76. CMB 2004 (vol 47 pp. 398)
 McKinnon, David

A Reduction of the BatyrevManin Conjecture for Kummer Surfaces
Let $V$ be a $K3$ surface defined over a number field $k$. The
BatyrevManin 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 $UZ_U$ 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 BatyrevManin conjecture for $V$ can be reduced to the
BatyrevManin 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, BatyrevManin conjecture, Kummer, surface, rational curve, abelian surface, height Categories:11G35, 14G05 

77. CMB 2004 (vol 47 pp. 264)
 McKinnon, David

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 

78. CMB 2004 (vol 47 pp. 271)
79. CMB 2004 (vol 47 pp. 22)
 Goto, Yasuhiro

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 

80. CMB 2003 (vol 46 pp. 546)
81. CMB 2003 (vol 46 pp. 575)
 Marshall, M.

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 

82. CMB 2003 (vol 46 pp. 495)
 Baragar, Arthur

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 

83. CMB 2003 (vol 46 pp. 321)
 Ballico, E.

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 

84. CMB 2003 (vol 46 pp. 429)
 Sastry, Pramathanath; Tong, Yue Lin L.

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{ast117} agrees with
$$
(2\pi i)^{n} (1)^{n(n1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb
C}) \to {\mathbb C}
$$
under the Dolbeault isomorphism.
Categories:14F10, 32A25, 14A15, 14F05, 18E30 

85. CMB 2003 (vol 46 pp. 400)
 Marshall, M.

Approximating Positive Polynomials Using Sums of Squares
The paper considers the relationship between positive polynomials,
sums of squares and the multidimensional moment problem in the
general context of basic closed semialgebraic sets in real $n$space.
The emphasis is on the noncompact 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 functionalanalytic side and extends these
results in a variety of ways.
Categories:14P10, 44A60 

86. CMB 2003 (vol 46 pp. 323)
 Chamberland, Marc

Characterizing TwoDimensional 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 (da) + b\alpha^2  c\beta^2 = 0.
\end{equation}
This paper shows how these theorems cannot be extended by constructing
a realanalytic map whose Jacobian eigenvalues are $\pm 1/2$ and does
not fit the previous form. This example is also used to construct
nonobvious solutions to nonlinear PDEs, including the MongeAmp\`ere
equation.
Keywords:Jacobian Conjecture, injectivity, MongeAmpÃ¨re equation Categories:26B10, 14R15, 35L70 

87. CMB 2003 (vol 46 pp. 204)
 Levy, Jason

Rationality and Orbit Closures
Suppose we are given a finitedimensional 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 

88. CMB 2003 (vol 46 pp. 140)
89. CMB 2002 (vol 45 pp. 686)
 Rauschning, Jan; Slodowy, Peter

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
6dimensional irreducible module of the symmetric group $S_5$. This
module splits with respect to the icosahedral group $A_5$ into the two
standard 3dimensional representations. The resulting linear
projections of $Q$ relate the action of $A_5$ on $Q$ to those on the
regular icosahedron.
Categories:14L24, 20B25 

90. CMB 2002 (vol 45 pp. 417)
 Kamiyama, Yasuhiko; Tsukuda, Shuichi

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 $n3$. 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 

91. CMB 2002 (vol 45 pp. 349)
 Coppens, Marc

Very Ample Linear Systems on BlowingsUp 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 blowingup 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:blowingup, projective space, very ample linear system, embeddings, Veronese map Categories:14E25, 14N05, 14N15 

92. CMB 2002 (vol 45 pp. 284)
 Sancho de Salas, Fernando

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 

93. CMB 2002 (vol 45 pp. 213)
 Gordon, B. Brent; Joshi, Kirti

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
nontrivial; 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 

94. CMB 2002 (vol 45 pp. 204)
 Fakhruddin, Najmuddin

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 

95. CMB 2002 (vol 45 pp. 89)
 Grant, David

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 

96. CMB 2001 (vol 44 pp. 491)
 Wang, Weiqiang

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 

97. CMB 2001 (vol 44 pp. 452)
 Ishihara, Hironobu

Some Adjunction Properties of Ample Vector Bundles
Let $\ce$ be an ample vector bundle of rank $r$ on a projective
variety $X$ with only logterminal singularities. We consider the
nefness of adjoint divisors $K_X + (tr) \det \ce$ 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 

98. CMB 2001 (vol 44 pp. 257)
 Abánades, Miguel A.

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 

99. CMB 2001 (vol 44 pp. 313)
 Reverter, Amadeu; Vila, Núria

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 

100. CMB 2001 (vol 44 pp. 223)