51. CMB 2009 (vol 53 pp. 247)
 Etingof, P.; Malcolmson, P.; Okoh, F.

Root Extensions and Factorization in Affine Domains
An integral domain R is IDPF (Irreducible Divisors of Powers Finite) if, for every nonzero element a in R, the ascending chain of nonassociate irreducible divisors in R of $a^{n}$ stabilizes on a finite set as n ranges over the positive integers, while R is atomic if every nonzero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension S of R is a \emph{root extension} or \emph{radical extension} if for each s in S, there exists a natural number $n(s)$ with $s^{n(s)}$ in R. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $(R,S)$ is governed by the relative sizes of the unit groups $\operatorname{U}(R)$ and $\operatorname{U}(S)$ and whether S is a root extension of R. The following results are deduced from these considerations. An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let R be a Noetherian domain with integral closure S. Suppose the conductor of S into R is nonzero. Then R is IDPF if and only if S is a root extension of R and $\operatorname{U}(S)/\operatorname{U}(R)$ is finite.
Categories:13F15, 14A25 

52. CMB 2009 (vol 53 pp. 218)
 Biswas, Indranil

Restriction of the Tangent Bundle of $G/P$ to a Hypersurface
Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.
Keywords:tangent bundle, homogeneous space, semistability, hypersurface Categories:14F05, 14J60, 14M15 

53. 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 space Categories:14R10, 14R20, 14R25, 13N15 

54. CMB 2009 (vol 52 pp. 493)
 Artebani, Michela

A OneDimensional 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 onedimensional family of $K3$ surfaces
associated to the locus of plane quartics with five nodes.
Keywords:genus three curves, $K3$ surfaces Categories:14J28, 14J50, 14J10 

55. 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, connection Categories:32L05, 14F05 

56. CMB 2009 (vol 52 pp. 224)
 Ghiloni, Riccardo

Equations and Complexity for the DuboisEfroymson 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 theorems Categories:14P05, 14P20 

57. 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 

58. CMB 2009 (vol 52 pp. 161)
59. 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 

60. 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
GelfandNaimark representation theorem for commutative $C^\ast$algebras.
A noncommutative version of GelfandNaimark 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 geometry Categories:16W80, 46L05, 46L89, 14P99 

61. 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 nonnegative
linear combinations of boundary divisors and the divisor of maps with
degenerate image.
Categories:14D20, 14E99, 14H10 

62. CMB 2008 (vol 51 pp. 283)
63. CMB 2008 (vol 51 pp. 125)
 PoloBlanco, 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 

64. 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 

65. 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
HodgeWitt. This is proved by generalizing to the case of
threefolds a wellknown criterion due to N.~Nygaard for surfaces to be HodgeWitt.
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, HodgeWitt, crystalline cohomology, slope spectral sequence, exotic torsion Categories:14F30, 14J30 

66. CMB 2007 (vol 50 pp. 486)
67. 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 $n+1$ semistable
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 

68. 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 

69. CMB 2007 (vol 50 pp. 215)
 Kloosterman, Remke

Elliptic $K3$ Surfaces with Geometric MordellWeil 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 MordellWeil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$surfaces with geometric MordellWeil ranks
$0,1,\dots, 14, 16, 17, 18$.
Categories:14J27, 14J28, 11G05 

70. CMB 2007 (vol 50 pp. 196)
 Fernández, Julio; González, Josep; Lario, JoanC.

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 genusthree case $N=5$.
Categories:11F03, 11F80, 14G05 

71. CMB 2007 (vol 50 pp. 161)
72. 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 quasiOre and is
even quasicommutative (\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 involution Categories:14P10, 16S30, 16W10 

73. 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, dialgebras Category:14M30 

74. 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 nonKummer, singular
K3 surface
are dense. We will also compute the entire NÃ©ronSeveri group of
this surface and find all low degree curves on it.
Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ronSeveri group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theory Categories:14G05, 14J28, 11D41 

75. CMB 2006 (vol 49 pp. 592)
 Sarti, Alessandra

Group Actions, Cyclic Coverings and Families of K3Surfaces
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 ADE singularities and one is
nonreduced. 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
3divisible sets, resp., 2divisible sets, of rational curves and
lattice theory, we describe explicitly the Picard lattices.
Categories:14J28, 14L30, 14E20, 14C22 
