101. CJM 2006 (vol 58 pp. 1000)
 Dhillon, Ajneet

On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of $\operatorname{SL}_n$
We compute some Hodge and Betti numbers of the moduli space of stable rank $r$,
degree $d$ vector bundles on a smooth projective curve. We
do not assume $r$ and $d$ are coprime.
In the process we equip the cohomology of an arbitrary algebraic stack with a
functorial mixed Hodge structure. This Hodge structure is
computed in the case of the moduli stack of rank $r$, degree
$d$ vector bundles on a curve. Our methods also yield a formula
for the Poincar\'e
polynomial of the moduli stack that is valid over any
ground field. In the last section we use the previous sections
to give a proof that the Tamagawa number of $\sln$ is one.
Categories:14H, 14L 

102. CJM 2006 (vol 58 pp. 476)
 Chipalkatti, Jaydeep

Apolar Schemes of Algebraic Forms
This is a note on the classical Waring's problem for algebraic forms.
Fix integers $(n,d,r,s)$, and let $\Lambda$ be a general $r$dimensional
subspace of degree $d$ homogeneous polynomials in $n+1$ variables. Let
$\mathcal{A}$ denote the variety of $s$sided polar polyhedra of $\Lambda$.
We carry out a casebycase study of the structure of $\mathcal{A}$ for several
specific values of $(n,d,r,s)$. In the first batch of examples, $\mathcal{A}$ is
shown to be a rational variety. In the second batch, $\mathcal{A}$ is a
finite set of which we calculate the cardinality.}
Keywords:Waring's problem, apolarity, polar polyhedron Categories:14N05, 14N15 

103. CJM 2006 (vol 58 pp. 262)
104. CJM 2006 (vol 58 pp. 93)
 Gordon, Julia

Motivic Haar Measure on Reductive Groups
We define a motivic analogue of the Haar measure for groups of the form
$G(k\llp t\rrp)$, where~$k$ is an algebraically closed field
of characteristic zero, and $G$ is a reductive algebraic group defined over
$k$.
A classical Haar measure on such groups does not
exist since they are not locally compact.
We use the theory of motivic integration introduced by M.~Kontsevich to
define an additive function on a certain natural Boolean algebra of subsets of
$G(k\llp t\rrp)$. This function takes values in the socalled dimensional
completion of
the Grothendieck ring of the category of varieties over the base
field. It is invariant under translations by all elements of $G(k\llp t\rrp)$,
and therefore we call it a motivic analogue of Haar measure.
We give an explicit construction of the motivic Haar measure, and then prove
that the result is independent of all the choices that are made in the process.
Keywords:motivic integration, reductive group Categories:14A15, 14L15 

105. CJM 2005 (vol 57 pp. 1178)
 Cutkosky, Steven Dale; Hà, Huy Tài; Srinivasan, Hema; Theodorescu, Emanoil

Asymptotic Behavior of the Length of Local Cohomology
Let $k$ be a field of characteristic 0, $R=k[x_1, \ldots, x_d]$ be a polynomial ring,
and $\mm$ its maximal homogeneous ideal. Let $I \subset R$ be a homogeneous ideal in
$R$. Let $\lambda(M)$ denote the length of an $R$module $M$. In this paper, we show
that
$$
\lim_{n \to \infty} \frac{\l\bigl(H^0_{\mathfrak{m}}(R/I^n)\bigr)}{n^d}
=\lim_{n \to \infty} \frac{\l\bigl(\Ext^d_R\bigl(R/I^n,R(d)\bigr)\bigr)}{n^d}
$$
always exists. This limit has been shown to be ${e(I)}/{d!}$ for $m$primary ideals
$I$ in a local CohenMacaulay ring, where $e(I)$ denotes the multiplicity
of $I$. But we find that this limit may not be rational in general. We give an example
for which the limit is an irrational number thereby showing that the lengths of these
extention modules may not have polynomial growth.
Keywords:powers of ideals, local cohomology, Hilbert function, linear growth Categories:13D40, 14B15, 13D45 

