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Search: MSC category 14M25 ( Toric varieties, Newton polyhedra [See also 52B20] )

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1. 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 systemCategories:14M25, 14T05, 14D20

2. CJM 2016 (vol 69 pp. 767)

Choi, Suyoung; Park, Hanchul
 Wedge Operations and Torus Symmetries II A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) 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 star-shaped simplicial sphere $K$ of dimension $n-1$ with $m$ vertices, the Picard number $\operatorname{Pic}(K)$ of $K$ is $m-n$. 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 mapCategories:57S25, 14M25, 52B11, 13F55, 18A10

3. CJM 2016 (vol 68 pp. 784)

Doran, Charles F.; Harder, Andrew
 Toric Degenerations and Laurent Polynomials Related to Givental's Landau-Ginzburg 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 Landau-Ginzburg 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 Landau-Ginzburg 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 Calabi-Yau varieties. Keywords:Fano varieties, Landau-Ginzburg models, Calabi-Yau varieties, toric varietiesCategories: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 group-theoretical 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 resolutionCategories: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 3-valent fan tropical curves in a fan tropical plane. Secondly, we prove that a generic non-singular tropical surface in tropical projective 3-space 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 theoryCategories:14T05, 14M25

6. CJM 2012 (vol 65 pp. 634)

Mezzetti, Emilia; Miró-Roig, 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 $(d-1)$-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 threefoldCategories: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 polytopesCategories:14J30, 14J30, 14M25, 52B20

8. CJM 2004 (vol 56 pp. 1094)

Thomas, Hugh
 Cycle-Level Intersection Theory for Toric Varieties This paper addresses the problem of constructing a cycle-level 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 non-singular), 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 theoryCategories: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 well-known 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 quasi-ordinaires 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 quasi-ordinaires. Cela fait intervenir d'une part une g\'en\'eralization de l'algorithme de Newton-Puiseux, 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 polytope-fibre 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
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