Expand all Collapse all | Results 126 - 147 of 147 |
126. CJM 2000 (vol 52 pp. 265)
On Orbit Closures of Symmetric Subgroups in Flag Varieties We study $K$-orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ and we
determine the isotropy groups. As a consequence, we describe the
closed (resp. affine) orbits in terms of $\theta$-stable
(resp. $\theta$-split) parabolic subgroups. We also describe the
decomposition of any $(K,P)$-double coset in $G$ into
$(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$-module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$-module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$-types.
Keywords:flag variety, symmetric subgroup Categories:14M15, 20G05 |
127. CJM 2000 (vol 52 pp. 348)
SingularitÃ©s quasi-ordinaires toriques et polyÃ¨dre de Newton du discriminant Nous \'etudions les polyn\^omes $F \in \C \{S_\tau\} [Y] $ \`a
coefficients dans l'anneau de germes de fonctions holomorphes au
point sp\'ecial d'une vari\'et\'e torique affine. Nous
g\'en\'eralisons \`a ce cas la param\'etrisation classique des
singularit\'es quasi-ordinaires. Cela fait intervenir d'une part
une g\'en\'eralization de l'algorithme de Newton-Puiseux, et
d'autre part une relation entre le poly\`edre de Newton du
discriminant de $F$ par rapport \`a $Y$ et celui de $F$ au moyen du
polytope-fibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous
permet enfin de calculer, sous des hypoth\`eses de non
d\'eg\'en\'erescence, les sommets du poly\`edre de Newton du
discriminant a partir de celui de $F$, et les coefficients
correspondants \`a partir des coefficients des exposants de $F$ qui
sont dans les ar\^etes de son poly\`edre de Newton.
Categories:14M25, 32S25 |
128. CJM 2000 (vol 52 pp. 123)
An Algorithm for Fat Points on $\mathbf{P}^2 Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 |
129. CJM 1999 (vol 51 pp. 1226)
Semi-Affine Coxeter-Dynkin Graphs and $G \subseteq \SU_2(C)$ The semi-affine Coxeter-Dynkin graph is introduced, generalizing
both the affine and the finite types.
Categories:20C99, 05C25, 14B05 |
130. CJM 1999 (vol 51 pp. 1123)
First Steps of Local Contact Algebra We consider germs of mappings of a line to contact space and
classify the first simple singularities up to the action of
contactomorphisms in the target space and diffeomorphisms of the
line. Even in these first cases there arises a new interesting
interaction of local commutative algebra with contact structure.
Keywords:contact manifolds, local contact algebra, Diracian, contactian Categories:53D10, 14B05 |
131. CJM 1999 (vol 51 pp. 1175)
Reflection Subquotients of Unitary Reflection Groups Let $G$ be a finite group generated by (pseudo-) reflections in a
complex vector space and let $g$ be any linear transformation which
normalises $G$. In an earlier paper, the authors showed how to
associate with any maximal eigenspace of an element of the coset
$gG$, a subquotient of $G$ which acts as a reflection group on the
eigenspace. In this work, we address the questions of
irreducibility and the coexponents of this subquotient, as well as
centralisers in $G$ of certain elements of the coset. A criterion
is also given in terms of the invariant degrees of $G$ for an
integer to be regular for $G$. A key tool is the investigation of
extensions of invariant vector fields on the eigenspace, which
leads to some results and questions concerning the geometry of
intersections of invariant hypersurfaces.
Categories:51F15, 20H15, 20G40, 20F55, 14C17 |
132. CJM 1999 (vol 51 pp. 936)
Galois Representations with Non-Surjective Traces Let $E$ be an elliptic curve over $\q$, and let $r$ be an integer.
According to the Lang-Trotter conjecture, the number of primes $p$
such that $a_p(E) = r$ is either finite, or is asymptotic to
$C_{E,r} {\sqrt{x}} / {\log{x}}$ where $C_{E,r}$ is a non-zero
constant. A typical example of the former is the case of rational
$\ell$-torsion, where $a_p(E) = r$ is impossible if $r \equiv 1
\pmod{\ell}$. We prove in this paper that, when $E$ has a rational
$\ell$-isogeny and $\ell \neq 11$, the number of primes $p$ such
that $a_p(E) \equiv r \pmod{\ell}$ is finite (for some $r$ modulo
$\ell$) if and only if $E$ has rational $\ell$-torsion over the
cyclotomic field $\q(\zeta_\ell)$. The case $\ell=11$ is special,
and is also treated in the paper. We also classify all those
occurences.
