Expand all Collapse all | Results 76 - 100 of 157 |
76. CJM 2008 (vol 60 pp. 875)
A Characterization of the Quantum Cohomology Ring of $G/B$ and Applications We observe that the small quantum product of the
generalized flag manifold $G/B$ is a product operation $\star$ on
$H^*(G/B)\otimes \bR[q_1,\dots, q_l]$ uniquely determined by the
facts
that: it is a deformation of the cup product on $H^*(G/B)$; it is
commutative, associative, and graded with respect to $\deg(q_i)=4$; it
satisfies a certain relation (of degree two); and the corresponding
Dubrovin connection is flat. Previously, we proved that these
properties alone imply the presentation of the ring $(H^*(G/B)\otimes
\bR[q_1,\dots, q_l],\star)$ in terms of generators and relations. In
this paper we use the above observations to give conceptually new
proofs of other fundamental results of the quantum Schubert calculus
for $G/B$: the quantum Chevalley formula of D. Peterson (see also
Fulton and Woodward ) and the ``quantization by standard
monomials" formula of Fomin, Gelfand, and Postnikov for
$G=\SL(n,\bC)$. The main idea of the proofs is the same as in
Amarzaya--Guest: from the quantum $\D$-module of $G/B$ one can
decode all information about the quantum cohomology of this space.
Categories:14M15, 14N35 |
77. CJM 2008 (vol 60 pp. 556)
Polarization of Separating Invariants We prove a characteristic free version of Weyl's theorem on
polarization. Our result is an exact analogue of Weyl's theorem, the
difference being that our statement is about separating invariants
rather than generating invariants. For the special case of finite
group actions we introduce the concept of \emph{cheap polarization},
and show that it is enough to take cheap polarizations of invariants
of just one copy of a representation to obtain separating vector
invariants for any number of copies. This leads to upper bounds on
the number and degrees of separating vector invariants of finite
groups.
Keywords:Jan Draisma, Gregor Kemper, David Wehlau Categories:13A50, 14L24 |
78. CJM 2008 (vol 60 pp. 532)
Local Bounds for Torsion Points on Abelian Varieties We say that an abelian variety over a $p$-adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$-rational torsion subgroup of a $g$-dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.
Categories:11G10, 14K15 |
79. CJM 2008 (vol 60 pp. 391)
The Geometry of the Weak Lefschetz Property and Level Sets of Points In a recent paper, F. Zanello showed that level Artinian algebras in 3
variables can fail to have the Weak Lefschetz Property (WLP), and can
even fail to have unimodal Hilbert function. We show that the same is
true for the Artinian reduction of reduced, level sets of points in
projective 3-space. Our main goal is to begin an understanding of how
the geometry of a set of points can prevent its Artinian reduction
from having WLP, which in itself is a very algebraic notion. More
precisely, we produce level sets of points whose Artinian reductions
have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the
worst case), but fail to have WLP. We also produce a level set of
points whose Artinian reduction fails to have unimodal Hilbert
function; our example is based on Zanello's example. Finally, we show
that a level set of points can have Artinian reduction that has WLP
but fails to have the Strong Lefschetz Property. While our
constructions are all based on basic double G-linkage, the
implementations use very different methods.
Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen--Macaulay, socle type, socle degree, Artinian reduction Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05 |
80. CJM 2008 (vol 60 pp. 297)
Transitive Factorizations in the Hyperoctahedral Group The classical Hurwitz enumeration problem has a presentation in terms of
transitive factorizations in the symmetric group. This presentation suggests
a generalization from type~$A$ to other
finite reflection groups and, in particular, to type~$B$.
We study this generalization both from a combinatorial and a geometric
point of view, with the prospect of providing a means of understanding more
of the structure of the moduli spaces of maps with an $\gS_2$-symmetry.
The type~$A$ case has been well studied and connects Hurwitz numbers
to the moduli space of curves. We conjecture an analogous setting for the
type~$B$ case that is studied here.
Categories:05A15, 14H10, 58D29 |
81. CJM 2008 (vol 60 pp. 379)
Finite Cohen--Macaulay Type and Smooth Non-Commutative Schemes A commutative local Cohen--Macaulay ring $R$ of finite Cohen--Macaulay type is known to be an isolated
singularity; that is, $\Spec(R) \setminus \{ \mathfrak {m} \}$ is smooth.
