Expand all Collapse all | Results 51 - 75 of 110 |
51. CMB 2006 (vol 49 pp. 592)
Group Actions, Cyclic Coverings and Families of K3-Surfaces In this paper we describe six pencils of $K3$-surfaces which have
large Picard number ($\rho=19,20$) and each contains precisely five
special fibers: four have A-D-E singularities and one is
non-reduced. In particular, we characterize these surfaces as cyclic
coverings of some $K3$-surfaces described in a recent paper by Barth
and the author.
In many cases, using
3-divisible sets, resp., 2-divisible sets, of rational curves and
lattice theory, we describe explicitly the Picard lattices.
Categories:14J28, 14L30, 14E20, 14C22 |
52. CMB 2006 (vol 49 pp. 560)
A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain non-Kummer, singular
K3 surface
are dense. We will also compute the entire N\'eron--Severi group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 |
53. CMB 2006 (vol 49 pp. 464)
A Note on Detecting Algebraic Cycles The purpose of this note is to show that the homologically trivial
cycles contructed by Clemens and their generalisations
due to Paranjape can be detected by the technique of
spreading out. More precisely, we associate to these cycles (and the
ambient varieties in which they live) certain families which arise
naturally and which are defined over $\bbC$ and show that these
cycles, along with their relations, can be detected in the singular
cohomology of the total space of these families.
Category:14C25 |
54. CMB 2006 (vol 49 pp. 270)
A Characterization of Products of Projective Spaces We give a characterization of products of projective spaces
using unsplit covering families of rational curves.
Keywords:Rational curves, Fano varieties Categories:14J40, 14J45 |
55. CMB 2006 (vol 49 pp. 196)
Another Proof of Totaro's Theorem on $E_8$-Torsors We give a short proof of Totaro's theorem that every$E_8$-torsor over
a field $k$ becomes trivial over a finiteseparable extension of $k$of
degree dividing $d(E_8)=2^63^25$.
Categories:11E72, 14M17, 20G15 |
56. CMB 2006 (vol 49 pp. 296)
On the Modularity of Three Calabi--Yau Threefolds With Bad Reduction at 11 This paper investigates the modularity of three
non-rigid Calabi--Yau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$-adic cohomology groups are shown to split into
two-dimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
two-dimensional 2-adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hesse-pencil, relating it to a newform of weight 4 and level
27.
Categories:14J32, 11F11, 11F23, 20C12 |
57. CMB 2006 (vol 49 pp. 11)
Going-Down Results for $C_{i}$-Fields We search for theorems that, given a $C_i$-field $K$ and a subfield $k$ of $K$, allow
us to conclude that $k$ is a $C_j$-field for some $j$. We give appropriate theorems in
the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$
is an algebraic extension. Here we are able to prove some results, and make conjectures. We
also point out the connection between these questions and Lang's conjecture on nonreal function
fields over a real closed field.
Keywords:$C_i$-fields, Lang's Conjecture Categories:12F, 14G |
58. CMB 2006 (vol 49 pp. 72)
Additive Riemann--Hilbert Problem in Line Bundles Over $\mathbb{CP}^1$ In this note we consider $\overline\partial$-problem in
line bundles over complex projective space $\mathbb{CP}^1$
and prove that the
equation can be solved for $(0,1)$ forms with compact support. As a
consequence, any Cauchy-Riemann function on a compact real hypersurface in
such line bundles is a jump of two holomorphic functions defined on the
sides of the hypersurface. In particular, the results can be applied to
$\mathbb{CP}^2$ since by removing a point from it we get a line bundle over
$\mathbb{CP}^1$.
Keywords:$\overline\partial$-problem, cohomology groups, line bundles Categories:32F20, 14F05, 32C16 |
59. CMB 2005 (vol 48 pp. 622)
Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any
hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em
i.e.,}
equivalent to a linear hyperplane up to an automorphism of $\C^n$.
