a nonnegative integer k. This result implies the equivalence of k-very ampleness and k-spannedness. We also give a complete classification of polarized abelian surfaces with k - very ample polarizations. Our results extend those of Bauer and Szemberg [BaSzl] and Ramanan [Ra].     

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Let X be a Hopf manifolds with an Abelian fundamental group.E is a holomorphic vectorbundle of rank r with trivial pull-back to W=C~n-{0}.We prove the existence of a non-vanishingsection of L(?)E for some line bundle on X and study the vector bundles filtration structure of E.These generalize the results of D.Mall about structure theorem of such a vector bundle E.     

Ampleness of intersections of translates of theta divisors in an abelian fourfold

We prove the ampleness of the cotangent bundle of the intersection of two general translates of a theta divosor of the Jacobian of a general curve of genus . From this, we deduce the same result in a general, principally polarized abelian variety of dimension .

     

Complex Finsler structures on tensor products

We give a complex Finsler structure T on E₁ ⊗ E₂ where E₁ is a negative complex vector bundle, and where E ₂ ^* is generated by global sections. The construction of T uses the Finsler structure on E₁, obtained by Kobayashi in [4]. We prove that this structure is strongly pseudoconvex and we show that it has negative curvature.