26. CJM 2016 (vol 69 pp. 338)
 Garbagnati, Alice

On K3 Surface Quotients of K3 or Abelian Surfaces
The aim of this paper is to prove that a K3 surface is the minimal
model of the quotient of an Abelian surface by a group $G$ (respectively
of a K3 surface by an Abelian group $G$) if and only if a certain
lattice is primitively embedded in its NÃ©ronSeveri group.
This allows one to describe the coarse moduli space of the K3
surfaces which are (rationally) $G$covered by Abelian or K3
surfaces (in the latter case $G$ is an Abelian group).
If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian
surface, this result was already known, so we extend it to the
other cases.
Moreover, we prove that a K3 surface $X_G$ is the minimal model
of the quotient of an Abelian surface by a group $G$ if and only
if a certain configuration of rational curves is present on $X_G$.
Again this result was known only in some special cases, in particular
if $G$ has order 2 or 3.
Keywords:K3 surfaces, Kummer surfaces, Kummer type lattice, quotient surfaces Categories:14J28, 14J50, 14J10 

27. CJM 2016 (vol 68 pp. 541)
 GarciaArmas, Mario

Strongly Incompressible Curves
Let $G$ be a finite group. A faithful $G$variety $X$ is called
strongly incompressible if every dominant $G$equivariant rational
map of $X$ onto another faithful $G$variety $Y$ is birational.
We settle the problem of existence of strongly incompressible
$G$curves for any finite group $G$ and any base field $k$ of
characteristic zero.
Keywords:algebraic curves, group actions, Galois cohomology Categories:14L30, 14E07, 14H37 

28. CJM 2016 (vol 68 pp. 504)
29. CJM 2016 (vol 68 pp. 361)
 Fité, Francesc; González, Josep; Lario, Joan Carles

Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent
Let $\mathcal{C}$ denote the Fermat curve over $\mathbb{Q}$ of prime
exponent $\ell$. The Jacobian $\operatorname{Jac}(\mathcal{C})$
of~$\mathcal{C}$ splits over $\mathbb{Q}$ as the product of Jacobians
$\operatorname{Jac}(\mathcal{C}_k)$, $1\leq k\leq \ell2$, where
$\mathcal{C}_k$ are curves obtained as quotients of $\mathcal{C}$ by
certain subgroups of automorphisms of $\mathcal{C}$. It is well known
that $\operatorname{Jac}(\mathcal{C}_k)$ is the power of an absolutely
simple abelian variety $B_k$ with complex multiplication. We call
degenerate those pairs $(\ell,k)$ for which $B_k$ has degenerate CM
type. For a nondegenerate pair $(\ell,k)$, we compute the SatoTate
group of $\operatorname{Jac}(\mathcal{C}_k)$, prove the generalized
SatoTate Conjecture for it, and give an explicit method to compute
the moments and measures of the involved distributions. Regardless of
$(\ell,k)$ being degenerate or not, we also obtain Frobenius
equidistribution results for primes of certain residue degrees in the
$\ell$th cyclotomic field. Key to our results is a detailed study of
the rank of certain generalized Demjanenko matrices.
Keywords:SatoTate group, Fermat curve, Frobenius distribution Categories:11D41, 11M50, 11G10, 14G10 

30. CJM 2016 (vol 68 pp. 280)
 da Silva, Genival; Kerr, Matt; Pearlstein, Gregory

Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution
We collect evidence in support of a conjecture of Griffiths,
Green
and Kerr
on the arithmetic of extension classes of
limiting
mixed Hodge structures arising from semistable degenerations
over
a number field. After briefly summarizing how a result of Iritani
implies this conjecture for a collection of hypergeometric
CalabiYau threefold examples studied by Doran and Morgan,
the authors investigate a sequence of (nonhypergeometric) examples
in dimensions $1\leq d\leq6$ arising from Katz's theory of the
middle
convolution.
A crucial role is played by the MumfordTate
group (which is $G_{2}$) of the family of 6folds, and the theory
of boundary components of MumfordTate domains.
Keywords:variation of Hodge structure, limiting mixed Hodge structure, CalabiYau variety, middle convolution, MumfordTate group Categories:14D07, 14M17, 17B45, 20G99, 32M10, 32G20 