Generalizing the approach adopted by Kaliman, V\'en\'ereau, and
Zaidenberg which
consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve
this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$
as in the title.
Keywords:variables, Abhyankar--Sathaye Embedding Problem Categories:14R10, 14R25 |
60. CMB 2005 (vol 48 pp. 547)
Degeneracy of 2-Forms and 3-Forms We study some global aspects of differential complex 2-forms and 3-forms
on complex manifolds.
We compute the cohomology classes represented by the sets of points
on a manifold where such a form degenerates in various senses,
together with other similar cohomological obstructions.
Based on these results and a formula for projective
representations, we calculate the degree of the projectivization
of certain orbits of the representation $\Lambda^k\C^n$.
Keywords:Classes of degeneracy loci, 2-forms, 3-forms, Thom polynomials, global singularity theory Categories:14N10, 57R45 |
61. CMB 2005 (vol 48 pp. 428)
Reduction of Elliptic Curves in Equal Characteristic~3 (and~2) and fibre type for elliptic curves
over discrete valued fields of equal characteristic~3.
Along the same lines, partial results are obtained
in equal characteristic~2.
Categories:14H52, 14K15, 11G07, 11G05, 12J10 |
62. CMB 2005 (vol 48 pp. 473)
Logarithms and the Topology of the Complement of a Hypersurface This paper is devoted to analysing the relation between the
logarithm of a non-constant holomorphic polynomial $Q(z)$ and
the topology of the complement of the hypersurface defined by
$Q(z)=0$.
Keywords:Logarithm, homology groups and periods Categories:32Q55, 14F45 |
63. CMB 2005 (vol 48 pp. 414)
Vector Fields and the Cohomology Ring of Toric Varieties Let $X$ be a smooth complex
projective variety with a holomorphic vector field with isolated
zero set $Z$. From the results of Carrell and Lieberman
there exists a filtration
$F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of
$\c$-valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X,
\c)$ as graded algebras. In this note, for a smooth projective
toric variety and a vector field generated by the action of a
$1$-parameter subgroup of the torus, we work out this filtration.
Our main result is an explicit connection between this filtration
and the polytope algebra of $X$.
Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra Categories:14M25, 52B20 |
64. CMB 2005 (vol 48 pp. 180)
Geometry and Arithmetic of Certain Double Octic Calabi--Yau Manifolds We study Calabi--Yau manifolds constructed as double coverings of
$\mathbb{P}^3$ branched along an octic surface. We give a list of 87
examples corresponding to arrangements of eight planes defined over
$\mathbb{Q}$. The Hodge numbers are computed for all examples. There are
10 rigid Calabi--Yau manifolds and 14 families with $h^{1,2}=1$. The
modularity conjecture is verified for all the rigid examples.
Keywords:Calabi--Yau, double coverings, modular forms Categories:14G10, 14J32 |
65. CMB 2005 (vol 48 pp. 237)
Indecomposable Higher Chow Cycles Let $X$ be a projective smooth variety over a field $k$.
In the first part we show that
an indecomposable element in $CH^2(X,1)$ can be lifted
to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function
field of 1 variable over $k$. We also show that if $X$ is the self-product
of an elliptic curve over $\Q$ then the $\Q$-vector space of
indecomposable cycles
$CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional.
In the second part we give a new
definition of the group of indecomposable cycles
of $CH^3(X,2)$ and give an example of non-torsion
cycle in this group.
Categories:14C25, 19D45 |
66. CMB 2005 (vol 48 pp. 203)
Non-Cohen--Macaulay Projective Monomial Curves with Positive ${h}$-Vector We find an infinite family of projective monomial
curves all of which have $h$-vector with no negative values and
are not Cohen-Macaulay.
