Ample vector bundles with zero loci of small Δ-genera

Lanteri, Antonio; Novelli, Carla

doi:10.1515/advgeom.2008.016

Cited by 3 publications

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“…Some arguments presented here are different from those in [10], and we carry out the proof from the beginning. Some arguments presented here are different from those in [10], and we carry out the proof from the beginning.…”

Section: Proof Of Theoremmentioning

confidence: 91%

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Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Lanteri¹,

Maeda²

2008

Math. Proc. Camb. Phil. Soc.

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Link to this article: http://journals.cambridge.org/abstract_S0305004107000813 How to cite this article: ANTONIO LANTERI and HIDETOSHI MAEDA (2008). Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles. AbstractLet E be an ample vector bundle of rank r 2 on a smooth complex projective variety X of dimension n such that there exists a global section of E whose zero locus Z is a smooth subvariety of dimension n − r 2 of X . Let H be an ample line bundle on X such that the restriction H Z of H to Z is very ample. Triplets (X, E, H ) with g(Z , H Z ) = 3 are classified, where g(Z , H Z ) is the sectional genus of (Z , H Z ).

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Section: Proof Of Theoremmentioning

confidence: 91%

“…Before we proceed to the proof of Theorem 1, we note that the fact that n −r = 3 has been shown in [10, lemma 2•5] under the assumption of Theorem 1. Some arguments presented here are different from those in [10], and we carry out the proof from the beginning. Now let us prove Theorem 1.…”

Section: Proof Of Theoremmentioning

confidence: 91%

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Lanteri¹,

Maeda²

2008

Math. Proc. Camb. Phil. Soc.

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Ample vector bundles of small Δ-genera

Lanteri¹,

Novelli²

2010

Journal of Algebra

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A natural notion of "delta-genus" for a generalized polarized manifold (X, E), strictly related to its associated scroll, is introduced and pairs (X, E) with low are classified. The stronger are the properties enjoyed by the vector bundle E, the larger are the values of attained by the results.© 2009 Elsevier Inc. All rights reserved. IntroductionLet X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X . In order to study polarized manifolds (X, L) Fujita introduced the -genus of (X, L), which is a nonnegative integer defined by the formulaThe theory developed around this invariant has been a powerful tool in characterizing polarized varieties with small enough [11]. As noticed in [11, p. 176] t here is not a good vector bundle version of the theory of -genus. This sentence motivated our interest in the subject.Let E be an ample vector bundle of rank r 2 on X . There are two obvious polarized varieties naturally associated with (X, E), namely (X, det E) and the scroll (P , H), where P = P X (E) and H is the tautological line bundle. One could be tempted to use their -genera to study (X, E). The natural expectation, however, is to have a new invariant capturing the vector bundle aspects in a better way, e.g. involving the rank r and all Chern classes of E.ing to the theory (see [11] for H ample, and [15,17,18] for H very ample), polarized manifolds with low -genus are rather special and include several special varieties arising from adjunction theory.Since we already know that (P , H) is a scroll, the investigation of scrolls admitting another relevant structure for adjunction theory (e.g. non-trivial reductions, quadric fibrations, del Pezzo and Mukai manifolds, etc.) plays a key role in our analysis. This investigation takes Section 2. Some results we prove to this end are of interest in themselves, e.g. see Propositions 2.8 and 2.12. Another point deserves to be stressed. The map associating (P , H) to (X, E) is not injective. Hence, in the reconstruction process of (X, E) from (P , H), one can meet admissible pairs (P , H) carrying distinct scroll structures. This happens in several instances, some of which are nontrivial, e.g. see Remarks 2.5, 2.10, 5.5 and 5.7. Finally, while the value of the -genus increases, new possible varieties arise as candidates for (P , H) (e.g. see the proof of Theorem 6.3); of course this makes it harder to analyze the compatibility of different structures on (P , H).The paper is organized as follows: in Section 1 we collect some background material; scrolls carrying a further structure are analysed in Section 2; the -genus of (X, E) is discussed in Section 3 for ample vector bundles; in Section 4 we consider ample vector bundles spanned by global sections, while Sections 5 and 6 are devoted to very ample vector bundles. 1 − 3c 2 1 c 2 + 2c 1 c 3 + c 2 2 − c 4 . Then the iterated procedure described above leads to a recursive expression for P (c 1 , . . . , c r ). The result is the following Lemma 1.1. The degree of (P , H) is given by d(P ...

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Varieties of small degree with respect to codimension and ample vector bundles

Lanteri

Novelli

2008

J. Math. Soc. Japan

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Ample vector bundles with zero loci of small Δ-genera

Cited by 3 publications

References 29 publications

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Ample vector bundles of small Δ-genera

Varieties of small degree with respect to codimension and ample vector bundles

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