26. CJM 2014 (vol 67 pp. 55)
 Barron, Tatyana; Kerner, Dmitry; Tvalavadze, Marina

On Varieties of Lie Algebras of Maximal Class
We study complex projective varieties that parametrize
(finitedimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinitedimensional case we concentrate our attention on
${\mathbb N}$graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.
Keywords:filiform Lie algebras, graded Lie algebras, projective varieties, topology, classification Categories:17B70, 14F45 

27. CJM 2014 (vol 66 pp. 961)
 Baird, Thomas

Moduli Spaces of Vector Bundles over a Real Curve: $\mathbb Z/2$Betti Numbers
Moduli spaces of real bundles over a real curve arise naturally
as Lagrangian submanifolds of the moduli space of semistable
bundles over a complex curve. In this paper, we adapt the methods
of AtiyahBott's ``YangMills over a Riemann Surface'' to compute
$\mathbb Z/2$Betti numbers of these spaces.
Keywords:cohomology of moduli spaces, holomorphic vector bundles Categories:32L05, 14P25 

28. CJM 2014 (vol 67 pp. 198)
 Murty, V. Kumar; Patankar, Vijay M.

Tate Cycles on Abelian Varieties with Complex Multiplication
We consider Tate cycles on an Abelian variety $A$ defined over
a sufficiently large number field $K$ and having complex
multiplication. We show that
there is an effective bound $C = C(A,K)$ so that
to check whether a given cohomology class is a Tate class on
$A$, it suffices to check the action of
Frobenius elements at primes $v$ of norm $ \leq C$.
We also show that for a set of primes $v$ of $K$ of density
$1$, the space of Tate cycles on the special fibre $A_v$ of the
NÃ©ron model of $A$ is isomorphic to the space of Tate cycles
on $A$ itself.
Keywords:Abelian varieties, complex multiplication, Tate cycles Categories:11G10, 14K22 

29. CJM 2014 (vol 67 pp. 667)
 Nishinou, Takeo

Toric Degenerations, Tropical Curve, and GromovWitten Invariants of Fano Manifolds
In this paper, we give a tropical method for computing GromovWitten
type invariants
of Fano manifolds of special type.
This method applies to those Fano manifolds which admit toric
degenerations
to toric Fano varieties with singularities allowing small resolutions.
Examples include (generalized) flag manifolds of type A, and
some moduli space
of rank two bundles on a genus two curve.
Keywords:Fano varieties, GromovWitten invariants, tropical curves Category:14J45 

30. CJM 2014 (vol 67 pp. 286)
 Bell, Jason P.; Lagarias, Jeffrey C.

A SkolemMahlerLech Theorem for Iterated Automorphisms of $K$algebras
This paper proves a commutative algebraic extension
of a generalized SkolemMahlerLech theorem due to the first
author.
Let $A$ be a finitely generated commutative $K$algebra
over a field of characteristic $0$, and let $\sigma$ be
a $K$algebra automorphism of $A$.
Given ideals $I$ and $J$ of $A$, we show that
the set $S$ of integers $m$ such that
$\sigma^m(I) \supseteq J$ is a finite union of
complete doubly infinite arithmetic progressions in $m$, up to
the addition of a finite set.
Alternatively, this result states that for an affine scheme
$X$ of finite type over $K$,
an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$
any two closed subschemes of $X$, the set
of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above.
The paper presents examples
showing that this result may fail to hold if the affine scheme
$X$ is
not of finite type, or if $X$ is of finite type but the field
$K$ has positive characteristic.
Keywords:automorphisms, endomorphisms, affine space, commutative algebras, SkolemMahlerLech theorem Categories:11D45, 14R10, 11Y55, 11D88 

