Expand all Collapse all | Results 101 - 125 of 153 |
101. CJM 2004 (vol 56 pp. 716)
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 |
102. CJM 2004 (vol 56 pp. 495)
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 |
103. CJM 2004 (vol 56 pp. 612)
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 |
104. CJM 2004 (vol 56 pp. 310)
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 |
105. CJM 2003 (vol 55 pp. 1100)
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 |
106. CJM 2003 (vol 55 pp. 897)
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 |
107. CJM 2003 (vol 55 pp. 766)
Homology TQFT's and the Alexander--Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory |
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 |
108. CJM 2003 (vol 55 pp. 839)
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 |
109. CJM 2003 (vol 55 pp. 693)
Une formule de Riemann-Roch 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 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 |
110. CJM 2003 (vol 55 pp. 561)
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 |
111. CJM 2003 (vol 55 pp. 649)
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 |
112. CJM 2003 (vol 55 pp. 609)
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 |
113. CJM 2003 (vol 55 pp. 533)
Automorphismes modÃ©rÃ©s 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 Category:14R10 |
114. CJM 2003 (vol 55 pp. 248)
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 $1\le d\le g-1$.
Category:14H99 |
115. CJM 2003 (vol 55 pp. 331)
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 |
116. CJM 2003 (vol 55 pp. 157)
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 |
117. CJM 2003 (vol 55 pp. 133)
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.
Category:14F35 |
118. CJM 2002 (vol 54 pp. 1319)
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 |
119. CJM 2002 (vol 54 pp. 595)
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 |
120. CJM 2002 (vol 54 pp. 554)
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 |
121. CJM 2002 (vol 54 pp. 352)
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 |
122. CJM 2002 (vol 54 pp. 55)
On the Milnor Fiber of a Quasi-ordinary Surface Singularity We verify a generalization of (3.3) from \cite{Le} proving
that the homotopy type of the Milnor fiber of a reduced
hypersurface singularity depends only on the embedded
topological type of the singularity. In particular, using
\cite{Za,Li1,Oh1,Gau} for irreducible quasi-ordinary germs,
it depends only on the normalized distinguished pairs of the
singularity. The main result of the paper provides an explicit
formula for the Euler-characteristic of the Milnor fiber in the
surface case.
Categories:14B05, 14E15, 32S55 |
123. CJM 2001 (vol 53 pp. 1309)
The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces A theorem of Donaldson on the existence of Hermitian-Einstein metrics
on stable holomorphic bundles over a compact K\"ahler surface is
extended to bundles which are parabolic along an effective divisor
with normal crossings. Orbifold methods, together with a suitable
approximation theorem, are used following an approach successful for
the case of Riemann surfaces.
Categories:14J17, 57R57 |
124. CJM 2001 (vol 53 pp. 923)
Decompositions of the Hilbert Function of a Set of Points in $\P^n$ Let $\H$ be the Hilbert function of some set of distinct points
in $\P^n$ and let $\alpha = \alpha (\H)$ be the least degree
of a hypersurface of $\P^n$ containing these points. Write $\alpha
= d_s + d_{s-1} + \cdots + d_1$ (where $d_i > 0$). We canonically
decompose $\H$ into $s$ other Hilbert functions $\H
\leftrightarrow (\H_s^\prime, \dots, \H_1^\prime)$ and show
how to find sets of distinct points $\Y_s, \dots, \Y_1$,
lying on reduced hypersurfaces of degrees $d_s, \dots, d_1$
(respectively) such that the Hilbert function of $\Y_i$ is
$\H_i^\prime$ and the Hilbert function of $\Y = \bigcup_{i=1}^s
\Y_i$ is $\H$. Some extremal properties of this canonical
decomposition are also explored.
Categories:13D40, 14M10 |
125. CJM 2001 (vol 53 pp. 834)
Zeta Functions and `Kontsevich Invariants' on Singular Varieties Let $X$ be a nonsingular algebraic variety in characteristic zero. To
an effective divisor on $X$ Kontsevich has associated a certain
motivic integral, living in a completion of the Grothendieck ring of
algebraic varieties. He used this invariant to show that birational
(smooth, projective) Calabi-Yau varieties have the same Hodge
numbers. Then Denef and Loeser introduced the invariant {\it motivic
(Igusa) zeta function}, associated to a regular function on $X$, which
specializes to both the classical $p$-adic Igusa zeta function and the
topological zeta function, and also to Kontsevich's invariant.
This paper treats a generalization to singular varieties. Batyrev
already considered such a `Kontsevich invariant' for log terminal
varieties (on the level of Hodge polynomials of varieties instead of
in the Grothendieck ring), and previously we introduced a motivic zeta
function on normal surface germs. Here on any $\bbQ$-Gorenstein
variety $X$ we associate a motivic zeta function and a `Kontsevich
invariant' to effective $\bbQ$-Cartier divisors on $X$ whose support
contains the singular locus of~$X$.
Keywords:singularity invariant, topological zeta function, motivic zeta function Categories:14B05, 14E15, 32S50, 32S45 |