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76. CJM 2007 (vol 59 pp. 1098)

Rodrigues, B.
Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions
In this paper we study ruled surfaces which appear as an exceptional surface in a succession of blowing-ups. In particular we prove that the $e$-invariant of such a ruled exceptional surface $E$ is strictly positive whenever its intersection with the other exceptional surfaces does not contain a fiber (of $E$). This fact immediately enables us to resolve an open problem concerning an intersection configuration on such a ruled exceptional surface consisting of three nonintersecting sections. In the second part of the paper we apply the non-vanishing of $e$ to the study of the poles of the well-known topological, Hodge and motivic zeta functions.

Categories:14E15, 14J26, 14B05, 14J17, 32S45

77. CJM 2007 (vol 59 pp. 1069)

Reydy, Carine
Quotients jacobiens : une approche algébrique
Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui repr\'esente ce germe et permet d'en calculer les invariants. On donne une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$ \`a partir du diagramme d'Eisenbud et Neumann de $fg$.

Keywords:Singularité, jacobien, quotient jacobien, polygone de Newton
Categories:14B05, 32S05, 32S50

78. CJM 2007 (vol 59 pp. 742)

Gil, Juan B.; Krainer, Thomas; Mendoza, Gerardo A.
Geometry and Spectra of Closed Extensions of Elliptic Cone Operators
We study the geometry of the set of closed extensions of index $0$ of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

Keywords:resolvents, manifolds with conical singularities, spectral theor, boundary value problems, Grassmannians
Categories:58J50, 35J70, 14M15

79. CJM 2007 (vol 59 pp. 488)

Bernardi, A.; Catalisano, M. V.; Gimigliano, A.; Idà, M.
Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties
We consider the $k$-osculating varieties $O_{k,n.d}$ to the (Veronese) $d$-uple embeddings of $\PP^n$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

Categories:14N15, 15A69

80. CJM 2007 (vol 59 pp. 372)

Maisner, Daniel; Nart, Enric
Zeta Functions of Supersingular Curves of Genus 2
We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic $2$ contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus $2$. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.

Categories:11G20, 14G15, 11G10

81. CJM 2007 (vol 59 pp. 36)

Develin, Mike; Martin, Jeremy L.; Reiner, Victor
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

Keywords:Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence
Categories:14M15, 05E05

82. 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

83. 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

84. 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

85. 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

86. 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

87. 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

88. 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

89. 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

90. 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

91. 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 $\mathcal{D}$.

92. 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

93. 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

94. 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

95. 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

96. 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

97. 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

98. 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

99. 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

100. 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
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