Category:14H52 |
133. CJM 1999 (vol 51 pp. 1089)
The Characteristic Numbers of Quartic Plane Curves The characteristic numbers of smooth plane quartics are computed
using intersection theory on a component of the moduli space of
stable maps. This completes the verification of Zeuthen's
prediction of characteristic numbers of smooth plane curves. A
short sketch of a computation of the characteristic numbers of
plane cubics is also given as an illustration.
Categories:14N10, 14D22 |
134. CJM 1999 (vol 51 pp. 771)
Stable Bi-Period Summation Formula and Transfer Factors This paper starts by introducing a bi-periodic summation formula
for automorphic forms on a group $G(E)$, with periods by a subgroup
$G(F)$, where $E/F$ is a quadratic extension of number fields. The
split case, where $E = F \oplus F$, is that of the standard trace
formula. Then it introduces a notion of stable bi-conjugacy, and
stabilizes the geometric side of the bi-period summation formula.
Thus weighted sums in the stable bi-conjugacy class are expressed
in terms of stable bi-orbital integrals. These stable integrals
are on the same endoscopic groups $H$ which occur in the case of
standard conjugacy.
The spectral side of the bi-period summation formula involves
periods, namely integrals over the group of $F$-adele points of
$G$, of cusp forms on the group of $E$-adele points on the group
$G$. Our stabilization suggests that such cusp forms---with non
vanishing periods---and the resulting bi-period distributions
associated to ``periodic'' automorphic forms, are related to
analogous bi-period distributions associated to ``periodic''
automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$.
This offers a sharpening of the theory of liftings, where periods
play a key role.
The stabilization depends on the ``fundamental lemma'', which
conjectures that the unit elements of the Hecke algebras on $G$ and
$H$ have matching orbital integrals. Even in stating this
conjecture, one needs to introduce a ``transfer factor''. A
generalization of the standard transfer factor to the bi-periodic
case is introduced. The generalization depends on a new definition
of the factors even in the standard case.
Finally, the fundamental lemma is verified for $\SL(2)$.
Categories:11F72, 11F70, 14G27, 14L35 |
135. CJM 1999 (vol 51 pp. 616)
Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory Let $L$ be a simple algebraic group and $P$ a parabolic subgroup
with Abelian unipotent radical $P^u$. Many familiar varieties
(determinantal varieties, their symmetric and skew-symmetric
analogues) arise as closures of $P$-orbits in $P^u$. We give a
unified invariant-theoretic treatment of various properties of
these orbit closures. We also describe the closures of the
conormal bundles of these orbits as the irreducible components of
some commuting variety and show that the polynomial algebra
$k[P^u]$ is a free module over the algebra of covariants.
Categories:14L30, 13A50 |
136. CJM 1998 (vol 50 pp. 1209)
A lower bound for $K_X L$ of quasi-polarized surfaces $(X,L)$ with non-negative Kodaira dimension Let $X$ be a smooth projective surface over the complex
number field and let $L$ be a nef-big divisor on $X$. Here we consider
the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$,
then $K_{X}L\geq 2q(X)-4$, where $q(X)$ is the irregularity of $X$. In
this paper, we prove that this conjecture is true if (1) the case in which
$\kappa(X)=0$ or $1$, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq
2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$,
and $L$ satisfies some conditions.
Keywords:Quasi-polarized surface, sectional genus Category:14C20 |
137. CJM 1998 (vol 50 pp. 1253)
Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety For an arbitrary finite Galois $p$-extension $L/K$ of
$\zp$-cyclotomic number fields of $\CM$-type with Galois group $G =
\Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$
are zero, we obtain unconditionally and explicitly the Galois
module structure of $\clases$, the minus part of the $p$-subgroup
of the class group of $L$. For an arbitrary finite Galois
$p$-extension $L/K$ of algebraic function fields of one variable
over an algebraically closed field $k$ of characteristic $p$ as its
exact field of constants with Galois group $G = \Gal(L/K)$ we
obtain unconditionally and explicitly the Galois module structure
of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated
to $L/k$.
Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure Categories:11R33, 11R23, 11R58, 14H40 |
138. CJM 1998 (vol 50 pp. 929)
Decomposition varieties in semisimple Lie algebras The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 |
139. CJM 1998 (vol 50 pp. 863)
Smooth formal embeddings and the residue complex Let $\pi\colon X \ar S$ be a finite type morphism of noetherian schemes.
A {\it smooth formal embedding\/} of $X$ (over $S$) is a bijective closed
immersion $X \subset \mfrak{X}$, where $\mfrak{X}$ is a noetherian
formal scheme, formally smooth over $S$. An example of such an embedding
is the formal completion $\mfrak{X} = Y_{/ X}$ where $X \subset Y$
is an algebraic embedding. Smooth formal embeddings can be used to
calculate algebraic De~Rham (co)homology.
Our main application is an explicit construction of the Grothendieck
residue complex when $S$ is a regular scheme. By definition the residue
complex is the Cousin complex of $\pi^{!} \mcal{O}_{S}$, as in \cite{RD}.
We start with I-C.~Huang's theory of pseudofunctors on modules with
$0$-dimensional support, which provides a graded sheaf $\bigoplus_{q}
\mcal{K}^{q}_{\,X / S}$. We then use smooth formal embeddings to obtain
the coboundary operator $\delta \colon\mcal{K}^{q}_{X / S} \ar
\mcal{K}^{q + 1}_{\,X / S}$. We exhibit a canonical isomorphism between
the complex $(\mcal{K}^{\bdot}_{\,X / S}, \delta)$ and the residue complex
of \cite{RD}. When $\pi$ is equidimensional of dimension $n$ and
generically smooth we show that $\mrm{H}^{-n} \mcal{K}^{\bdot}_{\,X/S}$
is canonically isomorphic to to the sheaf of regular differentials of
Kunz-Waldi \cite{KW}.
Another issue we discuss is Grothendieck Duality on a noetherian formal
scheme $\mfrak{X}$. Our results on duality are used in the construction
of $\mcal{K}^{\bdot}_{\,X / S}$.
Categories:14B20, 14F10, 14B15, 14F20 |
140. CJM 1998 (vol 50 pp. 829)
Conjugacy classes and nilpotent variety of a reductive monoid We continue in this paper our study of conjugacy classes
of a reductive monoid $M$. The main theorems establish a strong connection
with the Bruhat-Renner decomposition of $M$. We use our results to decompose
the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components.
We also identify a class of nilpotent elements that we call standard and prove
that the number of conjugacy classes of standard nilpotent elements is always
finite.
Categories:20G99, 20M10, 14M99, 20F55 |
141. CJM 1998 (vol 50 pp. 525)
Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of
scheme-theoretic
intersections of nilpotent orbit closures with the diagonal matrices.
Here $\mu'$ gives the Jordan block structure of the nilpotent matrix.
de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of
Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology
rings of the varieties constructed by
Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka
polynomial~$\Klmt(q)$ is the Hilbert series for the
multiplicity of the irreducible symmetric group representation indexed
by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$.
\LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition
of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with
non-negative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobenius-splitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of
the ideals of nilpotent orbit closures, arising from the inclusions of
the corresponding sets. We carry out the restriction to the diagonal
using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum
decomposition of the ideals yielding the $k[\Cmubar\cap
\hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of
the $q$-Kostka polynomials.
Keywords:$q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties Categories:05E10, 14M99, 20G05, 05E15 |
142. CJM 1998 (vol 50 pp. 581)
The homology of singular polygon spaces Let $M_n$ be the variety of spatial polygons $P= (a_1, a_2, \dots,
a_n)$ whose sides are vectors $a_i \in \text{\bf R}^3$ of length
$\vert a_i \vert=1 \; (1 \leq i \leq n),$ up to motion in
$\text{\bf R}^3.$ It is known that for odd $n$, $M_n$ is a
smooth manifold, while for even $n$, $M_n$ has cone-like singular
points. For odd $n$, the rational homology of $M_n$ was determined
by Kirwan and Klyachko [6], [9]. The purpose of this paper is to
determine the rational homology of $M_n$ for even $n$. For even
$n$, let ${\tilde M}_n$ be the manifold obtained from $M_n$ by the
resolution of the singularities. Then we also determine the
integral homology of ${\tilde M}_n$.
