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Search: MSC category 14D20 ( Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} )

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1. CMB Online first

Maican, Mario
Moduli of space sheaves with Hilbert polynomial $4m+1$
We investigate the moduli space of sheaves supported on space curves of degree $4$ and having Euler characteristic $1$. We give an elementary proof of the fact that this moduli space consists of three irreducible components.

Keywords:moduli of sheaves, semi-stable sheaves
Categories:14D20, 14D22

2. CMB Online first

Iena, Oleksandr; Leytem, Alain
On the singular sheaves in the fine Simpson moduli spaces of $1$-dimensional sheaves
In the Simpson moduli space $M$ of semi-stable sheaves with Hilbert polynomial $dm-1$ on a projective plane we study the closed subvariety $M'$ of sheaves that are not locally free on their support. We show that for $d\ge 4$ it is a singular subvariety of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted as a (partial) modification of $M\setminus M'$ by line bundles (on support).

Keywords:Simpson moduli spaces, coherent sheaves, vector bundles on curves, singular sheaves

3. CMB 2008 (vol 51 pp. 519)

Coskun, Izzet; Harris, Joe; Starr, Jason
The Effective Cone of the Kontsevich Moduli Space
In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, $\Kgnb{0,0}(\PP^r,d)$, stabilize when $r \geq d$. We give a complete characterization of the effective divisors on $\Kgnb{0,0}(\PP^d,d)$. They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

Categories:14D20, 14E99, 14H10

4. CMB 2007 (vol 50 pp. 427)

Mejía, Israel Moreno
On the Image of Certain Extension Maps.~I
Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.

Categories:14H60, 14F05, 14D20

5. CMB 2005 (vol 48 pp. 90)

Jeffrey, Lisa C.; Mare, Augustin-Liviu
Products of Conjugacy Classes in $SU(2)$
We obtain a complete description of the conjugacy classes $C_1,\dots,C_n$ in $SU(2)$ with the property that $C_1\cdots C_n=SU(2)$. The basic instrument is a characterization of the conjugacy classes $C_1,\dots,C_{n+1}$ in $SU(2)$ with $C_1\cdots C_{n+1}\ni I$, which generalizes a result of \cite{Je-We}.

Categories:14D20, 14P05

6. CMB 2002 (vol 45 pp. 417)

Kamiyama, Yasuhiko; Tsukuda, Shuichi
On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons
For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the sheaf of germs of holomorphic sections of the tangent bundle $TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all $q \geq 0$ and all odd $n$. In particular, we see that the moduli space of deformations of the complex structure on $M_n$ consists of a point. Thus the complex structure on $M_n$ is locally rigid.

Keywords:polygon space, complex structure
Categories:14D20, 32C35

7. CMB 2000 (vol 43 pp. 174)

Gantz, Christian; Steer, Brian
Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces
We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

Categories:14J27, 32L07, 14H60, 14D20

8. CMB 2000 (vol 43 pp. 162)

Foth, Philip
Moduli Spaces of Polygons and Punctured Riemann Spheres
The purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces $\P_r$ of polygons with fixed side lengths in the Euclidean space $\E^3$. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of $\P_r$. We also classify moduli spaces of pentagons.

Categories:14D20, 18G55, 14H10

9. CMB 1999 (vol 42 pp. 307)

Kapovich, Michael; Millson, John J.
On the Moduli Space of a Spherical Polygonal Linkage
We give a ``wall-crossing'' formula for computing the topology of the moduli space of a closed $n$-gon linkage on $\mathbb{S}^2$. We do this by determining the Morse theory of the function $\rho_n$ on the moduli space of $n$-gon linkages which is given by the length of the last side---the length of the last side is allowed to vary, the first $(n - 1)$ side-lengths are fixed. We obtain a Morse function on the $(n - 2)$-torus with level sets moduli spaces of $n$-gon linkages. The critical points of $\rho_n$ are the linkages which are contained in a great circle. We give a formula for the signature of the Hessian of $\rho_n$ at such a linkage in terms of the number of back-tracks and the winding number. We use our formula to determine the moduli spaces of all regular pentagonal spherical linkages.

Categories:14D20, 14P05

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