Projective Toric Varieties Research Articles

Self-dual projective toric varieties and their ideals

Self-dual projective toric varieties and their ideals

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

A short resolution of the diagonal for smooth projective toric varieties of Picard rank 2

Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties

This paper explores the possibility of constructing multivariate generating functions for all cohomology dimensions of all holomorphic line bundles on certain complex projective varieties of Fano, Calabi-Yau and general type in various dimensions and Picard numbers. Most of the results are conjectural and rely on explicit cohomology computations. We first propose a generating function for the Euler characteristic of all holomorphic line bundles on complete intersections in products of projective spaces and toric varieties. This generating function is constructed by expanding the Hilbert-Poincaré series associated with the coordinate ring of the variety around all possible combinations of zero and infinity and then summing up the resulting contributions with alternating signs. Similar generating functions are proposed for the individual cohomology dimensions of all holomorphic line bundles on certain complete intersections, including examples of Mori and non-Mori dream spaces. Surprisingly, the examples studied indicate that a single generating function encodes both the zeroth and all higher cohomologies upon expansion around different combinations of zero and infinity, raising the question whether such generating functions determine the variety uniquely.

Toric Sylvester forms

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety X with respect to the irrelevant ideal of X. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on X. In particular, we prove that toric Sylvester forms yield bases of some graded components of Isat/I, where I denotes an ideal generated by n+1 generic forms, n is the dimension of X and Isat is the saturation of I with respect to the irrelevant ideal of the Cox ring of X. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explore the computation of the toric residue of the product of two forms.

Computations of Gromov–Witten invariants of toric varieties

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah–Bott formula on the moduli space of genus 0 stable maps M‾0,m(X,β) where X is any smooth projective toric variety, and β is any effective 1-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety X, as well as the cohomology class β, must be defined using the package Oscar.jl.

Genus 0 logarithmic and tropical fixed‐domain counts for Hirzebruch surfaces

AbstractFor a non‐singular projective toric variety , the virtual logarithmic Tevelev degrees are defined as the virtual degree of the morphism from the moduli stack of logarithmic stable maps to the product . In this paper, after proving that Mikhalkin's correspondence theorem holds in genus 0 for logarithmic virtual Tevelev degrees, we use tropical methods to provide closed formulas for the case in which is a Hirzebruch surface. In order to do so, we explicitly list all the tropical curves contributing to the count.

Toric differential forms and periods of complete intersections

Let n be an even natural number. We compute the periods of any n2-dimensional complete intersection algebraic cycle inside an n-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting.

Deformation of pairs and Noether–Lefschetz loci in toric varieties

We continue our study of the Noether–Lefschetz loci in toric varieties and investigate deformation of pairs (V, X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a simplicial projective toric variety PΣ2k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}_{\\Sigma }^{2k+1}$$\\end{document}, with V⊂X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V\\subset X$$\\end{document}. The hypersurface X is supposed to be of Macaulay type, which means that its toric Jacobian ideal is Cox–Gorenstein, a property that generalizes the notion of Gorenstein ideal in the standard polynomial ring. Under some assumptions, we prove that the class λV∈Hk,k(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _V\\in H^{k,k}(X)$$\\end{document} deforms to an algebraic class if and only if it remains of type (k, k). Actually we prove that locally the Noether–Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work (Bruzzo and Montoya 15(2):682–694, 2021) and the main theorem proved by Dan (in: Analytic and Algebraic Geometry. Hindustan Book Agency, New Delhi, pp 107–115, 2017).

Cohomology of a real toric variety and shellability of posets arising from a graph

AbstractGiven a graph G without loops, the pseudograph associahedron PG is a smooth polytope, so there is a projective smooth toric variety XG corresponding to PG. Taking the real locus of XG, we have the projective smooth real toric variety $X^{\mathbb{R}}_G$. The integral cohomology groups of $X^{\mathbb{R}}_G$ can be computed by studying the topology of certain posets of even subgraphs of G; such a poset is neither pure nor shellable in general. We completely characterize the graphs whose posets of even subgraphs are always shellable. It follows that we get a family of projective smooth real toric varieties whose integral cohomology groups are torsion-free or have only 2-torsion.

