Expand all Collapse all | Results 76 - 100 of 111 |
76. CMB 2003 (vol 46 pp. 495)
Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two Let $V$ be an algebraic K3 surface defined over a number field $K$.
Suppose $V$ has Picard number two and an infinite group of
automorphisms $\mathcal{A} = \Aut(V/K)$. In this paper, we
introduce the notion of a vector height $\mathbf{h} \colon V \to
\Pic(V) \otimes \mathbb{R}$ and show the existence of a canonical
vector height $\widehat{\mathbf{h}}$ with the following properties:
\begin{gather*}
\widehat{\mathbf{h}} (\sigma P) = \sigma_* \widehat{\mathbf{h}} (P) \\
h_D (P) = \widehat{\mathbf{h}} (P) \cdot D + O(1),
\end{gather*}
where $\sigma \in \mathcal{A}$, $\sigma_*$ is the pushforward of
$\sigma$ (the pullback of $\sigma^{-1}$), and $h_D$ is a Weil
height associated to the divisor $D$. The bounded function implied
by the $O(1)$ does not depend on $P$. This allows us to attack
some arithmetic problems. For example, we show that the number of
rational points with bounded logarithmic height in an
$\mathcal{A}$-orbit satisfies
$$
N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q) Categories:11G50, 14J28, 14G40, 14J50, 14G05 |
77. CMB 2003 (vol 46 pp. 321)
Discreteness For the Set of Complex Structures On a Real Variety Let $X$, $Y$ be reduced and irreducible compact complex spaces and
$S$ the set of all isomorphism classes of reduced and irreducible
compact complex spaces $W$ such that $X\times Y \cong X\times W$.
Here we prove that $S$ is at most countable. We apply this result
to show that for every reduced and irreducible compact complex
space $X$ the set $S(X)$ of all complex reduced compact complex
spaces $W$ with $X\times X^\sigma \cong W\times W^\sigma$ (where
$A^\sigma$ denotes the complex conjugate of any variety $A$) is at
most countable.
Categories:32J18, 14J99, 14P99 |
78. CMB 2003 (vol 46 pp. 429)
The Grothendieck Trace and the de Rham Integral On a smooth $n$-dimensional complete variety $X$ over ${\mathbb C}$ we
show that the trace map ${\tilde\theta}_X \colon\break
H^n (X,\Omega_X^n)
\to {\mathbb C}$ arising from Lipman's version of Grothendieck duality
in \cite{ast-117} agrees with
$$
(2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb
C}) \to {\mathbb C}
$$
under the Dolbeault isomorphism.
Categories:14F10, 32A25, 14A15, 14F05, 18E30 |
79. CMB 2003 (vol 46 pp. 400)
Approximating Positive Polynomials Using Sums of Squares The paper considers the relationship between positive polynomials,
sums of squares and the multi-dimensional moment problem in the
general context of basic closed semi-algebraic sets in real $n$-space.
The emphasis is on the non-compact case and on quadratic module
representations as opposed to quadratic preordering presentations.
The paper clarifies the relationship between known results on the
algebraic side and on the functional-analytic side and extends these
results in a variety of ways.
Categories:14P10, 44A60 |
80. CMB 2003 (vol 46 pp. 323)
Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2
\rightarrow \mathbb{R}^2$
whose Jacobians have constant eigenvalues can be completely
characterized if either the eigenvalues are equal or $F$ is a
polynomial. Specifically, $F=(u,v)$ must take the form
\begin{gather*}
u = ax + by + \beta \phi(\alpha x + \beta y) + e \\
v = cx + dy - \alpha \phi(\alpha x + \beta y) + f
\end{gather*}
for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and
a $C^1$ function $\phi$ in one variable. If, in addition, the function
$\phi$ is not affine, then
\begin{equation}
\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.
\end{equation}
This paper shows how these theorems cannot be extended by constructing
a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does
not fit the previous form. This example is also used to construct
non-obvious solutions to nonlinear PDEs, including the Monge--Amp\`ere
equation.