31. CJM 2015 (vol 68 pp. 241)
 Allermann, Lars; Hampe, Simon; Rau, Johannes

On Rational Equivalence in Tropical Geometry
This article discusses the concept of rational equivalence
in tropical
geometry
(and replaces an older and imperfect version).
We give the basic definitions in the context of tropical varieties
without boundary points and prove some basic properties.
We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing
that they are isomorphic
to the group of fan cycles. The main step in the proof is of
independent interest:
We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated)
fan cycles. This also
proves that the intersection ring of tropical cycles is generated
in codimension 1 (by hypersurfaces).
Keywords:tropical geometry, rational equivalence Category:14T05 

32. CJM 2015 (vol 68 pp. 67)
33. CJM 2015 (vol 68 pp. 334)
 Demchenko, Oleg; Gurevich, Alexander

Kernels in the Category of Formal Group Laws
Fontaine described the category of formal groups over the ring
of Witt vectors over a finite field
of characteristic $p$ with the aid of triples consisting of the
module of logarithms,
the DieudonnÃ© module and the morphism from the former to the
latter. We propose
an explicit construction for the kernels in this category in
term of Fontaine's triples.
The construction is applied to the formal norm homomorphism in
the case of an unramified extension
of $\mathbb{Q}_p$ and of a totally ramified extension of degree less
or equal than $p$. A similar
consideration applied to a global extension allows us to establish
the existence of a strict
isomorphism between the formal norm torus and a formal group
law coming from $L$series.
Keywords:formal groups, $p$divisible groups, Dieudonne modules, norm tori Category:14L05 

34. CJM 2015 (vol 67 pp. 961)
 Abuaf, Roland; Boralevi, Ada

Orthogonal Bundles and SkewHamiltonian Matrices
Using properties of skewHamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r2)n\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n1$ and $n\ge 8$. We speculate that the result holds in
greater generality.
Keywords:orthogonal vector bundles, moduli spaces, skewHamiltonian matrices Categories:14J60, 15B99 

35. CJM 2015 (vol 67 pp. 1201)
 Aluffi, Paolo; Faber, Eleonore

Chern Classes of Splayed Intersections
We generalize the Chern class relation for the transversal intersection
of two nonsingular
varieties to a relation for possibly singular varieties, under
a splayedness assumption.
We show that the relation for the ChernSchwartzMacPherson classes
holds for two splayed hypersurfaces in a nonsingular variety,
and under a `strong splayedness' assumption for more
general subschemes. Moreover, the relation is shown to hold for
the ChernFulton classes
of any two splayed subschemes.
The main tool is a formula for Segre classes of splayed
subschemes. We also discuss the Chern class relation under the
assumption that one of the
varieties is a general very ample divisor.
Keywords:splayed intersection, ChernSchwartzMacPherson class, ChernFulton class, splayed blowup, Segre class Categories:14C17, 14J17 

36. CJM 2015 (vol 67 pp. 1109)
 Nohara, Yuichi; Ueda, Kazushi

Goldman Systems and Bending Systems
We show that the moduli space
of parabolic bundles on the projective line
and the polygon space are isomorphic,
both as complex manifolds
and symplectic manifolds equipped with structures of completely integrable systems,
if the stability parameters are
small.
Keywords:toric degeneration Categories:53D30, 14H60 

37. CJM 2015 (vol 68 pp. 24)
 Bonfanti, Matteo Alfonso; van Geemen, Bert

Abelian Surfaces with an Automorphism and Quaternionic Multiplication
We construct one dimensional families of Abelian surfaces with
quaternionic multiplication
which also have an automorphism of order three or four. Using Barth's
description of the moduli space of $(2,4)$polarized Abelian surfaces,
we find the Shimura curve parametrizing these Abelian surfaces in a
specific case.
We explicitly relate these surfaces to the Jacobians of genus two
curves studied by Hashimoto and Murabayashi.
We also describe a (Humbert) surface in Barth's moduli space which
parametrizes Abelian surfaces with real multiplication by
$\mathbf{Z}[\sqrt{2}]$.
Keywords:abelian surfaces, moduli, shimura curves Categories:14K10, 11G10, 14K20 

