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76. 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 $$N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q) Categories:11G50, 14J28, 14G40, 14J50, 14G05 77. CMB 2003 (vol 46 pp. 321) Ballico, E.  Discreteness For the Set of Complex Structures On a Real Variety Let X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X\times Y \cong X\times W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X\times X^\sigma \cong W\times W^\sigma (where A^\sigma denotes the complex conjugate of any variety A) is at most countable. Categories:32J18, 14J99, 14P99 78. CMB 2003 (vol 46 pp. 429) Sastry, Pramathanath; Tong, Yue Lin L.  The Grothendieck Trace and the de Rham Integral On a smooth n-dimensional complete variety X over {\mathbb C} we show that the trace map {\tilde\theta}_X \colon\break H^n (X,\Omega_X^n) \to {\mathbb C} arising from Lipman's version of Grothendieck duality in \cite{ast-117} agrees with$$ (2\pi i)^{-n} (-1)^{n(n-1)/2} \int_X \colon H^{2n}_{DR} (X,{\mathbb C}) \to {\mathbb C} $$under the Dolbeault isomorphism. Categories:14F10, 32A25, 14A15, 14F05, 18E30 79. CMB 2003 (vol 46 pp. 400) Marshall, M.  Approximating Positive Polynomials Using Sums of Squares The paper considers the relationship between positive polynomials, sums of squares and the multi-dimensional moment problem in the general context of basic closed semi-algebraic sets in real n-space. The emphasis is on the non-compact case and on quadratic module representations as opposed to quadratic preordering presentations. The paper clarifies the relationship between known results on the algebraic side and on the functional-analytic side and extends these results in a variety of ways. Categories:14P10, 44A60 80. CMB 2003 (vol 46 pp. 323) Chamberland, Marc  Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues Recent papers have shown that C^1 maps F\colon \mathbb{R}^2 \rightarrow \mathbb{R}^2 whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or F is a polynomial. Specifically, F=(u,v) must take the form \begin{gather*} u = ax + by + \beta \phi(\alpha x + \beta y) + e \\ v = cx + dy - \alpha \phi(\alpha x + \beta y) + f \end{gather*} for some constants a, b, c, d, e, f, \alpha, \beta and a C^1 function \phi in one variable. If, in addition, the function \phi is not affine, then $$\alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0.$$ 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 equationCategories:26B10, 14R15, 35L70 81. 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 82. 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 83. 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 84. 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 structureCategories:14D20, 32C35 85. 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 mapCategories:14E25, 14N05, 14N15 86. 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 87. 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 groupCategories:14J20, 14C25 88. 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 89. 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 90. 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 genusCategories:14J60, 14C20, 14F05, 14J40 91. 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 92. 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 93. 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, isogenyCategories:11R32, 11G05, 12F10, 14K02 94. 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 95. 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 96. CMB 2000 (vol 43 pp. 304) Darmon, Henri; Mestre, Jean-François  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) 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 98. 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 sumsCategories:11G05, 14H52

99. 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

100. CMB 2000 (vol 43 pp. 129)

Ballico, E.
 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
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