Keywords:Jacobian Conjecture, injectivity, Monge--AmpÃ¨re equation Categories:26B10, 14R15, 35L70 |
81. CMB 2003 (vol 46 pp. 204)
Rationality and Orbit Closures Suppose we are given a finite-dimensional vector space $V$ equipped
with an $F$-rational action of a linearly algebraic group $G$, with
$F$ a characteristic zero field. We conjecture the following: to each
vector $v\in V(F)$ there corresponds a canonical $G(F)$-orbit of
semisimple vectors of $V$. In the case of the adjoint action, this
orbit is the $G(F)$-orbit of the semisimple part of $v$, so this
conjecture can be considered a generalization of the Jordan
decomposition. We prove some cases of the conjecture.
Categories:14L24, 20G15 |
82. CMB 2003 (vol 46 pp. 140)
An Explicit Cell Decomposition of the Wonderful Compactification of a Semisimple Algebraic Group We determine an explicit cell decomposition of the wonderful
compactification of a semi\-simple algebraic group. To do this we first
identify the $B\times B$-orbits using the generalized Bruhat
decomposition of a reductive monoid. From there we show how each cell
is made up from $B\times B$-orbits.
Categories:14L30, 14M17, 20M17 |
83. CMB 2002 (vol 45 pp. 686)
An Aspect of Icosahedral Symmetry We embed the moduli space $Q$ of 5 points on the projective line
$S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the
6-dimensional irreducible module of the symmetric group $S_5$. This
module splits with respect to the icosahedral group $A_5$ into the two
standard 3-dimensional representations. The resulting linear
projections of $Q$ relate the action of $A_5$ on $Q$ to those on the
regular icosahedron.
Categories:14L24, 20B25 |
84. CMB 2002 (vol 45 pp. 417)
On Deformations of the Complex Structure on the Moduli Space of Spatial Polygons For an integer $n \geq 3$, let $M_n$ be the moduli space of spatial polygons
with $n$ edges. We consider the case of odd $n$. Then $M_n$ is a Fano
manifold of complex dimension $n-3$. Let $\Theta_{M_n}$ be the
sheaf of germs of holomorphic sections of the tangent bundle
$TM_n$. In this paper, we prove $H^q (M_n,\Theta_{M_n})=0$ for all
$q \geq 0$ and all odd $n$. In particular, we see that the moduli
space of deformations of the complex structure on $M_n$ consists of
a point. Thus the complex structure on $M_n$ is locally rigid.
Keywords:polygon space, complex structure Categories:14D20, 32C35 |
85. CMB 2002 (vol 45 pp. 349)
Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces Let $\mathbf{P}^n$ be the $n$-dimensional projective space over some
algebraically closed field $k$ of characteristic $0$. For an integer
$t\geq 3$ consider the invertible sheaf $O(t)$ on $\mathbf{P}^n$ (Serre
twist of the structure sheaf). Let $N = \binom{t+n}{n}$, the
dimension of the space of global sections of $O(t)$, and let $k$ be an
integer satisfying $0\leq k\leq N - (2n+2)$. Let $P_1,\dots,P_k$
be general points on $\mathbf{P}^n$ and let $\pi \colon X \to
\mathbf{P}^n$ be the blowing-up of $\mathbf{P}^n$ at those points.
Let $E_i = \pi^{-1} (P_i)$ with $1\leq i\leq k$ be the exceptional
divisor. Then $M = \pi^* \bigl( O(t) \bigr) \otimes O_X (-E_1 -
\cdots -E_k)$ is a very ample invertible sheaf on $X$.
Keywords:blowing-up, projective space, very ample linear system, embeddings, Veronese map Categories:14E25, 14N05, 14N15 |
86. CMB 2002 (vol 45 pp. 284)
Residue: A Geometric Construction A new construction of the ordinary residue of differential forms is
given. This construction is intrinsic, \ie, it is defined without
local coordinates, and it is geometric: it is constructed out of the
geometric structure of the local and global cohomology groups of the
differentials. The Residue Theorem and the local calculation then
follow from geometric reasons.
