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101. 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 case-by-case 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

102. CJM 2006 (vol 58 pp. 262)

Biswas, Indranil
Connections on a Parabolic Principal Bundle Over a Curve
The aim here is to define connections on a parabolic principal bundle. Some applications are given.

Keywords:parabolic bundle, holomorphic connection, unitary connection
Categories:53C07, 32L05, 14F05

103. 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 so-called 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

104. 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 Cohen--Macaulay 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

105. 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 Darboux-Givental' 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

106. 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 base-point 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

107. 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

108. 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 0-dimensional 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

109. CJM 2005 (vol 57 pp. 3)

Alberich-Carramiñana, 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 $\mathcal{D}$.

110. 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 Hodge-Tate 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 Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple {\em poly}\/logarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued 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 single-valued version of multiple polylogarithms.

Categories:14D07, 14D05, 33B30

111. CJM 2004 (vol 56 pp. 1145)

Daigle, Daniel; Russell, Peter
On Log $\mathbb Q$-Homology Planes and Weighted Projective Planes
We classify normal affine surfaces with trivial Makar-Limanov invariant and finite Picard group of the smooth locus, realizing them as open subsets of weighted projective planes. We also show that such a surface admits, up to conjugacy, one or two $G_a$-actions.

Categories:14R05, 14J26, 14R20

112. CJM 2004 (vol 56 pp. 1094)

Thomas, Hugh
Cycle-Level Intersection Theory for Toric Varieties
This paper addresses the problem of constructing a cycle-level 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 non-singular), 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

113. 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, Cohen-Macaulay, multi-projective space
Categories:13D40, 13D02, 13H10, 14A15

114. CJM 2004 (vol 56 pp. 495)

Gomi, Yasushi; Nakamura, Iku; Shinoda, Ken-ichi
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 Hilbert-Chow 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

115. 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 non-singular, 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

116. 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 16-th 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 algebro-geometric concepts of divisor and zero-cycle 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 integer-valued 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

117. 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

118. CJM 2003 (vol 55 pp. 897)

Archinard, Natália
Hypergeometric Abelian Varieties
In this paper, we construct abelian varieties associated to Gauss' and Appell--Lauricella 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 CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.

Categories:11, 14

119. CJM 2003 (vol 55 pp. 766)

Kerler, Thomas
Homology TQFT's and the Alexander--Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory
We develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. 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 Frohman--Nicas theory from the Hard--Lefschetz 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, Seiberg--Witten 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 Reshetikhin--Turaev 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 Reshetikhin--Turaev TQFT and such a homological TQFT.

Categories:57R56, 14D20, 16W30, 17B37, 18D35, 57M27

120. 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 2-cocycles 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

121. CJM 2003 (vol 55 pp. 693)

Borne, Niels
Une formule de Riemann-Roch équivariante 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 Riemann-Roch 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,, Riemann-Roch theorems
Categories:14L30, 14C40

122. CJM 2003 (vol 55 pp. 561)

Laface, Antonio; Ugaglia, Luca
Quasi-Homogeneous Linear Systems on $\mathbb{P}^2$ with Base Points of Multiplicity $5$
In this paper we consider linear systems of $\mathbb{P}^2$ with all but one of the base points of multiplicity $5$. We give an explicit way to evaluate the dimensions of such systems.

Categories:14C20, 14N05

123. CJM 2003 (vol 55 pp. 649)

Zucconi, Francesco
Surfaces with $p_{g}=q=2$ and an Irrational Pencil
We describe the irrational pencils on surfaces of general type with $p_{g}=q=2$.

Categories:14J29, 14J25, 14D06, 14D99

124. 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

125. CJM 2003 (vol 55 pp. 533)

Edo, Eric
Automorphismes modérés de l'espace affine
Le probl\`eme de Jung-Nagata ({\it cf.}\ [J], [N]) consiste \`a savoir s'il existe des automorphismes de $k[x,y,z]$ qui ne sont pas mod\'er\'es. Nous proposons une approche nouvelle de cette question, fond\'ee sur l'utilisation de la th\'eorie des automates et du polygone de Newton. Cette approche permet notamment de g\'en\'eraliser de fa\c con significative les r\'esultats de [A]. The Jung-Nagata's problem ({\it cf.}\ [J], [N]) asks if there exists non-tame (or wild) automorphisms of $k[x,y,z]$. We give a new way to attack this question, based on the automata theory and the Newton polygon. This new approch allows us to generalize significantly the results of [A].

Keywords:tame automorphisms, automata, Newton polygon
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