38. CJM 2015 (vol 67 pp. 696)
 Zhang, Tong

Geography of Irregular Gorenstein 3folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3folds of general type. We generalize the classical NoetherCastelnuovo type inequalities for irregular surfaces to irregular 3folds according to the Albanese dimension.
Keywords:3fold, geography, irregular variety Category:14J30 

39. CJM 2014 (vol 67 pp. 923)
 Pan, Ivan Edgardo; Simis, Aron

Cremona Maps of de JonquiÃ¨res Type
This paper is concerned with suitable generalizations of a plane de
JonquiÃ¨res map to higher dimensional space
$\mathbb{P}^n$ with $n\geq 3$.
For each given point of $\mathbb{P}^n$ there is a subgroup of the entire
Cremona group of dimension $n$
consisting of such maps.
One studies both geometric and grouptheoretical properties of this notion.
In the case where $n=3$ one describes an explicit set of generators of
the group and gives a homological characterization
of a basic subgroup thereof.
Keywords:Cremona map, de JonquiÃ¨res map, Cremona group, minimal free resolution Categories:14E05, 13D02, 13H10, 14E07, 14M05, 14M25 

40. CJM 2014 (vol 67 pp. 527)
 Brugallé, Erwan; Shaw, Kristin

Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory
We provide some new local obstructions to
approximating
tropical curves in
smooth tropical surfaces. These obstructions are based on
a
relation between tropical and complex intersection theories which is
also established here. We give
two applications of the methods developed in this paper.
First we classify all locally irreducible approximable 3valent fan tropical
curves in a
fan tropical plane.
Secondly, we prove that a generic nonsingular
tropical surface
in tropical projective 3space contains finitely
many approximable tropical lines
if
it is of degree 3, and contains no approximable tropical lines if
it is of degree 4 or more.
Keywords:tropical geometry, amoebas, approximation of tropical varieties, intersection theory Categories:14T05, 14M25 

41. CJM 2014 (vol 67 pp. 639)
 Gonzalez, Jose Luis; Karu, Kalle

Projectivity in Algebraic Cobordism
The algebraic cobordism group of a scheme is generated by cycles that
are proper morphisms from smooth quasiprojective varieties. We prove
that over a field of characteristic zero the quasiprojectivity
assumption can be omitted to get the same theory.
Keywords:algebraic cobordism, quasiprojectivity, cobordism cycles Categories:14C17, 14F43, 55N22 

42. CJM 2014 (vol 67 pp. 893)
43. CJM 2014 (vol 67 pp. 1219)
 Balwe, Chetan

$p$adic and Motivic Measure on Artin $n$stacks
We define a notion of $p$adic measure on Artin $n$stacks which are
of strongly finite type over the ring of $p$adic integers. $p$adic
measure on schemes can be evaluated by counting points on the
reduction of the scheme modulo $p^n$. We show that an analogous
construction works in the case of Artin stacks as well if we count the
points using the counting measure defined by ToÃ«n. As a consequence,
we obtain the result that the PoincarÃ© and Serre series of such
stacks are rational functions, thus extending Denef's result for
varieties. Finally, using motivic integration we show that as $p$
varies, the rationality of the Serre series of an Artin stack defined
over the integers is uniform with respect to $p$.
Keywords:padic integration, motivic integration, Artin stacks Categories:14E18, 14A20 

44. CJM 2014 (vol 67 pp. 848)
 Köck, Bernhard; Tait, Joseph

Faithfulness of Actions on RiemannRoch Spaces
Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the RiemannRoch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$invariant divisor on $X$ of
degree at least~$2g_X2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
Keywords:faithful action, RiemannRoch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology Categories:14H30, 30F30, 14L30, 14D15, 11R32 

45. 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 

46. 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 

47. 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 

48. 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 

49. 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 

50. 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 
