A Home-Made Hartshorne-Serre Correspondence

Arrondo, Enrique

doi:10.5209/rev_rema.2007.v20.n2.16502

Cited by 52 publications

(87 citation statements)

References 10 publications

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“…We observe that under the assumption thatX contains no fiber of the projection π, we have E = E| E restricted to every fiber of the projection π is a bundle of rank 2 with trivial first Chern class and either a nonvanishing section or a section vanishing in one point, i.e., it is either (1). It follows that π * (E ) is a vector bundle of rank 2 and R i π * E = 0.…”

Section: By the Projection Formula We Have E(−3ξ +(D − 12)h) Has A Smentioning

confidence: 99%

“…The proof that this classification includes all Calabi-Yau threefolds in P 6 of degree d ≤ 14 and a classification of Calabi-Yau threefolds contained in 5-dimensional quadrics is our main result. The classification is given by providing a list of vector bundles {E i } i∈I such that the considered Calabi-Yau threefolds are exactly the smooth threefolds which appear as Pfaffians Pf(σ ) for some σ ∈ H 0 ( 2 E i (1)) and i ∈ I . Let us point out that our list contains two distinct vector bundles corresponding to degree 14 Calabi-Yau threefolds.…”

Section: Problem 11 Classify the Calabi-yau Threefolds In Pmentioning

confidence: 99%

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Calabi–Yau threefolds in $${\mathbb {P}}^6$$ P 6

Kapustka

2015

Annali di Matematica

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We study the geometry of 3-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi-Yau threefolds in projective 6-space. Moreover, we prove that this classification includes all Calabi-Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi-Yau threefolds of degree at most 14 in P 6 . Keywords Calabi-Yau threefolds · Pfaffian varieties · Canonical surfaces Mathematics Subject Classification Primary: 14J32 IntroductionIt is conjectured that, when 2n ≥ N , there is a finite number of smooth families of smooth n-dimensional subvarieties of P N that are not of general type. This conjecture was inspired by [15] where the statement was formulated and proven in the case of surfaces in P 4 . In [9], the conjecture was proven in the case of threefolds in P 5 . Moreover, Schneider in [33] proved that the statement is true when 2n ≥ N + 2. In this context, it is a natural problem to classify families of smooth n-folds of small degree in P N for chosen n, N ∈ N satisfying 2n ≥ N . In the case of codimension 2 subvarieties, this problem was addressed by many authors (see [3,6,11,17]).The next step is to study codimension 3 subvarieties in P 6 . In this case, standard tools such as the Barth-Lefschetz theorem do not apply. However, some general structure theorems were recently developed. We say that a submanifolds X ⊂ P n is subcanonical when ω X = O X (k) for some k ∈ Z. A codimension 3 submanifolds X is called Pfaffian if it is the first nonzero degeneracy locus of a skew-symmetric morphism of vector bundles of odd rank E * (−t) ϕ − → E where t ∈ Z. In this case, X is given locally by the vanishing of 2u × 2u Pfaffians of an alternating map ϕ from the vector bundle E of odd rank 2u + 1 to its twisted dual. More precisely, if X is Pfaffian, then we have:where s = c 1 (E) + ut. Moreover, from [29, Sect.3], we have in this caseSince the choice of an alternating map ϕ is equivalent to the choice of a section σ ∈ H 0 ( 2 E(t)), we will use the notation Pf(σ ) for the variety described by the Pfaffians of the map corresponding to σ . Answering a question of Okonek (see [29]), Walter, in [37], proved that if n is not divisible by 4 then a locally Gorenstein codimension 3 submanifold of P n+3 is Pfaffian if and only if it is subcanonical. In the case when n = 4k, the last statement is not true; however, there is another structure theorem (see [16]).The nongeneral type subcanonical threefolds in P 6 are either well understood Fano threefolds or threefolds with trivial canonical class. A very natural class of varieties among varieties with trivial canonical class are Calabi-Yau threefolds, i.e., smooth threefolds X with K X = 0 and H 1 (X, O X ) = 0. In the paper, we shall sometimes also consider singular Calabi-Yau threefolds by which we mean complex projective threefolds with Gorenstein singularities, ω X = 0 and with h 1 (O X ) = 0.For Calabi-Yau threefolds, the theory of Pfaffians is more specific. For instance, Schreyer,...

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Section: By the Projection Formula We Have E(−3ξ +(D − 12)h) Has A Smentioning

confidence: 99%

Section: Problem 11 Classify the Calabi-yau Threefolds In Pmentioning

confidence: 99%

Calabi–Yau threefolds in $${\mathbb {P}}^6$$ P 6

Kapustka

2015

Annali di Matematica

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“…In particular in all the above cases condition is satisfied with the only exceptions of cases (c 1 , c 2 ) ∈ {(−1, 1), (0, 2), (1, 5)} when r = 4 and (c 1 , c 2 ) = (0, 1) when r = 3. Moreover, as shown in [15], it also holds that the vector bundles on X 4 with (c 1 , c 2 ) = (3,14) and the general one with (c 1 , c 2 ) = (2, 8) are generated by their global sections.…”

Section: Moreover All the Cases Arisementioning

confidence: 93%

“…Next we recall the generalized Hartshorne-Serre correspondence (see [3], [23]), which states that is possible to reverse this process, in the sense that one recovers the vector bundle from the surjection in (2.5): This gives, whenever we have an exact sequence like (2.4), the relation between the two first Chern classes of E and the degree of C. Moreover, the third Chern class of E is determined by the (arithmetic) genus of the curve C:…”

Section: Preliminaries and Basic Factsmentioning

confidence: 99%

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Curves and Vector Bundles on Quartic Threefolds

Arrondo¹,

Madonna²

2009

Journal of the Korean Mathematical Society

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Abstract. In this paper we study arithmetically Cohen-Macaulay (ACM for short) vector bundles E of rank k ≥ 3 on hypersurfaces X r ⊂ P 4 of degree r ≥ 1. We consider here mainly the case of degree r = 4, which is the first unknown case in literature. Under some natural conditions for the bundle E we derive a list of possible Chern classes (c 1 , c 2 , c 3 ) which may arise in the cases of rank k = 3 and k = 4, when r = 4 and we give several examples.

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Globally generated vector bundles on with

Anghel

Manolache

2013

Mathematische Nachrichten

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Abstract. One classifies the globally generated vector bundles on P n whith the first Chern class c 1 = 3 (in this paper n = 3, the case n = 3 being studied in [18]). The case c 1 = 1 is very easy, the case c 1 = 2 was done in [28], the case c 1 = 3, rank= 2 was settled in [13] and the case c 1 ≤ 5, rank=2 in [6]. Our work is based on Serre's theorem relating vector bundles of rank = 2 with codimension 2 lci subschemes and its generalization for higher ranks, considered firstly by Vogelaar in [33].

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A Home-Made Hartshorne-Serre Correspondence

Cited by 52 publications

References 10 publications

Calabi–Yau threefolds in $${\mathbb {P}}^6$$ P 6

Calabi–Yau threefolds in $${\mathbb {P}}^6$$ P 6

Curves and Vector Bundles on Quartic Threefolds

Globally generated vector bundles on with

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