31. CJM 2013 (vol 66 pp. 1250)
 Feigin, Evgeny; Finkelberg, Michael; Littelmann, Peter

Symplectic Degenerate Flag Varieties
A simple finite dimensional module $V_\lambda$ of a simple complex
algebraic group $G$ is naturally endowed with a filtration induced by the PBWfiltration
of $U(\mathrm{Lie}\, G)$. The associated graded space $V_\lambda^a$ is a module
for the group $G^a$, which can be roughly described as a semidirect product of a
Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_a^M$. In analogy
to the flag variety $\mathcal{F}_\lambda=G.[v_\lambda]\subset \mathbb{P}(V_\lambda)$,
we call the closure
$\overline{G^a.[v_\lambda]}\subset \mathbb{P}(V_\lambda^a)$
of the $G^a$orbit through the highest weight line the degenerate flag variety $\mathcal{F}^a_\lambda$.
In general this is a
singular variety, but we conjecture that it has many nice properties similar to
that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group.
The symplectic case is important for the conjecture
because it is the first known case where even for fundamental weights $\omega$ the varieties
$\mathcal{F}^a_\omega$ differ from $\mathcal{F}_\omega$. We give an explicit
construction of the varieties $Sp\mathcal{F}^a_\lambda$ and construct desingularizations,
similar to the BottSamelson resolutions in the classical case. We prove that $Sp\mathcal{F}^a_\lambda$
are normal locally complete intersections with terminal and rational singularities.
We also show that these varieties are Frobenius split. Using the above mentioned results, we
prove an analogue of the BorelWeil theorem and obtain a $q$character formula
for the characters of irreducible $Sp_{2n}$modules via the AtiyahBottLefschetz fixed
points formula.
Keywords:Lie algebras, flag varieties, symplectic groups, representations Categories:14M15, 22E46 

32. CJM 2013 (vol 66 pp. 1305)
 Koskivirta, JeanStefan

Congruence Relations for Shimura Varieties Associated with $GU(n1,1)$
We prove the congruence relation for the mod$p$ reduction of Shimura
varieties associated to a unitary similitude group $GU(n1,1)$ over
$\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$
is even was obtained by T. Wedhorn and O. B?ltel, as a special case
of a result of B. Moonen, when the $\mu$ordinary locus of the $p$isogeny
space is dense. This condition fails in our case. We show that every
supersingular irreducible component of the special fiber of $p\textrm{}\mathscr{I}sog$
is annihilated by a degree one polynomial in the Frobenius element
$F$, which implies the congruence relation.
Keywords:Shimura varieties, congruence relation Categories:11G18, 14G35, 14K10 

33. CJM 2013 (vol 66 pp. 505)
 Arapura, Donu

Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes
Suppose that $Y$ is a cyclic cover of projective space branched over
a hyperplane arrangement $D$, and that $U$ is the complement of the
ramification locus in $Y$. The first theorem implies that the
BeilinsonHodge conjecture holds for $U$ if certain multiplicities of
$D$ are coprime to the degree of the cover. For instance this applies
when $D$ is reduced with normal crossings. The second theorem shows
that when $D$ has normal crossings and the degree of the cover is a
prime number, the generalized Hodge conjecture holds for any toroidal
resolution of $Y$. The last section contains some partial extensions
to more general nonabelian covers.
Keywords:Hodge cycles, hyperplane arrangements Category:14C30 

34. CJM 2013 (vol 66 pp. 1225)
35. CJM 2013 (vol 66 pp. 924)
 Stankewicz, James

Twists of Shimura Curves
Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an AtkinLehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of JordanLivnÃ© on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.
Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves Categories:11G18, 14G35, 11G15, 11G10 

36. CJM 2013 (vol 65 pp. 1125)
 Vandenbergen, Nicolas

On the Global Structure of Special Cycles on Unitary Shimura Varieties
In this paper, we study the reduced loci of special cycles on local
models of the Shimura variety for $\operatorname{GU}(1,n1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of BruhatTits
theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.
Keywords:Shimura varieties, local models, special cycles Category:14G35 

37. CJM 2013 (vol 66 pp. 1167)
 Rotger, Victor; de VeraPiquero, Carlos

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves
The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an AtkinLehner quotient over $\mathbb{Q}$.
Keywords:Shimura curves, rational points, Galois representations, Hasse principle, BrauerManin obstruction Categories:11G18, 14G35, 14G05 

38. CJM 2012 (vol 66 pp. 3)
 Abdesselam, Abdelmalek; Chipalkatti, Jaydeep

On Hilbert Covariants
Let $F$ denote a binary form of order $d$ over the
complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant
$\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an
order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give
defining equations for the image variety $X$ of an embedding $\mathbf{P}^r
\hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of
the Hilbert covariant; and simultaneously situate it into a wider class of
covariants called the GÃ¶ttingen covariants, all of which vanish on
$X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$
defines $X$ as a scheme. Finally, we exhibit a generalisation of the
GÃ¶ttingen covariants to $n$ary forms using the classical Clebsch transfer principle.
Keywords:binary forms, covariants, $SL_2$representations Categories:14L30, 13A50 

39. 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 

40. 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 

41. 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 

42. 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 

43. 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 

44. 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 

45. 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 

46. 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 

47. 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 

48. 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 

49. 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 

50. 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 