Keywords:singular polygon space, homology Categories:14D20, 57N65 |
143. CJM 1998 (vol 50 pp. 378)
Equivariant polynomial automorphism of $\Theta$-representations We show that every equivariant polynomial automorphism of a
$\Theta$-repre\-sen\-ta\-tion and of the reduction of an irreducible
$\Theta$-representation is a multiple of the identity.
Categories:14L30, 14L27 |
144. CJM 1997 (vol 49 pp. 1281)
Pieri's formula via explicit rational equivalence Pieri's formula describes the intersection product of a Schubert
cycle by a special Schubert cycle on a Grassmannian.
We present a new geometric proof,
exhibiting an explicit chain of rational equivalences
from a suitable sum of distinct Schubert cycles
to the intersection of a Schubert cycle with a special
Schubert cycle. The geometry of these rational equivalences
indicates a link to a combinatorial proof of Pieri's formula using
Schensted insertion.
Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion Categories:14M15, 05E10 |
145. CJM 1997 (vol 49 pp. 675)
Some adjunction-theoretic properties of codimension two non-singular subvarities of quadrics We make precise the structure of the first two reduction morphisms
associated with codimension two non-singular subvarieties
of non-singular quadrics $\Q^n$, $n\geq 5$.
We give a coarse classification of the same class of subvarieties
when they are assumed not to be of log-general-type.}
Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non log-general-type, quadric, reduction, special, variety. Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10 |
146. CJM 1997 (vol 49 pp. 749)
Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups Following a method outlined by Greenberg, root
number computations give a conjectural lower bound for the ranks of
certain Mordell-Weil groups of elliptic curves. More specifically,
for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and
$E$ an elliptic curve over {\bf Q}, define the motive $E \otimes
\rho$, where $\rho$ is any irreducible representation of
$\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in
the conjectural functional equation for the $L$-function of $E
\otimes \rho$ is easily computible, and a `$-1$' implies, by the
Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in
$E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$
gives a conjectural lower bound of
$$
p^{2n} - p^{2n - 1} - p - 1
$$
for the rank of $E(\PQ_{n})$.
Categories:11G05, 14G10 |
147. CJM 1997 (vol 49 pp. 417)
Characteristic cycles in Hermitian symmetric spaces
We give explicit combinatorial expresssions for the characteristic
cycles associated to certain canonical sheaves on Schubert varieties
$X$ in the classical Hermitian symmetric spaces: namely the
intersection homology sheaves $IH_X$ and the constant sheaves $\Bbb
C_X$. The three main cases of interest are the Hermitian symmetric
spaces for groups of type $A_n$ (the standard Grassmannian), $C_n$
(the Lagrangian Grassmannian) and $D_n$. In particular we find that
$CC(IH_X)$ is irreducible for all Schubert varieties $X$ if and only
if the associated Dynkin diagram is simply laced. The result for
Schubert varieties in the standard Grassmannian had been established
earlier by Bressler, Finkelberg and Lunts, while the computations in
the $C_n$ and $D_n$ cases are new.
Our approach is to compute $CC(\Bbb C_X)$ by a direct geometric
method, then to use the combinatorics of the Kazhdan-Lusztig
polynomials (simplified for Hermitian symmetric spaces) to compute
$CC(IH_X)$. The geometric method is based on the fundamental formula
$$CC(\Bbb C_X) = \lim_{r\downarrow 0} CC(\Bbb C_{X_r}),$$ where the
$X_r \downarrow X$ constitute a family of tubes around the variety
$X$. This formula leads at once to an expression for the coefficients
of $CC(\Bbb C_X)$ as the degrees of certain singular maps between
spheres.
Categories:14M15, 22E47, 53C65 |