Category:14A25 |
87. CMB 2002 (vol 45 pp. 213)
Griffiths Groups of Supersingular Abelian Varieties The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over
an algebraically closed field is defined to be the group of homologically
trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of
algebraically trivial algebraic cycles. The main result of this paper is
that the Griffiths group $\Gr^2 (A_{\bar{k}})$ of a supersingular
abelian variety $A_{\bar{k}}$ over the algebraic closure of a finite
field of characteristic $p$ is at most a $p$-primary torsion group.
As a corollary the same conclusion holds for supersingular Fermat
threefolds. In contrast, using methods of C.~Schoen it is also
shown that if the Tate conjecture is valid for all smooth
projective surfaces and all finite extensions of the finite ground
field $k$ of characteristic $p>2$, then the Griffiths group of any ordinary
abelian threefold $A_{\bar{k}}$ over the algebraic closure of $k$ is
non-trivial; in fact, for all but a finite number of primes $\ell\ne p$ it
is the case that $\Gr^2 (A_{\bar{k}}) \otimes \Z_\ell \neq 0$.
Keywords:Griffiths group, Beauville conjecture, supersingular Abelian variety, Chow group Categories:14J20, 14C25 |
88. CMB 2002 (vol 45 pp. 204)
On the Chow Groups of Supersingular Varieties We compute the rational Chow groups of supersingular abelian varieties
and some other related varieties, such as supersingular Fermat
varieties and supersingular $K3$ surfaces. These computations are
concordant with the conjectural relationship, for a smooth projective
variety, between the structure of Chow groups and the coniveau
filtration on the cohomology.
Categories:14C25, 14K99 |
89. CMB 2002 (vol 45 pp. 89)
On Gunning's Prime Form in Genus $2$ Using a classical generalization of Jacobi's derivative formula, we
give an explicit expression for Gunning's prime form in genus 2 in
terms of theta functions and their derivatives.
Categories:14K25, 30F10 |
90. CMB 2001 (vol 44 pp. 452)
Some Adjunction Properties of Ample Vector Bundles Let $\ce$ be an ample vector bundle of rank $r$ on a projective
variety $X$ with only log-terminal singularities. We consider the
nefness of adjoint divisors $K_X + (t-r) \det \ce$ when $t \ge \dim X$
and $t>r$. As an application, we classify pairs $(X,\ce)$ with
$c_r$-sectional genus zero.
Keywords:ample vector bundle, adjunction, sectional genus Categories:14J60, 14C20, 14F05, 14J40 |
91. CMB 2001 (vol 44 pp. 491)
Resolution of Singularities of Null Cones We give canonical resolutions of singularities of several cone
varieties arising from invariant theory. We establish a connection
between our resolutions and resolutions of singularities of closure of
conjugacy classes in classical Lie algebras.
Categories:14L35, 22G |
92. CMB 2001 (vol 44 pp. 257)
Algebraic Homology For Real Hyperelliptic and Real Projective Ruled Surfaces Let $X$ be a reduced nonsingular quasiprojective scheme over ${\mathbb
R}$ such that the set of real rational points $X({\mathbb R})$ is dense
in $X$ and compact. Then $X({\mathbb R})$ is a real algebraic variety.
Denote by $H_k^{\alg}(X({\mathbb R}), {\mathbb Z}/2)$ the group of
homology classes represented by Zariski closed $k$-dimensional
subvarieties of $X({\mathbb R})$. In this note we show that $H_1^{\alg}
(X({\mathbb R}), {\mathbb Z}/2)$ is a proper subgroup of
$H_1(X({\mathbb R}), {\mathbb Z}/2)$ for a nonorientable hyperelliptic
surface $X$. We also determine all possible groups $H_1^{\alg}(X({\mathbb R}),
{\mathbb Z}/2)$ for a real ruled surface $X$ in connection with the previously
known description of all possible topological configurations of $X$.
Categories:14P05, 14P25 |
93. CMB 2001 (vol 44 pp. 313)
Images of mod $p$ Galois Representations Associated to Elliptic Curves We give an explicit recipe for the determination of the images
associated to the Galois action on $p$-torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.
