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

Lanteri, Antonio; Maeda, Hidetoshi

doi:10.1017/s0305004107000813

Cited by 6 publications

(15 citation statements)

References 13 publications

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“…Therefore r = 2, according to (5.6.1). It follows that (X, E) is as in case (4) in view of [15,Theorem 4.2 and Proposition 4.7] and [22,Lemma 4].…”

Section: Theorem 51 Let X E (P H) and D Be As In 13 Assume Tmentioning

confidence: 93%

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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“…Therefore r = 2, according to (5.6.1). It follows that (X, E) is as in case (4) in view of [15,Theorem 4.2 and Proposition 4.7] and [22,Lemma 4].…”

Section: Theorem 51 Let X E (P H) and D Be As In 13 Assume Tmentioning

confidence: 93%

Ample vector bundles of small Δ-genera

Lanteri¹,

Novelli²

2010

Journal of Algebra

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“…Three cases lead us to (X, E, H ) in the Theorem. Another case can be ruled out by relying on our results in [18]. The final case deserves special care.…”

Section: Introductionmentioning

confidence: 93%

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Lanteri

Maeda

2007

Geom Dedicata

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Let 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 ≥ 3 of X . Let H be an ample line bundle on X such that its restriction H Z to Z is very ample. Triplets (X, E, H ) are classified under the assumption that (Z , H Z ) has a smooth bielliptic curve section of genus ≥ 3 with H n−r c r (E) ≤ 8.

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“…Cases (IVi)-(IV-iv) are listed as Cases (2), (3), (5) and (6) respectively in [21,Section 1]. Notice that the extra assumption that |H| embeds Z used in [21, Section 1] to study our Case (IV-ii) has been recently removed [22,Section 1]. Relying on this result, these four possibilities lead to the triplets (X, E, H) as in (4), (6)-(9) of the statement or to the following situation: n − r ≤ 5, and there exists a surjective morphism ϕ : X −→ P 1 whose general fiber F is a smooth quadric hypersurface Q n−1 in P n with H F ∼ = O Q n−1 (1) and E F ∼ = O Q n−1 (1) ⊕r .…”

Section: Case (D)mentioning

confidence: 99%

Ample vector bundles with zero loci of small Δ-genera

Lanteri¹,

Novelli²

2008

Advg

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Abstract. Let X be a smooth complex projective variety endowed with an ample vector bundle E admitting a global section whose zero locus is a smooth subvariety Z of the expected dimension, and let H be an ample line bundle on X, whose restriction HZ to Z is very ample. Triplets (X, E, H) are studied and classified under the assumption that the delta genus of (Z, HZ ) is either small (≤ 3) or small in comparison with the corank of E or the degree.

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

Cited by 6 publications

References 13 publications

Ample vector bundles of small Δ-genera

Ample vector bundles of small Δ-genera

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Ample vector bundles with zero loci of small Δ-genera

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