106. CJM 2005 (vol 57 pp. 1314)
 Zhitomirskii, M.

Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the DarbouxGivental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 

107. CJM 2005 (vol 57 pp. 724)
 Purnaprajna, B. P.

Some Results on Surfaces of General Type
In this article we prove some new results on projective normality, normal
presentation and higher syzygies for surfaces of general type, not
necessarily smooth, embedded by adjoint linear series. Some of the
corollaries of more general results include: results on property $N_p$
associated to $K_S \otimes B^{\otimes n}$ where $B$ is basepoint free and
ample divisor with $B\otimes K^*$ {\it nef}, results for pluricanonical
linear systems and results giving effective bounds for adjoint linear series
associated to ample bundles. Examples in the last section show that the results
are optimal.
Categories:13D02, 14C20, 14J29 

108. CJM 2005 (vol 57 pp. 338)
 Lange, Tanja; Shparlinski, Igor E.

Certain Exponential Sums and Random Walks on Elliptic Curves
For a given elliptic curve $\E$, we obtain an upper bound
on the discrepancy of sets of
multiples $z_sG$ where $z_s$ runs through a sequence
$\cZ=\(z_1, \dots, z_T\)$
such that $k z_1,\dots, kz_T $ is a permutation of
$z_1, \dots, z_T$, both sequences taken modulo $t$, for
sufficiently many distinct values of $k$ modulo $t$.
We apply this result to studying an analogue of the power generator
over an elliptic curve. These results are elliptic curve analogues
of those obtained for multiplicative groups of finite fields and
residue rings.
Categories:11L07, 11T23, 11T71, 14H52, 94A60 

109. CJM 2005 (vol 57 pp. 400)
 Sabourin, Sindi

Generalized $k$Configurations
In this paper, we find configurations of points in $n$dimensional
projective space ($\proj ^n$) which simultaneously generalize both
$k$configurations and reduced 0dimensional complete intersections.
Recall that $k$configurations in $\proj ^2$ are disjoint unions of
distinct points on lines and in $\proj ^n$ are inductively disjoint
unions of $k$configurations on hyperplanes, subject to certain
conditions. Furthermore, the Hilbert function of a $k$configuration
is determined from those of the smaller $k$configurations. We call
our generalized constructions $k_D$configurations, where $D=\{ d_1,
\ldots ,d_r\}$ (a set of $r$ positive integers with repetition
allowed) is the type of a given complete intersection in $\proj ^n$.
We show that the Hilbert function of any $k_D$configuration can be
obtained from those of smaller $k_D$configurations. We then provide
applications of this result in two different directions, both of which
are motivated by corresponding results about $k$configurations.
Categories:13D40, 14M10 

110. CJM 2005 (vol 57 pp. 3)
 AlberichCarramiñana, Maria; Roé, Joaquim

Enriques Diagrams and Adjacency of Planar Curve Singularities
We study adjacency of equisingularity types of planar complex
curve singularities
in terms of their Enriques diagrams. The goal is, given two equisingularity
types, to determine whether one of them is adjacent to the other. For linear
adjacency a complete answer is obtained, whereas for arbitrary (analytic)
adjacency a necessary condition and a sufficient condition are
proved. We also obtain new examples of exceptional deformations,
{\em i.e.,} singular curves of type
$\mathcal{D}'$ that can be deformed to a curve of type $\mathcal{D}$ without
$\mathcal{D}'$ being adjacent to $\mathcal{D}$.


111. CJM 2004 (vol 56 pp. 1308)
 Zhao, Jianqiang

Variations of Mixed Hodge Structures of Multiple Polylogarithms
It is well known that multiple polylogarithms give rise to
good unipotent variations of mixed HodgeTate structures.
In this paper we shall {\em explicitly} determine these structures
related to multiple logarithms and some other multiple polylogarithms
of lower weights. The purpose of this explicit construction
is to give some important applications: First we study the limit of
mixed HodgeTate structures and make a conjecture relating the variations
of mixed HodgeTate structures of multiple logarithms to those of
general multiple {\em poly}\/logarithms. Then following
Deligne and Beilinson we describe an
approach to defining the singlevalued
real analytic version of the multiple polylogarithms which
generalizes the wellknown result of Zagier on
classical polylogarithms. In the process we find some interesting
identities relating singlevalued multiple polylogarithms of the
same weight $k$ when $k=2$ and 3. At the end of this paper,
motivated by Zagier's conjecture we pose
a problem which relates the special values of multiple
Dedekind zeta functions of a number field to the singlevalued
version of multiple polylogarithms.
Categories:14D07, 14D05, 33B30 