Keywords:Galois groups, elliptic curves, Galois representation, isogeny Categories:11R32, 11G05, 12F10, 14K02 |
94. CMB 2001 (vol 44 pp. 223)
Extending the Archimedean Positivstellensatz to the Non-Compact Case A generalization of Schm\"udgen's Positivstellensatz is given which holds
for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not).
The proof is an extension of W\"ormann's proof.
Categories:12D15, 14P10, 44A60 |
95. CMB 2000 (vol 43 pp. 312)
On the Prime Ideals in a Commutative Ring If $n$ and $m$ are positive integers, necessary and sufficient
conditions are given for the existence of a finite commutative ring $R$
with exactly $n$ elements and exactly $m$ prime ideals. Next,
assuming the Axiom of Choice, it is proved that if $R$ is a
commutative ring and $T$ is a commutative $R$-algebra which is
generated by a set $I$, then each chain of prime ideals of $T$ lying
over the same prime ideal of $R$ has at most $2^{|I|}$ elements. A
polynomial ring example shows that the preceding result is
best-possible.
Categories:13C15, 13B25, 04A10, 14A05, 13M05 |
96. CMB 2000 (vol 43 pp. 304)
Courbes hyperelliptiques Ã multiplications rÃ©elles et une construction de Shih Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos
\frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place
au-dessus de $p$. Lorsque l'image de $(2 - 2\cos \frac{2\pi}{r})$
dans $\F$ n'est pas un carr\'e, nous donnons une construction
g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de
Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement
de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3,
sec.~6.3], et son existence est pr\'edite par le crit\`ere de
rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction
s'obtient en tordant la representation galoisienne associ\'ee aux
points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a
multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et
Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un
corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien.
Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que
l'on retrouve quand $r = 2$ et $r = 3$.
Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos
\frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place
above $p$. When the image of $(2 - 2\cos \frac{2\pi}{r})$ in $\F$ is
not a square, we describe a geometric construction of a regular
extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension
corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$''
in the sense of [3, sec.~6.3], and its existence is predicted by the
rigidity criterion of Belyi, Fried, Thompson and Matzat. Its
construction is obtained by twisting the mod $p$ galois representation
attached to a family of abelian varieties with real multiplications by
$K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian
varieties are defined in general over a quadratic field, and are
isogenous to their galois conjugate. Our construction generalises a
method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.
Categories:11G30, 14H25 |
97. CMB 2000 (vol 43 pp. 162)
Moduli Spaces of Polygons and Punctured Riemann Spheres The purpose of this note is to give a simple combinatorial
construction of the map from the canonically compactified moduli
spaces of punctured complex projective lines to the moduli spaces
$\P_r$ of polygons with fixed side lengths in the Euclidean space
$\E^3$. The advantage of this construction is that one can obtain a
complete set of linear relations among the cycles that generate
homology of $\P_r$. We also classify moduli spaces of pentagons.
Categories:14D20, 18G55, 14H10 |
98. CMB 2000 (vol 43 pp. 239)
On the Number of Divisors of the Quadratic Form $m^2+n^2$ For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 |
99. CMB 2000 (vol 43 pp. 174)
Stable Parabolic Bundles over Elliptic Surfaces and over Riemann Surfaces We show that the use of orbifold bundles enables some questions to
be reduced to the case of flat bundles. The identification of
moduli spaces of certain parabolic bundles over elliptic surfaces
is achieved using this method.
Categories:14J27, 32L07, 14H60, 14D20 |
100. CMB 2000 (vol 43 pp. 129)
Maximal Subbundles of Rank 2 Vector Bundles on Projective Curves Let $E$ be a stable rank 2 vector bundle on a smooth projective
curve $X$ and $V(E)$ be the set of all rank~1 subbundles of $E$
with maximal degree. Here we study the varieties (non-emptyness,
irreducibility and dimension) of all rank~2 stable vector bundles,
$E$, on $X$ with fixed $\deg(E)$ and $\deg(L)$, $L \in V(E)$ and
such that $\card \bigl( V(E) \bigr) \geq 2$ (resp. $\card \bigl(
V(E) \bigr) = 2$).
Category:14H60 |