51. CJM 2012 (vol 65 pp. 823)
 Guardo, Elena; Harbourne, Brian; Van Tuyl, Adam

Symbolic Powers Versus Regular Powers of Ideals of General Points in $\mathbb{P}^1 \times \mathbb{P}^1$
Recent work of EinLazarsfeldSmith and HochsterHuneke
raised the problem of which symbolic powers of an ideal
are contained in a given ordinary power of the ideal.
BocciHarbourne developed methods to address this problem,
which involve asymptotic numerical characters of
symbolic powers of the ideals. Most of the work
done up to now has been done for ideals defining 0dimensional
subschemes of projective space.
Here we focus on certain subschemes given by
a union of lines in $\mathbb{P}^3$ which can also be viewed
as points in $\mathbb{P}^1 \times \mathbb{P}^1$.
We also obtain results on the
closely related problem, studied by Hochster and by LiSwanson, of
determining situations for which
each symbolic power of an ideal is an ordinary power.
Keywords:symbolic powers, multigraded, points Categories:13F20, 13A15, 14C20 

52. CJM 2012 (vol 65 pp. 1020)
 Goulden, I. P.; GuayPaquet, Mathieu; Novak, Jonathan

Monotone Hurwitz Numbers in Genus Zero
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the JucysMurphy elements, and have arisen in recent work on the the asymptotic expansion of the HarishChandraItzyksonZuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone joincut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.
Keywords:Hurwitz numbers, matrix models, enumerative geometry Categories:05A15, 14E20, 15B52 

53. CJM 2012 (vol 65 pp. 905)
 Thompson, Alan

Explicit Models for Threefolds Fibred by K3 Surfaces of Degree Two
We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two.
Keywords:threefold, fibration, K3 surface Categories:14J30, 14D06, 14E30, 14J28 

54. CJM 2012 (vol 65 pp. 634)
 Mezzetti, Emilia; MiróRoig, Rosa M.; Ottaviani, Giorgio

Laplace Equations and the Weak Lefschetz Property
We prove that $r$ independent homogeneous polynomials of the same degree $d$
become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety
whose $(d1)$osculating spaces have dimension smaller than expected. This gives an equivalence
between an algebraic notion (called Weak Lefschetz Property)
and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case,
some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefold Categories:13E10, 14M25, 14N05, 14N15, 53A20 

55. CJM 2012 (vol 65 pp. 575)
 Kallel, Sadok; Taamallah, Walid

The Geometry and Fundamental Group of Permutation Products and Fat Diagonals
Permutation products and their various ``fat diagonal'' subspaces are
studied from the topological and geometric point of view. We describe
in detail the stabilizer and orbit stratifications related to the
permutation action, producing a sharp upper bound for its depth and
then paying particular attention to the geometry of the diagonal
stratum. We write down an expression for the fundamental group of any
permutation product of a connected space $X$ having the homotopy type
of a CW complex in terms of $\pi_1(X)$ and $H_1(X;\mathbb{Z})$. We then
prove that the fundamental group of the configuration space of
$n$points on $X$, of which multiplicities do not exceed $n/2$,
coincides with $H_1(X;\mathbb{Z})$. Further results consist in giving
conditions for when fat diagonal subspaces of manifolds can be
manifolds again. Various examples and homological calculations are
included.
Keywords:symmetric products, fundamental group, orbit stratification Categories:14F35, 57F80 

56. CJM 2012 (vol 65 pp. 961)
 Aholt, Chris; Sturmfels, Bernd; Thomas, Rekha

A Hilbert Scheme in Computer Vision
Multiview geometry is the study of
twodimensional images of threedimensional scenes, a foundational subject in computer vision.
We determine a universal GrÃ¶bner basis for the multiview ideal of $n$ generic cameras.
As the cameras move, the multiview varieties vary in a family of dimension $11n15$.
This family is the distinguished component of a multigraded Hilbert scheme
with a unique Borelfixed point.
We present a combinatorial study
of ideals lying on that Hilbert scheme.
Keywords:multigraded Hilbert Scheme, computer vision, monomial ideal, Groebner basis, generic initial ideal Categories:14N, 14Q, 68 

57. CJM 2012 (vol 64 pp. 1222)
 Bobiński, Grzegorz

Normality of Maximal Orbit Closures for Euclidean Quivers
Let $\Delta$ be an Euclidean quiver. We prove that the closures of
the maximal orbits in the varieties of representations of $\Delta$
are normal and CohenMacaulay (even complete intersections).
Moreover, we give a generalization of this result for the tame
concealedcanonical algebras.
Keywords:normal variety, complete intersection, Euclidean quiver, concealedcanonical algebra Categories:16G20, 14L30 

58. 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 

59. 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 

60. 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 

61. 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 

62. 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 

63. 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 

64. 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 

65. 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 

66. 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 

67. 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 

68. 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 

69. 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 

70. 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 

71. 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 

72. 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 

73. 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 

74. 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 

75. 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 
