The Adjunction Theory of Complex Projective Varieties

Beltrametti, Mauro C.; Sommese, Andrew J.

doi:10.1515/9783110871746

Cited by 274 publications

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“…Beltrametti and Sommese (1995) while discussing the -ampleness of the linear system of hyperplane sections for different . This notion was further developed in a recent paper (Chiantini and Ciliberto 2010).…”

Section: Remarkmentioning

confidence: 99%

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

Nenashev

Shapiro

2017

Arnold Math J.

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The space Pol dCP d of all complex-valued binary forms of degree d (considered up to a constant factor) has a standard stratification, each stratum of which contains all forms whose set of multiplicities of their distinct roots is given by a fixed partition μ d. For each such stratum S μ , we introduce its secant degeneracy index μ which is the minimal number of projectively dependent pairwise distinct points on S μ , i.e., points whose projective span has dimension smaller than μ − 1. In what follows, we discuss the secant degeneracy index μ and the secant degeneracy index μ of the closureS μ .

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Section: Remarkmentioning

confidence: 99%

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

Nenashev

Shapiro

2017

Arnold Math J.

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“…is a linear P 4 -bundle, π : M → C, over a smooth curve C (see [3,Section 7.2]). The first two cases contradict our present assumption Pic(M) ∼ = Pic(A) ∼ = Pic(Y ) ⊕ Z.…”

Section: Proof Of Proposition 01 Let L := O M (A) and Let K Be The mentioning

confidence: 99%

A Note on P1-Bundles as Hyperplane Sections

Beltrametti¹,

Fania

Sommese

2005

Kyushu J. Math.

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Abstract. Let M be a five-dimensional manifold polarized by a very ample line bundle L. We show that a smooth A ∈ |L| cannot be a holomorphic P 1 -bundle over a smooth projective 3-fold Y , unless Y ∼ = P 3 and A ∼ = P 1 × P 3 . IntroductionThe aim of this note is to provide a proof of the following result. The proof is based on Lemma 1.1 and on a result due to Lanteri and Struppa [9].To give a motivation for this result, recall the general conjecture that for a P dbundle p : A → B, over a projective manifold B of dimension b, which is an ample divisor on a manifold M, the condition d ≥ b − 1 should be true (see [3, Section 5.5]). Our result gives some more evidence to the conjecture, proving it in the special case when d = 1, b = 3 and A is a very ample divisor.Note also that a much weaker version of Proposition 0.1 was claimed in [5, p. 216] and used in an essential way in [2].We work on the complex field C and use the standard terminology in algebraic geometry. In particular, we use the additive notation for line bundles, and we denote the canonical bundle of a projective manifold M by K M .We refer to [3] and [2] for any background material we need.

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“…By [2], Theorem 2.3.11 we get that P ic(X) Z[L], so that, if X is not of general type, then K X = eL, with e ≤ 0. The adjunction formula gives c 1 (N ) = (e + n)L. By plugging into (5) we get 1 2…”

Section: Let Us Choose a General Line ⊆ |L|mentioning

confidence: 99%

A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

Cataldo

1997

Trans. Amer. Math. Soc.

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Abstract. We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics Q n which are not of general type, for n = 5 and n ≥ 7. We prove a similar statement also for the case of higher codimension. IntroductionThere are only finitely many families of codimension two nonsingular subvarieties not of general type of the projective spaces P n , for n ≥ 4; see [7] and [4]. More generally, a similar statement holds for the case of higher codimension; see [16].In this paper we concentrate on the case of codimension two subvarieties of quadrics. Our main result is Theorem 4.3: there are only finitely many families of nonsingular codimension two subvarieties not of general type in the quadrics Q n , n = 4, 5 or n ≥ 7. The case n = 4 is proved in [1], §6. The case of n = 5 is at the heart of the paper; the main tools are the semipositivity of the normal bundles of nonsingular subvarieties of quadrics, the double point formula, the generalized Hodge index theorem, bounds for the genus of curves on Q 3 , Proposition 1.5 and Corollary 3.4. The case n = 6 is still open 1 . The case of codimension two with n ≥ 7 is covered by Theorem 2.1, which hinges upon the result of [16]; it also gives a finiteness result in codimension bigger than two in the same spirit as [16].The paper is organized as follows. Section 1 records, for the reader's convenience, some results used in the paper. A generalization of a lifting criterion of Roth's is contained in section 1.1; we shall need the particular case expressed by Proposition 1.4. Section 2 deals with higher codimensional cases. Sections 3 and 4 are modeled on [4]. Section 3 contains the lengthy proof of Theorem 3.1 and of its Corollary 3.4. Section 4 contains the proof of Theorem 4.3. Notation and conventions.Our basic reference is [10]. We work over any algebraically closed field of characteristic zero. A quadric Q n , here, is a nonsingular hypersurface of degree two in the projective space P n+1 . Little or no distinction is made between line bundles, associated sheaves of sections and Cartier divisors; moreover the additive and tensor notation are both used.

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The Adjunction Theory of Complex Projective Varieties

Cited by 274 publications

References 0 publications

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

Secant Degeneracy Index of the Standard Strata in The Space of Binary Forms

A Note on P1-Bundles as Hyperplane Sections

A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

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