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1. CMB 2008 (vol 51 pp. 519)
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 |
2. CMB 2007 (vol 50 pp. 427)
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 |
3. CMB 2005 (vol 48 pp. 90)
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 |
4. CMB 2002 (vol 45 pp. 417)
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 |
5. CMB 2000 (vol 43 pp. 174)
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 |
6. CMB 2000 (vol 43 pp. 162)
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 |
7. CMB 1999 (vol 42 pp. 307)
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 |