51. CJM 2012 (vol 65 pp. 721)
 Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul

Tameness of Complex Dimension in a Real Analytic Set
Given a real analytic set $X$ in a complex manifold and a positive
integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which
there exists a germ of a complex analytic set of dimension $d$ contained in $X$.
It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$.
Keywords:complex dimension, finite type, semianalytic set, tameness Categories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15 

52. CJM 2012 (vol 65 pp. 544)
 Deitmar, Anton; Horozov, Ivan

Iterated Integrals and Higher Order Invariants
We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The nonuniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 

53. CJM 2012 (vol 65 pp. 195)
 Penegini, Matteo; Polizzi, Francesco

Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$
We classify minimal surfaces of general type with $p_g=q=2$ and
$K^2=6$ whose Albanese map is a generically finite double cover.
We show that the corresponding moduli space is the disjoint union
of three generically smooth irreducible components
$\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of
dimension $4$, $4$, $3$, respectively.
Keywords:surface of general type, abelian surface, Albanese map Categories:14J29, 14J10 

54. CJM 2012 (vol 65 pp. 120)
 Francois, Georges; Hampe, Simon

Universal Families of Rational Tropical Curves
We introduce the notion of families of $n$marked
smooth rational tropical curves over smooth tropical varieties and
establish a onetoone correspondence between (equivalence classes of)
these families and morphisms
from smooth tropical varieties into the moduli space of $n$marked
abstract rational tropical curves $\mathcal{M}_{n}$.
Keywords:tropical geometry, universal family, rational curves, moduli space Categories:14T05, 14D22 

55. CJM 2011 (vol 64 pp. 1090)
 Rosso, Daniele

Classic and Mirabolic RobinsonSchenstedKnuth Correspondence for Partial Flags
In this paper we first generalize to the case of
partial flags a result proved both by Spaltenstein and by Steinberg
that relates the relative position of two complete flags and the
irreducible components of the flag variety in which they lie, using
the RobinsonSchenstedKnuth correspondence. Then we use this result
to generalize the mirabolic RobinsonSchenstedKnuth correspondence
defined by Travkin, to the case of two partial flags and a line.
Keywords:partial flag varieties, RSK correspondence Categories:14M15, 05A05 

56. CJM 2011 (vol 64 pp. 1248)
 Gärtner, Jérôme

Darmon's Points and Quaternionic Shimura Varieties
In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a GrossKohnenZagier theorem for Darmon's
points.
Keywords:elliptic curves, StarkHeegner points, quaternionic Shimura varieties Categories:11G05, 14G35, 11F67, 11G40 

57. CJM 2011 (vol 64 pp. 3)
 Boissière, Samuel

Automorphismes naturels de l'espace de Douady de points sur une surface
On Ã©tablit quelques rÃ©sultats gÃ©nÃ©raux relatifs Ã la taille
du groupe d'automorphismes de l'espace de Douady de points sur une
surface, puis on Ã©tudie quelques propriÃ©tÃ©s des automorphismes
provenant d'un automorphisme de la surface, en particulier leur action
sur la cohomologie et la classification de leurs points fixes.
Keywords:SchÃ©ma de Hilbert, automorphismes, points fixes Category:14C05 

58. CJM 2011 (vol 64 pp. 1122)
 Seveso, Marco Adamo

$p$adic $L$functions and the Rationality of Darmon Cycles
Darmon cycles are a higher weight analogue of StarkHeegner points. They
yield local cohomology classes in the Deligne representation associated with a
cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$.
They are conjectured to be the restriction of global cohomology classes in
the BlochKato Selmer group defined over narrow ring class fields attached
to a real quadratic field. We show that suitable linear combinations of them
obtained by genus characters satisfy these conjectures. We also prove $p$adic GrossZagier type formulas, relating the derivatives of $p$adic $L$functions of the weight variable attached to imaginary (resp. real)
quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express
the second derivative of the MazurKitagawa $p$adic $L$function of the
weight variable in terms of a global cycle defined over a quadratic
extension of $\mathbb{Q}$.
Categories:11F67, 14G05 

59. CJM 2011 (vol 64 pp. 845)
 Helm, David; Katz, Eric

Monodromy Filtrations and the Topology of Tropical Varieties
We study the topology of tropical varieties that arise from a certain
natural class of varieties. We use the theory of tropical
degenerations to construct a natural, ``multiplicityfree''
parameterization of $\operatorname{Trop}(X)$ by a topological space
$\Gamma_X$ and give a geometric interpretation of the cohomology of
$\Gamma_X$ in terms of the action of a monodromy operator on the
cohomology of $X$. This gives bounds on the Betti numbers of
$\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the
topology of $\operatorname{Trop}(X)$. We also obtain a description of
the top power of the monodromy operator acting on middle cohomology of
$X$ in terms of the volume pairing on $\Gamma_X$.
Categories:14T05, 14D06 