112. CJM 2004 (vol 56 pp. 1145)
113. CJM 2004 (vol 56 pp. 1094)
 Thomas, Hugh

CycleLevel Intersection Theory for Toric Varieties
This paper addresses the problem of constructing a
cyclelevel intersection theory for toric varieties.
We show that by making one global choice,
we can determine a cycle representative
for the intersection of an equivariant Cartier divisor with an invariant
cycle on a toric variety. For a toric variety
defined by a fan in $N$, the choice consists of giving an
inner product or a complete flag for $M_\Q=
\Qt \Hom(N,\mathbb{Z})$, or more
generally giving for each cone $\s$ in the fan a linear subspace of
$M_\Q$ complementary to $\s^\perp$, satisfying certain compatibility
conditions.
We show that these intersection cycles have properties analogous to the
usual intersections modulo rational equivalence.
If $X$ is simplicial (for instance, if $X$ is nonsingular),
we obtain a commutative ring structure
to the invariant cycles of $X$ with rational
coefficients. This ring structure determines cycles representing
certain characteristic classes of the toric variety.
We also discuss
how to define intersection cycles that require no choices,
at the expense of increasing
the size of the coefficient field.
Keywords:toric varieties, intersection theory Categories:14M25, 14C17 

114. CJM 2004 (vol 56 pp. 716)
 Guardo, Elena; Van Tuyl, Adam

Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions
We study the Hilbert functions of fat points in $\popo$.
If $Z \subseteq \popo$ is an arbitrary fat point scheme, then
it can be shown that for every $i$ and $j$ the values of the Hilbert
function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for
$l \gg 0$. We show how to determine these eventual values
by using only the multiplicities of the points, and the
relative positions of the points in $\popo$. This enables
us to compute all but a finite number values of $H_{Z}$
without using the coordinates of points.
We also characterize the ACM fat point schemes
sing our description of the eventual behaviour. In fact,
n the case that $Z \subseteq \popo$ is ACM, then
the entire Hilbert function and its minimal free resolution
depend solely on knowing the eventual values of the Hilbert function.
Keywords:Hilbert function, points, fat points, CohenMacaulay, multiprojective space Categories:13D40, 13D02, 13H10, 14A15 

115. CJM 2004 (vol 56 pp. 495)
 Gomi, Yasushi; Nakamura, Iku; Shinoda, Kenichi

Coinvariant Algebras of Finite Subgroups of $\SL(3,C)$
For most of the finite subgroups of $\SL(3,\mathbf{C})$, we give explicit formulae for
the Molien series of the coinvariant algebras, generalizing McKay's formulae
\cite{M99} for subgroups of $\SU(2)$. We also study the $G$orbit Hilbert scheme
$\Hilb^G(\mathbf{C}^3)$ for any finite subgroup $G$ of $\SO(3)$, which is known to be a
minimal (crepant) resolution of the orbit space $\mathbf{C}^3/G$. In this case the fiber
over the origin of the HilbertChow morphism from $\Hilb^G(\mathbf{C}^3)$ to $\mathbf{C}^3/G$
consists of finitely many smooth rational curves, whose planar dual graph is
identified with a certain subgraph of the representation graph of $G$. This is
an $\SO(3)$ version of the McKay correspondence in the $\SU(2)$ case.
Keywords:Hilbert scheme, Invariant theory, Coinvariant algebra,, McKay quiver, McKay correspondence Categories:14J30, 14J17 

