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101. 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

102. 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

103. 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{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

104. 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 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

105. CMB 2003 (vol 46 pp. 323)

Chamberland, Marc
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--Ampère equation
Categories:26B10, 14R15, 35L70

106. 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

107. CMB 2003 (vol 46 pp. 204)

Levy, Jason
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

108. CMB 2003 (vol 46 pp. 140)

Renner, Lex E.
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

109. 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 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

110. CMB 2002 (vol 45 pp. 349)

Coppens, Marc
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

111. 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 $n-3$. 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

112. 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

113. 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.


114. 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 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

115. 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

116. 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

117. 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 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

118. 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

119. 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

120. CMB 2001 (vol 44 pp. 223)

Marshall, M.
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

121. CMB 2000 (vol 43 pp. 312)

Dobbs, David E.
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

122. CMB 2000 (vol 43 pp. 304)

Darmon, Henri; Mestre, Jean-François
Courbes hyperelliptiques à multiplications réelles 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

123. CMB 2000 (vol 43 pp. 162)

Foth, Philip
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

124. CMB 2000 (vol 43 pp. 239)

Yu, Gang
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

125. CMB 2000 (vol 43 pp. 174)

Gantz, Christian; Steer, Brian
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
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