60. CJM 2011 (vol 64 pp. 805)
 Chapon, François; Defosseux, Manon

Quantum Random Walks and Minors of Hermitian Brownian Motion
Considering quantum random walks, we construct discretetime
approximations of the eigenvalues processes of minors of Hermitian
Brownian motion. It has been recently proved by Adler, Nordenstam, and
van Moerbeke that the process of eigenvalues of
two consecutive minors of a Hermitian Brownian motion is a Markov
process; whereas, if one considers more than two consecutive minors,
the Markov property fails. We show that there are analog results in
the noncommutative counterpart and establish the Markov property of
eigenvalues of some particular submatrices of Hermitian Brownian
motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process Categories:46L53, 60B20, 14L24 

61. CJM 2011 (vol 63 pp. 1345)
 Jardine, J. F.

Pointed Torsors
This paper gives a characterization of homotopy fibres of inverse
image maps on groupoids of torsors that are induced by geometric
morphisms, in terms of both pointed torsors and pointed cocycles,
suitably defined. Cocycle techniques are used to give a complete
description of such fibres, when the underlying geometric morphism is
the canonical stalk on the classifying topos of a profinite group
$G$. If the torsors in question are defined with respect to a constant
group $H$, then the path components of the fibre can be identified with
the set of continuous maps from the profinite group $G$ to the group
$H$. More generally, when $H$ is not constant, this set of path components
is the set of continuous maps from a proobject in sheaves of
groupoids to $H$, which proobject can be viewed as a ``Grothendieck
fundamental groupoid".
Keywords:pointed torsors, pointed cocycles, homotopy fibres Categories:18G50, 14F35, 55B30 

62. CJM 2011 (vol 64 pp. 123)
 Lee, JaeHyouk

Gosset Polytopes in Picard Groups of del Pezzo Surfaces
In this article, we study the correspondence between the geometry of
del Pezzo surfaces $S_{r}$ and the geometry of the $r$dimensional Gosset
polytopes $(r4)_{21}$. We construct Gosset polytopes $(r4)_{21}$ in
$\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify
divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a1)$simplexes ($a\leq
r$), $(r1)$simplexes and $(r1)$crosspolytopes of the polytope $(r4)_{21}$.
Then we explain how these classes correspond to skew $a$lines($a\leq r$),
exceptional systems, and rulings, respectively.
As an application, we work on the monoidal transform for lines to study the
local geometry of the polytope $(r4)_{21}$. And we show that the Gieser transformation
and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and
$4_{21}$, respectively.
Categories:51M20, 14J26, 22E99 

63. CJM 2011 (vol 64 pp. 409)
 Rainer, Armin

Lifting Quasianalytic Mappings over Invariants
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blowups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 

64. CJM 2011 (vol 64 pp. 81)
 David, C.; Wu, J.

Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication,
and for each prime
$p$ of good reduction, let $n_E(p) =  E(\mathbb F_p) $. For any integer
$b$, we consider elliptic pseudoprimes to the base
$b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq
x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E,
b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such
that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called
elliptic curve pseudoprimes). Motivated by cryptography applications,
we address the problem of finding upper bounds for
$Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$,
generalising some of the literature for the classical pseudoprimes
to this new setting.
Keywords:RosserIwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Categories:11N36, 14H52 

65. CJM 2011 (vol 63 pp. 1058)
 Easton, Robert W.

$S_3$covers of Schemes
We analyze flat $S_3$covers of schemes, attempting to create
structures parallel to those found in the abelian and triple cover
theories. We use an initial local analysis as a guide in finding a
global description.
Keywords:nonabelian groups, permutation group, group covers, schemes Category:14L30 

66. CJM 2011 (vol 63 pp. 992)
 Bruin, Nils; Doerksen, Kevin

The Arithmetic of Genus Two Curves with (4,4)Split Jacobians
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$torsion, and we derive the relation their absolute invariants
satisfy.
As an intermediate step, we give a general description of Richelot
isogenies between Jacobians of genus $2$ curves, where previously only
Richelot isogenies with kernels that are pointwise defined over the
base field were considered.
Our main tool is a Galois theoretic characterization of genus $2$
curves admitting multiple Richelot isogenies.
Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curves Categories:11G30, 14H40 

