1. CMB 2016 (vol 59 pp. 865)
 Pal, Sarbeswar

Moduli of Rank 2 Stable Bundles and Hecke Curves
Let $X$ be smooth projective curve of arbitrary genus $g \gt 3$
over complex numbers. In this short note we will show that the
moduli
space of rank $2$ stable vector bundles with determinant isomorphic
to $L_x$, where $L_x$ denote the line bundle corresponding to
a point $x \in X$ is isomorphic to certain lines in the moduli
space of Sequivalence classes of semistable bundles of rank
2 with
trivial determinant.
Keywords:Hecke curve, (0,1) stable bundle Category:14D21 

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 semistable sheaves with
Hilbert polynomial $dm1$ 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 Category:14D20 

3. CMB 2016 (vol 59 pp. 858)
 Osserman, Brian

Stability of Vector Bundles on Curves and Degenerations
We introduce a weaker notion of (semi)stability for vector bundles
on
reducible curves which does not depend on a choice of polarization,
and
which suffices for many applications of degeneration techniques.
We explore the basic
properties of this alternate notion of (semi)stability. In a
complementary
direction, we record a proof of the existence of semistable extensions
of vector bundles in suitable degenerations.
Keywords:vector bundle, stability, degeneration Categories:14D06, 14H60 

4. CMB 2013 (vol 57 pp. 439)
 Yang, YanHong

The Fixed Point Locus of the Verschiebung on $\mathcal{M}_X(2, 0)$ for Genus2 Curves $X$ in Charateristic $2$
We prove that for every ordinary genus$2$ curve $X$ over a finite
field $\kappa$ of characteristic $2$ with
$\textrm{Aut}(X/\kappa)=\mathbb{Z}/2\mathbb{Z} \times S_3$, there exist
$\textrm{SL}(2,\kappa[\![s]\!])$representations of $\pi_1(X)$ such
that the image of $\pi_1(\overline{X})$ is infinite. This result
produces a family of examples similar to Laszlo's counterexample
to de Jong's question regarding the finiteness of the geometric
monodromy of representations of the fundamental group.
Keywords:vector bundle, Frobenius pullback, representation, etale fundamental group Categories:14H60, 14D05, 14G15 

5. CMB 2011 (vol 55 pp. 26)
 Bertin, Marie José

A Mahler Measure of a $K3$ Surface Expressed as a Dirichlet $L$Series
We present another example of a $3$variable polynomial defining a $K3$hypersurface
and having a logarithmic Mahler measure expressed in terms of a Dirichlet
$L$series.
Keywords:modular Mahler measure, EisensteinKronecker series, $L$series of $K3$surfaces, $l$adic representations, LivnÃ© criterion, RankinCohen brackets Categories:11, 14D, 14J 

6. CMB 2011 (vol 54 pp. 472)
 Iacono, Donatella

A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps
We study infinitesimal deformations of holomorphic maps of
compact, complex, KÃ¤hler manifolds. In particular, we describe a
generalization of Bloch's semiregularity map that annihilates
obstructions to deform holomorphic maps with fixed codomain.
Keywords:semiregularity map, obstruction theory, functors of Artin rings, differential graded Lie algebras Categories:13D10, 14D15, 14B10 

7. 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 nonnegative
linear combinations of boundary divisors and the divisor of maps with
degenerate image.
Categories:14D20, 14E99, 14H10 

8. 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 $n+1$ semistable
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 

9. CMB 2005 (vol 48 pp. 90)
 Jeffrey, Lisa C.; Mare, AugustinLiviu

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{JeWe}.
Categories:14D20, 14P05 

10. 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 $n3$. 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 

11. CMB 2000 (vol 43 pp. 174)
12. 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 

13. CMB 1999 (vol 42 pp. 307)
 Kapovich, Michael; Millson, John J.

On the Moduli Space of a Spherical Polygonal Linkage
We give a ``wallcrossing'' 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 sidethe length of the last side is
allowed to vary, the first $(n  1)$ sidelengths 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 backtracks and the winding
number. We use our formula to determine the moduli spaces of all
regular pentagonal spherical linkages.
Categories:14D20, 14P05 
