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51. CMB 2009 (vol 52 pp. 535)

Daigle, Daniel; Kaliman, Shulim
 A Note on Locally Nilpotent Derivations\\ and Variables of $k[X,Y,Z]$ We strengthen certain results concerning actions of $(\Comp,+)$ on $\Comp^{3}$ and embeddings of $\Comp^{2}$ in $\Comp^{3}$, and show that these results are in fact valid over any field of characteristic zero. Keywords:locally nilpotent derivations, group actions, polynomial automorphisms, variable, affine spaceCategories:14R10, 14R20, 14R25, 13N15

52. CMB 2009 (vol 52 pp. 493)

Artebani, Michela
 A One-Dimensional Family of $K3$ Surfaces with a $\Z_4$ Action The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of $\Z_4$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes. Keywords:genus three curves, $K3$ surfacesCategories:14J28, 14J50, 14J10

53. CMB 2009 (vol 52 pp. 175)

Biswas, Indranil
 Connections on a Parabolic Principal Bundle, II In \emph{Connections on a parabolic principal bundle over a curve, I} we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence. Keywords:Parabolic bundle, Atiyah exact sequence, connectionCategories:32L05, 14F05

54. CMB 2009 (vol 52 pp. 224)

Ghiloni, Riccardo
 Equations and Complexity for the Dubois--Efroymson Dimension Theorem Let $\R$ be a real closed field, let $X \subset \R^n$ be an irreducible real algebraic set and let $Z$ be an algebraic subset of $X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence of an irreducible algebraic subset of $X$ of codimension $1$ containing~$Z$. We improve this dimension theorem as follows. Indicate by $\mu$ the minimum integer such that the ideal of polynomials in $\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials of degree $\leq \mu$. We prove the following two results: \begin{inparaenum}[\rm(1)] \item There exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$ such that $X \cap P^{-1}(0)$ is an irreducible algebraic subset of $X$ of codimension $1$ containing~$Z$. \item Let $F$ be a polynomial in $\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq \max\{d,\mu+1\}$ such that, for each $t \in (-1,1) \setminus \{0\}$, the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic subset of $X$ of codimension $1$ containing~$Z$. \end{inparaenum} Result (1) and a slightly different version of result~(2) are valid over any algebraically closed field also. Keywords:Irreducible algebraic subvarieties, complexity of algebraic varieties, Bertini's theoremsCategories:14P05, 14P20

55. CMB 2009 (vol 52 pp. 200)

Gatto, Letterio; Santiago, Ta\'\i se
 Schubert Calculus on a Grassmann Algebra The ({\em classical}, {\em small quantum}, {\em equivariant}) cohomology ring of the grassmannian $G(k,n)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ ( Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$. Categories:14N15, 14M15

56. CMB 2009 (vol 52 pp. 161)

Arcara, D.; Lee, Y.-P.
 A New Tautological Relation in $\overline{\mathcal{M}}_{3,1}$ via the Invariance Constraint A new tautological relation of $\overline{\mathcal{M}}_{3,1}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work. Category:14H10

57. CMB 2009 (vol 52 pp. 117)

Poulakis, Dimitrios
 On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$ Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points. Categories:11G30, 14G05, 14G25

58. CMB 2009 (vol 52 pp. 39)

Cimpri\v{c}, Jakob
 A Representation Theorem for Archimedean Quadratic Modules on $*$-Rings We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution. Keywords:Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometryCategories:16W80, 46L05, 46L89, 14P99

59. CMB 2008 (vol 51 pp. 519)

Coskun, Izzet; Harris, Joe; Starr, Jason
 The Effective Cone of the Kontsevich Moduli Space In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, $\Kgnb{0,0}(\PP^r,d)$, stabilize when $r \geq d$. We give a complete characterization of the effective divisors on $\Kgnb{0,0}(\PP^d,d)$. They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image. Categories:14D20, 14E99, 14H10

60. CMB 2008 (vol 51 pp. 283)

Ravindra, G. V.
 The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing. Keywords:Noether--Lefschetz, algebraic cycles, Picard numberCategories:14C15, 14C25