67. CJM 2011 (vol 63 pp. 1388)
 Misamore, Michael D.

Nonabelian $H^1$ and the Ãtale Van Kampen Theorem
Generalized Ã©tale homotopy progroups $\pi_1^{\operatorname{Ã©t}}(Ä{C}, x)$
associated with pointed, connected, small Grothendieck
sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois
theory and the theory of pointed torsors for discrete
groups is explained.
Applications include new rigorous proofs of some folklore results
around $\pi_1^{\operatorname{Ã©t}}(Ã©t(X), x)$, a description of
Grothendieck's short exact sequence for Galois descent in terms of
pointed torsor trivializations, and a new Ã©tale
van Kampen theorem that gives a simple statement about a pushout
square of progroups that works for covering
families that do not necessarily consist exclusively of
monomorphisms. A corresponding van Kampen result for
Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows.
Keywords:Ã©tale homotopy theory, simplicial sheaves Categories:18G30, 14F35 

68. CJM 2011 (vol 63 pp. 755)
 Chu, Kenneth C. K.

On the Geometry of the Moduli Space of Real Binary Octics
The moduli space of smooth real binary octics has five connected
components. They parametrize the real binary octics whose defining
equations have $0,\dots,4$ complexconjugate pairs of roots
respectively. We show that each of these five components has a real
hyperbolic structure in the sense that each is isomorphic as a
realanalytic manifold to the quotient of an open dense subset of
$5$dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an
arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are
commensurable to discrete hyperbolic reflection groups, and the
Vinberg diagrams of the latter are computed.
Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm Categories:32G13, 32G20, 14D05, 14D20 

69. CJM 2011 (vol 63 pp. 878)
 Howard, Benjamin; Manon, Christopher; Millson, John

The Toric Geometry of Triangulated Polygons in Euclidean Spac
Speyer and Sturmfels associated GrÃ¶bner toric
degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$
of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each
trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations
induce toric
degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the
space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line.
Our goal in this paper is to give a
geometric (Euclidean polygon) description of the toric fibers
and describe the action of the
compact part of the torus
as "bendings of polygons".
We prove the conjecture of Foth and Hu that
the toric fibers are homeomorphic
to the spaces defined by Kamiyama and Yoshida.
Categories:14L24, 53D20 

70. CJM 2011 (vol 63 pp. 616)
 Lee, Edward

A Modular Quintic CalabiYau Threefold of Level 55
In this note we search the parameter space of HorrocksMumford quintic
threefolds and locate a CalabiYau threefold that is modular, in the
sense that the $L$function of its middledimensional cohomology is
associated with a classical modular form of weight 4 and level 55.
Keywords: CalabiYau threefold, nonrigid CalabiYau threefold, twodimensional Galois representation, modular variety, HorrocksMumford vector bundle Categories:14J15, 11F23, 14J32, 11G40 

71. CJM 2011 (vol 63 pp. 481)
 Baragar, Arthur

The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal
representations of the ample or KÃ¤hler cone for surfaces in a
certain class of $K3$ surfaces. The class includes surfaces
described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a
sufficiently large number field $K$ that have a line parallel to
one of the axes and have Picard number four. We relate the
Hausdorff dimension of this fractal to the asymptotic growth of
orbits of curves under the action of the surface's group of
automorphisms. We experimentally estimate the Hausdorff dimension
of the fractal to be $1.296 \pm .010$.
Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 

72. CJM 2010 (vol 63 pp. 86)
73. CJM 2010 (vol 62 pp. 1293)
 Kasprzyk, Alexander M.

Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is
given. As an application of this technique, we present a
classification of the toric Fano threefolds with at worst canonical
singularities. Up to isomorphism, there are $674,\!688$ such
varieties.
Keywords:toric, Fano, threefold, canonical singularities, convex polytopes Categories:14J30, 14J30, 14M25, 52B20 

74. CJM 2010 (vol 62 pp. 1201)
75. CJM 2010 (vol 62 pp. 1131)
 Kleppe, Jan O.

Moduli Spaces of Reflexive Sheaves of Rank 2
Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of
dimension at least two and let $X \subset Y$ be the zero set of a section
$\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the
functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform
$\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two
forgetful maps between the functors, we prove that the scheme structure of
\emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$
at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves
on $Y$ become closely related. Using this relationship, we get criteria for the
dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $
{\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose
deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $
(\emph{e.g.,} of diameter at most 2),
we get necessary and sufficient conditions of unobstructedness that coincide
in the diameter one case. The conditions are further equivalent to the
vanishing of certain graded Betti numbers of the free graded minimal
resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible
component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter
one is reduced (generically smooth) and we compute its dimension. We also
determine a good lower bound for the dimension of any component of ${\rm
M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small''
deficiency module $M$.
Keywords:moduli space, reflexive sheaf, Hilbert scheme, space curve, Buchsbaum sheaf, unobstructedness, cup product, graded Betti numbers.xdvi Categories:14C05, qqqqq14D22, 14F05, 14J10, 14H50, 14B10, 13D02, 13D07 
