Notes on varieties of codimension 3 in ℙ N

Okonek, Christian

doi:10.1007/bf02567467

Cited by 29 publications

(47 citation statements)

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“…To date, liaison theory has not proved as useful in constructing examples as it has in codimension two. Perhaps the most interesting application of liaison to codimension three is the paper [32]. We believe that the principal reason that liaison has not proved useful in codimension three (apart from [32]) is that it is too restrictive to consider only complete intersections.…”

Section: Linkagementioning

confidence: 99%

“…Perhaps the most interesting application of liaison to codimension three is the paper [32]. We believe that the principal reason that liaison has not proved useful in codimension three (apart from [32]) is that it is too restrictive to consider only complete intersections. The main obstacle to extending this is that with present tools, it is hard to find an arithmetically Gorenstein ideal contained in a given ideal (of the same codimension), apart from the complete intersections (where one simply has to look for a regular sequence of the right length).…”

Section: Linkagementioning

confidence: 99%

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A construction of codimension three arithmetically Gorenstein subschemes of projective space

Migliore

Peterson

1997

Trans. Amer. Math. Soc.

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Abstract. This paper presents a construction method for a class of codimension three arithmetically Gorenstein subschemes of projective space. These schemes are obtained from degeneracy loci of sections of certain specially constructed rank three reflexive sheaves. In contrast to the structure theorem of Buchsbaum and Eisenbud, we cannot obtain every arithmetically Gorenstein codimension three subscheme by our method. However, certain geometric applications are facilitated by the geometric aspect of this construction, and we discuss several examples of this in the final section.

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Section: Linkagementioning

confidence: 99%

Section: Linkagementioning

confidence: 99%

A construction of codimension three arithmetically Gorenstein subschemes of projective space

Migliore

Peterson

1997

Trans. Amer. Math. Soc.

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“…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]).…”

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confidence: 99%

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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“…2 has special fiber given bŷ Theorem 10.2 (Buchsbaum-Eisenbud) [Buchsbaum, Eisenbud, 1977], [Okonek, 1994], A Pfaffian subscheme X of P n K has a locally free resolution…”

Section: Letŷmentioning

confidence: 99%

Mirror Symmetry and Tropical Geometry

Castaño-Bernard¹,

Soibelman²,

Zharkov³

2010

Contemporary Mathematics

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Using tropical geometry we propose a mirror construction for monomial degenerations of Calabi-Yau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for Calabi-Yau hypersurfaces and by Batyrev and Borisov for Calabi-Yau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rødland for the degree 14 Calabi-Yau threefold in P 6 defined by the Pfaffians of a general linear 7 × 7 skew-symmetric matrix.We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Gröbner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples.

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Notes on varieties of codimension 3 in ℙ N

Cited by 29 publications

References 25 publications

A construction of codimension three arithmetically Gorenstein subschemes of projective space

A construction of codimension three arithmetically Gorenstein subschemes of projective space

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

Mirror Symmetry and Tropical Geometry

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