61. CMB 2008 (vol 51 pp. 114)

Petrov, V.; Semenov, N.; Zainoulline, K.
 Zero Cycles on a Twisted Cayley Plane Let $k$ be a field of characteristic not $2,3$. Let $G$ be an exceptional simple algebraic group over $k$ of type $\F$, $^1{\E_6}$ or $\E_7$ with trivial Tits algebras. Let $X$ be a projective $G$-homogeneous variety. If $G$ is of type $\E_7$, we assume in addition that the respective parabolic subgroup is of type $P_7$. The main result of the paper says that the degree map on the group of zero cycles of $X$ is injective. Categories:20G15, 14C15

62. CMB 2008 (vol 51 pp. 125)

Polo-Blanco, Irene; Top, Jaap
 Explicit Real Cubic Surfaces The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L.~Schl\"afli in 1858. Using this, explicit examples of surfaces of every possible type are given. Categories:14J25, 14J80, 14P25, 14Q10

63. CMB 2007 (vol 50 pp. 567)

Joshi, Kirti
 Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications. Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsionCategories:14F30, 14J30

64. CMB 2007 (vol 50 pp. 486)

Cynk, S.; Hulek, K.
 Higher-Dimensional Modular\\Calabi--Yau Manifolds We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity. Categories:14G10, 14J32, 11G40

65. CMB 2007 (vol 50 pp. 427)

Mejía, Israel Moreno
 On the Image of Certain Extension Maps.~I Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective. Categories:14H60, 14F05, 14D20

66. CMB 2007 (vol 50 pp. 243)

Langlands, Robert P.
 Un nouveau point de repÃ¨re dans la thÃ©orie des formes automorphes Dans le papier Beyond Endoscopy une id\'ee pour entamer la fonctorialit\'e en utilisant la formule des traces a \'et\'e introduite. Maints probl\emes, l'existence d'une limite convenable de la formule des traces, est eqquiss\'ee dans cette note informelle mais seulement pour $GL(2)$ et les corps des fonctions rationelles sur un corps fini et en ne pas resolvant bon nombre de questions. Categories:32N10, 14xx

67. CMB 2007 (vol 50 pp. 161)

Arapura, Donu; Kang, Su-Jeong
 Functoriality of the Coniveau Filtration It is shown that the coniveau filtration on the cohomology of smooth projective varieties is preserved up to shift by pushforwards, pullbacks and products. Category:14C30

68. CMB 2007 (vol 50 pp. 196)

Fernández, Julio; González, Josep; Lario, Joan-C.
 Plane Quartic Twists of $X(5,3)$ Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$ defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$ realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case $N=5$. Categories:11F03, 11F80, 14G05

69. CMB 2007 (vol 50 pp. 215)

Kloosterman, Remke
 Elliptic $K3$ Surfaces with Geometric Mordell--Weil Rank $15$ We prove that the elliptic surface $y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil rank $15$. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell--Weil ranks $0,1,\dots, 14, 16, 17, 18$. Categories:14J27, 14J28, 11G05

70. CMB 2007 (vol 50 pp. 126)

Ongay, Fausto
 $\varphi$-Dialgebras and a Class of Matrix Coquecigrues'' Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we construct examples of `coquecigrues,'' in the sense of Loday, that is to say, manifolds whose tangent structure at a distinguished point coincides with that of the Leibniz algebra. We discuss some possible implications and generalizations of this construction. Keywords:Leibniz algebras, dialgebrasCategory:14M30

71. CMB 2007 (vol 50 pp. 105)

Klep, Igor
 On Valuations, Places and Graded Rings Associated to $*$-Orderings We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups. Keywords:$*$--orderings, valuations, rings with involutionCategories:14P10, 16S30, 16W10

72. CMB 2006 (vol 49 pp. 592)

Sarti, Alessandra
 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

73. CMB 2006 (vol 49 pp. 560)

Luijk, Ronald van
 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 NÃ©ron-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 theoryCategories:14G05, 14J28, 11D41

74. CMB 2006 (vol 49 pp. 464)

Ravindra, G. V.
 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

75. CMB 2006 (vol 49 pp. 270)

Occhetta, Gianluca
 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 varietiesCategories:14J40, 14J45
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