116. CJM 2004 (vol 56 pp. 612)
 Pál, Ambrus

Solvable Points on Projective Algebraic Curves
We examine the problem of finding rational points defined over
solvable extensions on algebraic curves defined over general fields.
We construct nonsingular, geometrically irreducible projective curves
without solvable points of genus $g$, when $g$ is at least $40$, over
fields of arbitrary characteristic. We prove that every smooth,
geometrically irreducible projective curve of genus $0$, $2$, $3$ or
$4$ defined over any field has a solvable point. Finally we prove
that every genus $1$ curve defined over a local field of
characteristic zero with residue field of characteristic $p$ has a
divisor of degree prime to $6p$ defined over a solvable extension.
Categories:14H25, 11D88 

117. CJM 2004 (vol 56 pp. 310)
 Llibre, Jaume; Schlomiuk, Dana

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order
In this article we determine the global geometry of the planar
quadratic differential systems with a weak focus of third order. This
class plays a significant role in the context of Hilbert's 16th
problem. Indeed, all examples of quadratic differential systems with
at least four limit cycles, were obtained by perturbing a system in
this family. We use the algebrogeometric concepts of divisor and
zerocycle to encode global properties of the systems and to give
structure to this class. We give a theorem of topological
classification of such systems in terms of integervalued affine
invariants. According to the possible values taken by them in this
family we obtain a total of $18$ topologically distinct phase
portraits. We show that inside the class of all quadratic systems
with the topology of the coefficients, there exists a neighborhood of
the family of quadratic systems with a weak focus of third order and
which may have graphics but no polycycle in the sense of \cite{DRR}
and no limit cycle, such that any quadratic system in this
neighborhood has at most four limit cycles.
Categories:34C40, 51F14, 14D05, 14D25 

118. CJM 2003 (vol 55 pp. 1100)
 Khesin, Boris; Rosly, Alexei

Polar Homology
For complex projective manifolds we introduce polar homology
groups, which are holomorphic analogues of the homology groups in
topology. The polar $k$chains are subvarieties of complex
dimension $k$ with meromorphic forms on them, while the boundary
operator is defined by taking the polar divisor and the Poincar\'e
residue on it. One can also define the corresponding analogues for the
intersection and linking numbers of complex submanifolds, which have the
properties similar to those of the corresponding topological notions.
Keywords:Poincar\' e residue, holomorphic linking Categories:14C10, 14F10, 58A14 

119. CJM 2003 (vol 55 pp. 897)
 Archinard, Natália

Hypergeometric Abelian Varieties
In this paper, we construct abelian varieties associated to Gauss' and
AppellLauricella hypergeometric series.
Abelian varieties of this kind and the algebraic curves we define
to construct them were considered by several authors in settings
ranging from monodromy groups (Deligne, Mostow), exceptional sets
(Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to
CMtype (Cohen, Shiga, Wolfart) and modularity (Darmon).
Our contribution is to provide a complete, explicit and selfcontained
geometric construction.
Categories:11, 14 

120. CJM 2003 (vol 55 pp. 766)
 Kerler, Thomas

Homology TQFT's and the AlexanderReidemeister Invariant of 3Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skeintheoretical algorithm to compute the
Alexander polynomial of a 3manifold from a surgery presentation
employing the methods used in the construction of quantum invariants
of 3manifolds. As a prerequisite we establish and prove a rather
unexpected equivalence between the topological quantum field theory
constructed by Frohman and Nicas using the homology of
$U(1)$representation varieties on the one side and the
combinatorially constructed Hennings TQFT based on the quasitriangular
Hopf algebra $\mathcal{N} = \mathbb{Z}/2 \ltimes \bigwedge^*
\mathbb{R}^2$ on the other side. We find that both TQFT's are $\SL
(2,\mathbb{R})$equivariant functors and, as such, are isomorphic.
The $\SL (2,\mathbb{R})$action in the Hennings construction comes
from the natural action on $\mathcal{N}$ and in the case of the
FrohmanNicas theory from the HardLefschetz decomposition of the
$U(1)$moduli spaces given that they are naturally K\"ahler. The
irreducible components of this TQFT, corresponding to simple
representations of $\SL(2,\mathbb{Z})$ and $\Sp(2g,\mathbb{Z})$, thus
yield a large family of homological TQFT's by taking sums and products.
We give several examples of TQFT's and invariants that appear to fit
into this family, such as Milnor and Reidemeister Torsion,
SeibergWitten theories, Casson type theories for homology circles
{\it \`a la} Donaldson, higher rank gauge theories following Frohman
and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of
ReshetikhinTuraev theories over the cyclotomic integers $\mathbb{Z}
[\zeta_p]$. We also conjecture that the Hennings TQFT for
quantum$\mathfrak{sl}_2$ is the product of the ReshetikhinTuraev
TQFT and such a homological TQFT.
Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27 

