scholarly journals Nearest points on toric varieties

2018 ◽  
Vol 122 (2) ◽  
pp. 213 ◽  
Author(s):  
Martin Helmer ◽  
Bernd Sturmfels
Keyword(s):  
Toric Variety ◽  
Euclidean Distance ◽  
Toric Varieties ◽  
Data Point ◽  
Nearest Points

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.

scholarly journals Codimension bounds for the Noether–Lefschetz components for toric varieties

2021 ◽  
Author(s):  
Ugo Bruzzo ◽  
William D. Montoya
Keyword(s):  
Irreducible Component ◽  
Toric Variety ◽  
Toric Varieties ◽  
Ample Divisor ◽  
Smooth Hypersurface

AbstractFor a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ i ∗ : H p ( P Σ ) → H p ( X ) induced by the inclusion is injective for $$p=\dim X$$ p = dim X and an isomorphism for $$p<\dim X-1$$ p < dim X - 1 . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ NL β as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ β such that $$i^*$$ i ∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ dim P Σ = 2 k + 1 and $$k\beta -\beta _0=n\eta $$ k β - β 0 = n η , $$n\in \mathbb {N}$$ n ∈ N , where $$\eta $$ η is the class of a 0-regular ample divisor, and $$\beta _0$$ β 0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ β satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ n + 1 ⩽ codim Z ⩽ h k - 1 , k + 1 ( X ) .

scholarly journals On the Hodge conjecture for quasi-smooth intersections in toric varieties

2021 ◽  
Author(s):  
Ugo Bruzzo ◽  
William Montoya
Keyword(s):  
Complete Intersection ◽  
Toric Variety ◽  
Toric Varieties ◽  
Picard Group ◽  
Hodge Conjecture ◽  
Smooth Hypersurface ◽  
Suitable Notion

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

scholarly journals Brick Manifolds and Toric Varieties of Brick Polytopes

10.37236/5038 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Moment Map ◽  
Twisted Product ◽  
General Fiber ◽  
Moment Polytope ◽  
The Moment ◽  
Subword Complex ◽  
Do So

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

scholarly journals Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties

2020 ◽  
Vol 71 (2) ◽  
pp. 395-438
Author(s):  
Jack Smith
Keyword(s):  
Toric Variety ◽  
Mirror Symmetry ◽  
Complex Structure ◽  
Toric Varieties ◽  
Quantum Cohomology ◽  
Closed String ◽  
Generalized Jacobian ◽  
Novikov Ring

Abstract We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety $X$, by showing directly that the Kodaira–Spencer map of Fukaya–Oh–Ohta–Ono defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, $X$ need not be compact. The proof is based on the purely algebraic fact that a class of generalized Jacobian rings associated to $X$ are free as modules over the Novikov ring. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

scholarly journals Cycle-Level Intersection Theory for Toric Varieties

2004 ◽  
Vol 56 (5) ◽  
pp. 1094-1120
Author(s):  
Hugh Thomas
Keyword(s):  
Toric Variety ◽  
Linear Subspace ◽  
Toric Varieties ◽  
Cartier Divisor ◽  
Ring Structure ◽  
Intersection Theory ◽  
Inner Product ◽  
Characteristic Classes ◽  
Level Intersection ◽  
Complete Flag

AbstractThis 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 inN, the choice consists of giving an inner product or a complete flag forMQ= ℚ ⊗ Hom(N, ℤ), or more generally giving for each cone σ in the fan a linear subspace ofMQcomplementary to σ⊥, satisfying certain compatibility conditions. We show that these intersection cycles have properties analogous to the usual intersections modulo rational equivalence. IfXis simplicial (for instance, ifXis non-singular), we obtain a commutative ring structure to the invariant cycles ofXwith 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.

scholarly journals Polar degrees and closest points in codimension two

2019 ◽  
Vol 18 (05) ◽  
pp. 1950095
Author(s):  
Martin Helmer ◽  
Bernt Ivar Utstøl Nødland
Keyword(s):  
Toric Variety ◽  
Euclidean Distance ◽  
Integer Matrix ◽  
Codimension Two ◽  
Geometric Methods ◽  
The Matrix ◽  
Matrix Formula ◽  
Combinatorial Methods

Suppose that [Formula: see text] is a toric variety of codimension two defined by an [Formula: see text] integer matrix [Formula: see text], and let [Formula: see text] be a Gale dual of [Formula: see text]. In this paper, we compute the Euclidean distance degree and polar degrees of [Formula: see text] (along with other associated invariants) combinatorially working from the matrix [Formula: see text]. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix [Formula: see text] in the codimension two case.

scholarly journals A-hypergeometric systems and relative cohomology

2020 ◽  
Vol 31 (13) ◽  
pp. 2050113
Author(s):  
Tsung-Ju Lee ◽  
Dingxin Zhang
Keyword(s):  
Toric Variety ◽  
Cohomology Group ◽  
Toric Varieties ◽  
Solution Space ◽  
Complete Intersections ◽  
Relative Cohomology ◽  
Rank One

We investigate the solution space to certain [Formula: see text]-hypergeometric [Formula: see text]-modules, which were defined and studied by Gelfand, Kapranov, and Zelevinsky. We show that the solution space can be identified with certain relative cohomology group of the toric variety determined by [Formula: see text], which generalizes the results of Huang, Lian, Yau and Zhu. As a corollary, we also prove the existence of rank one points for complete intersections in toric varieties.

On the integral cohomology of toric varieties

2015 ◽  
Vol 15 (02) ◽  
pp. 1650032
Author(s):  
Jin Hong Kim
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Cohomology Algebra ◽  
Linear Relations ◽  
Integral Cohomology ◽  
The Ideal

It is known that the integral cohomology algebra of any smooth compact toric variety XΣ associated to a complete regular fan Σ is isomorphic to the Stanley–Reisner algebra ℤ[Σ] modulo the ideal JΣ generated by linear relations determined by Σ. The aim of this paper is to show how to determine the integral cohomology algebra of a toric variety (in particular, a projective toric variety) associated to a certain simplicial fan. As a consequence, we confirm our expectation that for a certain simplicial fan the integral cohomology algebra is also given by the same formula as in a complete regular fan.

scholarly journals GENERALIZED SPECIAL LAGRANGIAN TORUS FIBRATION FOR CALABI–YAU HYPERSURFACES IN TORIC VARIETIES I

2007 ◽  
Vol 09 (02) ◽  
pp. 201-216 ◽  
Author(s):  
WEI-DONG RUAN
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Dimensional Torus ◽  
Lagrangian Torus ◽  
Special Lagrangian ◽  
Large Complex

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi–Yau hypersurfaces in toric varieties near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi–Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit X0. In this paper, we will deal with the region near the smooth top dimensional torus fibers of X0 and its mirror dual situation — the region near the 0-dimensional fibers of X0.

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