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126. 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 $\mathcal{D}$.

127. 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

128. 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

129. 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

130. 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

131. 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

132. 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

133. 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

134. 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

135. 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

136. 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

137. 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

138. 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

139. 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

140. 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

141. 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

142. 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

143. CJM 2003 (vol 55 pp. 248)

Dhillon, Ajneet
A Generalized Torelli Theorem
Given a smooth projective curve $C$ of positive genus $g$, Torelli's theorem asserts that the pair $\bigl( J(C),W^{g-1} \bigr)$ determines $C$. We show that the theorem is true with $W^{g-1}$ replaced by $W^d$ for each $d$ in the range $1\le d\le g-1$.


144. CJM 2003 (vol 55 pp. 331)

Savitt, David
The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$
We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.

Categories:11G20, 14H25

145. CJM 2003 (vol 55 pp. 133)

Shimada, Ichiro
On the Zariski-van Kampen Theorem
Let $f \colon E\to B$ be a dominant morphism, where $E$ and $B$ are smooth irreducible complex quasi-projective varieties, and let $F_b$ be the general fiber of $f$. We present conditions under which the homomorphism $\pi_1 (F_b)\to \pi_1 (E)$ induced by the inclusion is injective.


146. CJM 2003 (vol 55 pp. 157)

Shimada, Ichiro
Zariski Hyperplane Section Theorem for Grassmannian Varieties
Let $\phi \colon X\to M$ be a morphism from a smooth irreducible complex quasi-projective variety $X$ to a Grassmannian variety $M$ such that the image is of dimension $\ge 2$. Let $D$ be a reduced hypersurface in $M$, and $\gamma$ a general linear automorphism of $M$. We show that, under a certain differential-geometric condition on $\phi(X)$ and $D$, the fundamental group $\pi_1 \bigl( (\gamma \circ \phi)^{-1} (M\setminus D) \bigr)$ is isomorphic to a central extension of $\pi_1 (M\setminus D) \times \pi_1 (X)$ by the cokernel of $\pi_2 (\phi) \colon \pi_2 (X) \to \pi_2 (M)$.

Categories:14F35, 14M15

147. CJM 2002 (vol 54 pp. 1319)

Yekutieli, Amnon
The Continuous Hochschild Cochain Complex of a Scheme
Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$ of topological $\mathcal{O}_X$-modules, called the complete Hochschild chain complex of $X$. To any $\mathcal{O}_X$-module $\mathcal{M}$---not necessarily quasi-coherent---we assign the complex $\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ of continuous Hochschild cochains with values in $\mathcal{M}$. Our first main result is that when $X$ is smooth over $\mathbb{K}$ there is a functorial isomorphism $$ \mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R \mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) $$ in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, where $X^2 := X \times_{\mathbb{K}} X$. The second main result is that if $X$ is smooth of relative dimension $n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps $\pi \colon \widehat{\mathcal{C}}^{-q} (X) \to \Omega^q_{X/ \mathbb{K}}$ induce a quasi-isomorphism $$ \mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/ \mathbb{K}} [q], \mathcal{M} \Bigr) \to \mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl( \widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr). $$ When $\mathcal{M} = \mathcal{O}_X$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem. Combining the two results above we deduce a decomposition of the global Hochschild cohomology $$ \Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong \bigoplus_q \H^{i-q} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X} \mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M} \Bigr), $$ where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf.

Keywords:Hochschild cohomology, schemes, derived categories
Categories:16E40, 14F10, 18G10, 13H10

148. CJM 2002 (vol 54 pp. 554)

Hausen, Jürgen
Equivariant Embeddings into Smooth Toric Varieties
We characterize embeddability of algebraic varieties into smooth toric varieties and prevarieties. Our embedding results hold also in an equivariant context and thus generalize a well-known embedding theorem of Sumihiro on quasiprojective $G$-varieties. The main idea is to reduce the embedding problem to the affine case. This is done by constructing equivariant affine conoids, a tool which extends the concept of an equivariant affine cone over a projective $G$-variety to a more general framework.

Categories:14E25, 14C20, 14L30, 14M25

149. CJM 2002 (vol 54 pp. 595)

Nahlus, Nazih
Lie Algebras of Pro-Affine Algebraic Groups
We extend the basic theory of Lie algebras of affine algebraic groups to the case of pro-affine algebraic groups over an algebraically closed field $K$ of characteristic 0. However, some modifications are needed in some extensions. So we introduce the pro-discrete topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine algebraic group $G$ over $K$, which is discrete in the finite-dimensional case and linearly compact in general. As an example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in $\mathcal{L}(G)$. We also discuss the Hopf algebra of representative functions $H(L)$ of a residually finite dimensional Lie algebra $L$. As an example, we show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$ is connected, then the canonical Hopf algebra morphism from $K[G]$ into $H(L)$ is injective if and only if $L$ is algebraically dense in $\mathcal{L}(G)$.

Categories:14L, 16W, 17B45

150. CJM 2002 (vol 54 pp. 352)

Haines, Thomas J.
On Connected Components of Shimura Varieties
We study the cohomology of connected components of Shimura varieties $S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\olomega$ on the group of connected components of $S_{K^p}$ we define an operator $L(\omega)$ on the cohomology groups with compact supports $H^i_c (S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual trace of the composition of $L(\omega)$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted} (twisted) orbital integrals (where $\omega$-{\em weighted} means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\olomega$). As the crucial step, we define and study a new invariant $\alpha_1 (\gamma_0; \gamma, \delta)$ which is a refinement of the invariant $\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

Categories:14G35, 11F70
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