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On Buchsbaum bundles on quadric hypersurfaces

100%
Ballico E., Malaspina F., Valabrega P., Valenzano M.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1361-1379
EN
Let E be an indecomposable rank two vector bundle on the projective space ℙn, n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q n ⊂ ℙn+1, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q 3, k ≥ 2, we prove two boundedness results.
2
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ACM bundles, quintic threefolds and counting problems

100%
Mohan Kumar N., Rao A.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1380-1392
EN
We review some facts about rank two arithmetically Cohen-Macaulay bundles on quintic threefolds. In particular, we separate them into seventeen natural classes, only fourteen of which can appear on a general quintic. We discuss some enumerative problems arising from these.
3
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Instanton bundles on Fano threefolds

81%
Kuznetsov A.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1198-1231
EN
We introduce the notion of an instanton bundle on a Fano threefold of index 2. For such bundles we give an analogue of a monadic description and discuss the curve of jumping lines. The cases of threefolds of degree 5 and 4 are considered in a greater detail.
4
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Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

81%
Bruzzo U., Markushevich D., Tikhomirov A.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1232-1245
EN
Symplectic instanton vector bundles on the projective space ℙ3 constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I n;r of rank-2r symplectic instanton vector bundles on ℙ3 with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I n;r* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).
5
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Derived category of toric varieties with small Picard number

81%
Costa L., Miró-Roig R.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1280-1291
EN
This paper aims to construct a full strongly exceptional collection of line bundles in the derived category D b(X), where X is the blow up of ℙn−r ×ℙr along a multilinear subspace ℙn−r−1×ℙr−1 of codimension 2 of ℙn−r ×ℙr. As a main tool we use the splitting of the Frobenius direct image of line bundles on toric varieties.
6
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Rank-two vector bundles on Hirzebruch surfaces

52%
Aprodu M., Brînzănescu V., Marchitan M.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1321-1330
EN
We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.
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