Category:14H45 |
67. CMB 2005 (vol 48 pp. 90)
Products of Conjugacy Classes in $SU(2)$ We obtain a complete description of the conjugacy classes
$C_1,\dots,C_n$ in $SU(2)$ with the property that $C_1\cdots
C_n=SU(2)$. The basic instrument is a characterization of the
conjugacy classes $C_1,\dots,C_{n+1}$ in $SU(2)$ with $C_1\cdots
C_{n+1}\ni I$, which generalizes a result of \cite{Je-We}.
Categories:14D20, 14P05 |
68. CMB 2004 (vol 47 pp. 566)
Algebraicity of some Weil Hodge Classes We show that the Prym map for 4-th cyclic \'etale covers of curves
of genus 4 is a dominant morphism to a Shimura variety for a family
of Abelian 6-folds of Weil type. According to the result of Schoen,
this implies algebraicity of Weil classes for this family.
Category:14C30 |
69. CMB 2004 (vol 47 pp. 398)
A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces Let $V$ be a $K3$ surface defined over a number field $k$. The
Batyrev-Manin conjecture for $V$ states that for every nonempty open
subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating
rational curves such that the density of rational points on $U-Z_U$ is
strictly less than the density of rational points on $Z_U$. Thus,
the set of rational points of $V$ conjecturally admits a stratification
corresponding to the sets $Z_U$ for successively smaller sets $U$.
In this paper, in the case that $V$ is a Kummer surface, we prove that
the Batyrev-Manin conjecture for $V$ can be reduced to the
Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$
induced by multiplication by $m$ on the associated abelian surface
$A$. As an application, we use this to show that given some restrictions
on $A$, the set of rational points of $V$ which lie on rational curves
whose preimages have geometric genus 2 admits a stratification of
Keywords:rational points, Batyrev-Manin conjecture, Kummer, surface, rational curve, abelian surface, height Categories:11G35, 14G05 |
70. CMB 2004 (vol 47 pp. 264)
Counting Rational Points on Ruled Varieties In this paper, we prove a general result computing the number of rational points
of bounded height on a projective variety $V$ which is covered by lines. The
main technical result used to achieve this is an upper bound on the number of
rational points of bounded height on a line. This upper bound is such that it
can be easily controlled as the line varies, and hence is used to sum the counting
functions of the lines which cover the original variety $V$.
Categories:11G50, 11D45, 11D04, 14G05 |
71. CMB 2004 (vol 47 pp. 271)
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms |
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms We study the interplay between canonical heights and endomorphisms of an abelian
variety $A$ over a number field $k$. In particular we show that whenever the ring
of endomorphisms defined over $k$ is strictly larger than $\Z$ there will
be $\Q$-linear relations among the values of a canonical height pairing evaluated
at a basis modulo torsion of $A(k)$.
Categories:11G10, 14K15 |
72. CMB 2004 (vol 47 pp. 22)
A Note on the Height of the Formal Brauer Group of a $K3$ Surface Using weighted Delsarte surfaces, we give examples of $K3$ surfaces
in positive characteristic whose formal Brauer groups have height
equal to $5$, $8$ or $9$. These are among the four values of the
height left open in the article of Yui \cite{Y}.
Keywords:formal Brauer groups, $K3$ surfaces in positive, characteristic, weighted Delsarte surfaces Categories:14L05, 14J28 |
73. CMB 2003 (vol 46 pp. 546)
$L$-Series of Certain Elliptic Surfaces In this paper, we study the modularity of certain elliptic surfaces
by determining their $L$-series through their monodromy groups.
Categories:14J27, 11M06 |
74. CMB 2003 (vol 46 pp. 575)
Optimization of Polynomial Functions This paper develops a refinement of Lasserre's algorithm for
optimizing a polynomial on a basic closed semialgebraic set via
semidefinite programming and addresses an open question concerning the
duality gap. It is shown that, under certain natural stability
assumptions, the problem of optimization on a basic closed set reduces
to the compact case.
Categories:14P10, 46L05, 90C22 |
75. 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 |