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1
Content available remote

Recent results on quiver sheaves

100%
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 • ← • → •.
2
Content available remote

Rank-two vector bundles on Hirzebruch surfaces

100%
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.
3
Content available remote

The Brauer group of desingularization of moduli spaces of vector bundles over a curve

100%
Biswas I., Hogadi A., Holla Y.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1300-1305
EN
Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧r E = L. We show that the Brauer group of any desingularization of M C(r; L) is trivial.
4
Content available remote

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.
5
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Morse index of a cyclic polygon

100%
Panina G., Zhukova A.
Open Mathematics
|
2011
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tom 9
|
nr 2
364-377
EN
It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper we give an explicit formula of the Morse index for the signed area of a cyclic configuration. We show that it depends not only on the combinatorics of a cyclic configuration, but also on its metric properties.
6
Content available remote

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.
7
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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).
8
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Unramified Brauer group of the moduli spaces of PGLr(ℂ)-bundles over curves

88%
Biswas I., Hogadi A., Holla Y.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1157-1163
EN
Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.
9
Content available remote

Critical configurations of planar robot arms

63%
Khimshiashvili G., Panina G., Siersma D., Zhukova A.
Open Mathematics
|
2013
|
tom 11
|
nr 3
519-529
EN
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.
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