1. CJM Online first
 Choi, Suyoung; Park, Hanchul

Wedge operations and torus symmetries II
A fundamental idea in toric topology is that classes of manifolds
with wellbehaved torus actions (simply, toric spaces) are classified
by pairs of simplicial complexes and (nonsingular) characteristic
maps. The authors in their previous paper provided a new way
to find all characteristic maps on a simplicial complex $K(J)$
obtainable by a sequence of wedgings from $K$. The main idea
was that characteristic maps on $K$ theoretically determine all
possible characteristic maps on a wedge of $K$.
In this work, we further develop our previous work for classification
of toric spaces. For a starshaped simplicial sphere $K$ of dimension
$n1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is
$mn$. We refer to $K$ as a seed if $K$ cannot be obtained
by wedgings. First, we show that, for a fixed positive integer
$\ell$, there are at most finitely many seeds of Picard number
$\ell$ supporting characteristic maps. As a corollary, the conjecture
proposed by V.V. Batyrev in 1991 is solved affirmatively.
Second, we investigate a systematic method to find all characteristic
maps on $K(J)$ using combinatorial objects called (realizable)
puzzles that only depend on a seed $K$.
These two facts lead to a practical way to classify the toric
spaces of fixed Picard number.
Keywords:puzzle, toric variety, simplicial wedge, characteristic map Categories:57S25, 14M25, 52B11, 13F55, 18A10 

2. CJM Online first
 Manon, Christopher

Toric geometry of $SL_2(\mathbb{C})$ free group character varieties from outer space
Culler and Vogtmann defined a simplicial space $O(g)$ called
outer space to study the outer automorphism group
of the free group $F_g$. Using representation theoretic methods,
we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g,
SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety
of $F_g,$ reproving a result of Morgan and Shalen. Then we show
that every point $v$ contained in a maximal cell of $O(g)$ defines
a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to
a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there
is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g,
SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a
symplectomorphism on a dense, open subset of $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ with respect to natural symplectic structures
on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this
way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g,
SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$,
and we show that each $v$ defines a topological decomposition
of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition
of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that
the valuations coming from the closure of a maximal cell in $O(g)$
all arise as divisorial valuations built from an associated projective
compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$
Keywords:character variety, outer space, analytification, compactification, integrable system Categories:14M25, 14T05, 14D20 

3. CJM 2016 (vol 68 pp. 784)
 Doran, Charles F.; Harder, Andrew

Toric Degenerations and Laurent Polynomials Related to Givental's LandauGinzburg Models
For an appropriate class of Fano complete intersections in toric
varieties, we prove that there is a concrete relationship between
degenerations to specific toric subvarieties and expressions
for Givental's LandauGinzburg models as Laurent polynomials.
As a result, we show that Fano varieties presented as complete
intersections in partial flag manifolds admit degenerations to
Gorenstein toric weak Fano varieties, and their Givental LandauGinzburg
models can be expressed as corresponding Laurent polynomials.
We also use this to show that all of the Laurent polynomials
obtained by Coates, Kasprzyk and Prince by the so called Przyjalkowski
method correspond to toric degenerations of the corresponding
Fano variety. We discuss applications to geometric transitions
of CalabiYau varieties.
Keywords:Fano varieties, LandauGinzburg models, CalabiYau varieties, toric varieties Categories:14M25, 14J32, 14J33, 14J45 

4. 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 

5. 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 

6. 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 

7. 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 

8. CJM 2004 (vol 56 pp. 1094)
 Thomas, Hugh

CycleLevel Intersection Theory for Toric Varieties
This paper addresses the problem of constructing a
cyclelevel intersection theory for toric varieties.
We show that by making one global choice,
we can determine a cycle representative
for the intersection of an equivariant Cartier divisor with an invariant
cycle on a toric variety. For a toric variety
defined by a fan in $N$, the choice consists of giving an
inner product or a complete flag for $M_\Q=
\Qt \Hom(N,\mathbb{Z})$, or more
generally giving for each cone $\s$ in the fan a linear subspace of
$M_\Q$ complementary to $\s^\perp$, satisfying certain compatibility
conditions.
We show that these intersection cycles have properties analogous to the
usual intersections modulo rational equivalence.
If $X$ is simplicial (for instance, if $X$ is nonsingular),
we obtain a commutative ring structure
to the invariant cycles of $X$ with rational
coefficients. This ring structure determines cycles representing
certain characteristic classes of the toric variety.
We also discuss
how to define intersection cycles that require no choices,
at the expense of increasing
the size of the coefficient field.
Keywords:toric varieties, intersection theory Categories:14M25, 14C17 

9. CJM 2002 (vol 54 pp. 554)
 Hausen, Jürgen

Equivariant Embeddings into Smooth Toric Varieties
We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a wellknown embedding theorem
of Sumihiro on quasiprojective $G$varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$variety to
a more general framework.
Categories:14E25, 14C20, 14L30, 14M25 

10. CJM 2000 (vol 52 pp. 348)
 González Pérez, P. D.

SingularitÃ©s quasiordinaires toriques et polyÃ¨dre de Newton du discriminant
Nous \'etudions les polyn\^omes $F \in \C \{S_\tau\} [Y] $ \`a
coefficients dans l'anneau de germes de fonctions holomorphes au
point sp\'ecial d'une vari\'et\'e torique affine. Nous
g\'en\'eralisons \`a ce cas la param\'etrisation classique des
singularit\'es quasiordinaires. Cela fait intervenir d'une part
une g\'en\'eralization de l'algorithme de NewtonPuiseux, et
d'autre part une relation entre le poly\`edre de Newton du
discriminant de $F$ par rapport \`a $Y$ et celui de $F$ au moyen du
polytopefibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous
permet enfin de calculer, sous des hypoth\`eses de non
d\'eg\'en\'erescence, les sommets du poly\`edre de Newton du
discriminant a partir de celui de $F$, et les coefficients
correspondants \`a partir des coefficients des exposants de $F$ qui
sont dans les ar\^etes de son poly\`edre de Newton.
Categories:14M25, 32S25 