121. CJM 2003 (vol 55 pp. 839)
 Lee, Min Ho

Cohomology of Complex Torus Bundles Associated to Cocycles
Equivariant holomorphic maps of Hermitian symmetric domains into
Siegel upper half spaces can be used to construct families of
abelian varieties parametrized by locally symmetric spaces, which
can be regarded as complex torus bundles over the parameter spaces.
We extend the construction of such torus bundles using 2cocycles of
discrete subgroups of the semisimple Lie groups associated to the
given symmetric domains and investigate some of their properties.
In particular, we determine their cohomology along the fibers.
Categories:14K10, 14D06, 14F99 

122. CJM 2003 (vol 55 pp. 693)
 Borne, Niels

Une formule de RiemannRoch Ã©quivariante pour les courbes
Soit $G$ un groupe fini agissant sur une courbe alg\'ebrique
projective et lisse $X$ sur un corps alg\'ebriquement clos $k$. Dans
cet article, on donne une formule de RiemannRoch pour la
caract\'eristique d'Euler \'equivariante d'un $G$faisceau inversible
$\mathcal{L}$, \`a valeurs dans l'anneau $R_k (G)$ des caract\`eres du
groupe $G$. La formule donn\'ee a un bon comportement fonctoriel en
ce sens qu'elle rel\`eve la formule classique le long du morphisme
$\dim \colon R_k (G) \to \mathbb{Z}$, et est valable m\^eme pour une
action sauvage. En guise d'application, on montre comment calculer
explicitement le caract\`ere de l'espace des sections globales d'une
large classe de $G$faisceaux inversibles, en s'attardant sur le cas
particulier d\'elicat du faisceau des diff\'erentielles sur la courbe.
Keywords:group actions on varieties or schemes,, RiemannRoch theorems Categories:14L30, 14C40 

123. CJM 2003 (vol 55 pp. 609)
 Moraru, Ruxandra

Integrable Systems Associated to a Hopf Surface
A Hopf surface is the quotient of the complex surface $\mathbb{C}^2
\setminus \{0\}$ by an infinite cyclic group of dilations of
$\mathbb{C}^2$. In this paper, we study the moduli spaces
$\mathcal{M}^n$ of stable $\SL (2,\mathbb{C})$bundles on a Hopf
surface $\mathcal{H}$, from the point of view of symplectic geometry.
An important point is that the surface $\mathcal{H}$ is an elliptic
fibration, which implies that a vector bundle on $\mathcal{H}$ can be
considered as a family of vector bundles over an elliptic curve. We
define a map $G \colon \mathcal{M}^n \rightarrow \mathbb{P}^{2n+1}$
that associates to every bundle on $\mathcal{H}$ a divisor, called the
graph of the bundle, which encodes the isomorphism class of the bundle
over each elliptic curve. We then prove that the map $G$ is an
algebraically completely integrable Hamiltonian system, with respect
to a given Poisson structure on $\mathcal{M}^n$. We also give an
explicit description of the fibres of the integrable system. This
example is interesting for several reasons; in particular, since the
Hopf surface is not K\"ahler, it is an elliptic fibration that does
not admit a section.
Categories:14J60, 14D21, 14H70, 14J27 

124. CJM 2003 (vol 55 pp. 649)
125. CJM 2003 (vol 55 pp. 561)