This paper proves a non-commutative analogue. Namely, if $A$ is a
(non-commutative) graded Artin--Schelter \CM\ algebra which is fully
bounded Noetherian and
has finite Cohen--Macaulay type, then the non-commutative projective scheme determined by
$A$ is smooth.
Keywords:Artin--Schelter Cohen--Macaulay algebra, Artin--Schelter Gorenstein algebra, Auslander's theorem on finite Cohen--Macaulay type, Cohen--Macaulay ring, fully bounded Noetherian algebra, isolated singularity, maximal Cohen--Macaulay module, non-commutative Categories:14A22, 16E65, 16W50 |
82. CJM 2008 (vol 60 pp. 140)
On the Geometry of $p$-Typical Covers in Characteristic $p$ For $p$ a prime, a $p$-typical cover of a connected scheme on which $p=0$ is a finite
\'etale cover whose monodromy group (\emph{i.e.,} the Galois group of its
normal closure) is a $p$-group.
The geometry of such covers exhibits some unexpectedly pleasant
behaviors; building on work of Katz, we demonstrate some of these.
These include a criterion for when a morphism induces an isomorphism of
the $p$\nobreakdash-typi\-cal quotients of the \'etale fundamental groups,
and a decomposition theorem for $p$-typical covers of polynomial rings
over an algebraically closed field.
Category:14F35 |
83. CJM 2008 (vol 60 pp. 109)
Affine Lines on Affine Surfaces and the Makar--Limanov Invariant A smooth affine surface $X$ defined over the complex field $\C$ is an $\ML_0$ surface if the
Makar--Limanov invariant $\ML(X)$ is trivial. In this paper we study the topology and geometry of
$\ML_0$ surfaces. Of particular interest is the question: Is every curve $C$ in $X$ which is isomorphic
to
the affine line a fiber component of an $\A^1$-fibration
on $X$? We shall show that the answer is affirmative if the Picard number
$\rho(X)=0$, but negative in case $\rho(X) \ge 1$. We shall also study the ascent and descent of
the $\ML_0$ property under proper maps.
Categories:14R20, 14L30 |
84. CJM 2008 (vol 60 pp. 64)
Classification of Linear Weighted Graphs Up to Blowing-Up and Blowing-Down We classify linear weighted graphs up to the
blowing-up and blowing-down operations which are relevant for the
study of algebraic surfaces.
Keywords:weighted graph, dual graph, blowing-up, algebraic surface Categories:14J26, 14E07, 14R05, 05C99 |
85. CJM 2007 (vol 59 pp. 981)
The Chen--Ruan Cohomology of Weighted Projective Spaces In this paper we study the Chen--Ruan cohomology ring of weighted
projective spaces. Given a weighted projective space ${\bf
P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted
sectors and the corresponding degree shifting numbers. The main
result of this paper is that the obstruction bundle over any
3\nobreakdash-multi\-sector is a direct sum of line bundles which we use to
compute the orbifold cup product. Finally we compute the
Chen--Ruan cohomology ring of weighted projective space ${\bf
P}^{5}_{1,2,2,3,3,3}$.
Keywords:Chen--Ruan cohomology, twisted sectors, toric varieties, weighted projective space, localization Categories:14N35, 53D45 |
86. CJM 2007 (vol 59 pp. 1098)
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 |
87. CJM 2007 (vol 59 pp. 1069)
Quotients jacobiens : une approche algÃ©brique 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:SingularitÃ©, jacobien, quotient jacobien, polygone de Newton Categories:14B05, 32S05, 32S50 |
88. CJM 2007 (vol 59 pp. 742)
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 |
89. CJM 2007 (vol 59 pp. 488)
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 |
90. CJM 2007 (vol 59 pp. 372)
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 |
91. CJM 2007 (vol 59 pp. 36)
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 |
92. CJM 2006 (vol 58 pp. 1000)
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 |
93. CJM 2006 (vol 58 pp. 476)
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 |
94. CJM 2006 (vol 58 pp. 262)
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 |
95. CJM 2006 (vol 58 pp. 93)
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 |
96. CJM 2005 (vol 57 pp. 1178)
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 |
97. CJM 2005 (vol 57 pp. 1314)
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 |
98. CJM 2005 (vol 57 pp. 724)
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 |
99. CJM 2005 (vol 57 pp. 338)
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 |
100. CJM 2005 (vol 57 pp. 400)
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 |