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1
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A note on a Brill-Noether locus over a non-hyperelliptic curve of genus 4

100
Huh S.
Open Mathematics
|
2009
|
tom 7
|
nr 4
617-622
EN
We prove that a certain Brill-Noether locus over a non-hyperelliptic curve C of genus 4, is isomorphic to the Donagi-Izadi cubic threefold in the case when the pencils of the two trigonal line bundles of C coincide.
2
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SUX(r, L) is separably unirational

100
Hein G.
Open Mathematics
|
2009
|
tom 7
|
nr 1
59-60
EN
We show that the moduli space of SUX (r, L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational.
3
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Λ-modules and holomorphic Lie algebroid connections

100
Tortella P.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1422-1441
EN
Let X be a complex smooth projective variety, and G a locally free sheaf on X. We show that there is a one-to-one correspondence between pairs (Λ, Ξ), where Λ is a sheaf of almost polynomial filtered algebras over X satisfying Simpson’s axioms and $ \equiv :Gr\Lambda \to Sym \bullet
_{\mathcal{O}_X } \mathcal{G}$ is an isomorphism, and pairs (L, Σ), where L is a holomorphic Lie algebroid structure on $\mathcal{G}$ and Σ is a class in F 1 H 2(L, ℂ), the first Hodge filtration piece of the second cohomology of L. As an application, we construct moduli spaces of semistable flat L-connections for any holomorphic Lie algebroid L. Particular examples of these are given by generalized holomorphic bundles for any generalized complex structure associated to a holomorphic Poisson manifold.
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4
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The Picard group of a coarse moduli space of vector bundles in positive characteristic

100
Hoffmann N.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1306-1313
EN
Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M r,Lss denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M r,Lss) = ℤ, identify the ample generator, and deduce
that M r,Lss is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive characteristic.
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5
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Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ3

100
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).
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6
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The role of the Ellingsrud-Strømme construction in the classification of instantons

100
Gruson L., Han F.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1188-1197
EN
We review a construction of Ellingsrud-Strømme relating instantons of charge n on the ordinary projective space and theta-characteristics on a plane curve of degree n with some extra-structure.
7
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Elliptic curves on spinor varieties

76
Perrin N.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1393-1406
EN
We prove irreducibility of the scheme of morphisms, of degree large enough, from a smooth elliptic curve to spinor varieties. We give an explicit bound on the degree.
8
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Bubble tree compactification of moduli spaces of vector bundles on surfaces

76
Markushevich D., Tikhomirov A., Trautmann G.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1331-1355
EN
We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli
spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c 1 = 0, c 1 = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.
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9
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Symplectic structures on moduli spaces of framed sheaves on surfaces

76
Sala F.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1455-1471
EN
We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed
sheaves on some birationally ruled surfaces.
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10
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On the Brill-Noether theory for K3 surfaces

76
Leyenson M.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1486-1540
EN
Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for
curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.
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11
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Recent results on quiver sheaves

64
Laudin A., Schmitt A.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1246-1279
EN
In this article, we survey recent work on the construction and geometry of representations of a quiver in the category of coherent sheaves on a projective algebraic manifold. We will also prove new results in the case of the quiver • ← • → •.
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