Orbifold Kähler–Einstein metrics on projective toric varieties

AbstractIn this short note, we investigate the existence of orbifold Kähler–Einstein metrics on toric varieties. In particular, we show that every ‐factorial normal projective toric variety allows an orbifold Kähler–Einstein metric. Moreover, we characterize ‐stability of ‐factorial toric pairs of Picard number one in terms of the log Cox ring and the universal orbifold cover.

Simplicial generation of Chow rings of matroids

We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincaré duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge–Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of the work of Adiprasito et al. (2018). Our work avoids the use of “flips,” the key technical tool employed in that work.

Newton–Okounkov polytopes of flag varieties and marked chain-order polytopes

Marked chain-order polytopes are convex polytopes constructed from a marked poset. They give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand–Tsetlin poset of type A A , and realize the associated marked chain-order polytopes as Newton–Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand–Tsetlin polytopes and Feigin–Fourier–Littelmann–Vinberg polytopes as Newton–Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation. The basis is naturally parametrized by the set of lattice points in a marked chain-order polytope.

Projective toric varieties of codimension 2 with maximal Castelnuovo–Mumford regularity

The Eisenbud–Goto conjecture states that reg X ≤ deg X − codim X + 1 for a nondegenerate irreducible projective variety X over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when X is a projective toric variety of codimension 2. We classify the projective toric varieties of codimension 2 having maximal regularity, that is, for which equality holds in the Eisenbud–Goto bound. We also give combinatorial characterizations of the arithmetically Cohen–Macaulay toric varieties of maximal regularity in characteristic 0.

Curve classes on Calabi–Yau complete intersections in toric varieties

AbstractWe prove the integral Hodge conjecture for curve classes on smooth varieties of dimension at least three constructed as a complete intersection of ample hypersurfaces in a smooth projective toric variety, such that the anticanonical divisor is the restriction of a nef divisor. In particular, this includes the case of smooth anticanonical hypersurfaces in toric Fano varieties. In fact, using results of Casagrande and the toric minimal model program, we prove that in each case, is generated by classes of rational curves.

Euler-symmetric projective toric varieties and additive actions

Let Ga be the additive group of the field of complex numbers ℂ. We say that an irreducible algebraic variety X of dimension n admits an additive action if there is a regular action of the group Gan=Ga×…×Ga (n times) on X with an open orbit. In 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification of Euler-symmetric varieties and proved that any Euler-symmetric variety admits an additive action. In this paper we show that in the case of projective toric varieties the converse is also true. More precisely, a projective toric variety admitting an additive action is an Euler-symmetric variety with respect to any linearly normal embedding into a projective space. Also we discuss some properties of Euler-symmetric projective toric varieties.

Non-exactness of toric Poisson structures

We prove that a Poisson structure on a projective toric variety which is invariant by the torus action and whose symplectic leaves are the torus orbits is not exact. This is deduced from a geometric criterion for non-exactness of Poisson structures with a finite number of symplectic leaves.

Irrational Toric Varieties and Secondary Polytopes

The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration \({{\mathcal {A}}}\) of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of \({{\mathcal {A}}}\). For this, we develop a theory of irrational toric varieties associated to arbitrary fans. When the fan is rational, the irrational toric variety is the nonnegative part of the corresponding classical toric variety. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope.

Generation of jets and Fujita's jet ampleness conjecture on toric varieties

Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors , but also their higher order analogues called jets. We give sharp bounds guaranteeing that a line bundle on a projective toric variety is k -jet ample in terms of its intersection numbers with the invariant curves, in terms of the lattice lengths of the edges of its polytope , in terms of the higher concavity of its piecewise linear function and in terms of its Seshadri constant. For example, the tensor power k + n − 2 of an ample line bundle on a projective toric variety of dimension n ≥ 2 always generates all k -jets, but might not generate all ( k + 1 ) -jets. As an application, we prove the k -jet generalizations of Fujita's conjectures on toric varieties with arbitrary singularities.

Aspects of functoriality in homological mirror symmetry for toric varieties

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.

β-Density function on the class group of projective toric varieties

We prove the existence of a compactly supported, continuous (except at finitely many points) function $g_{I, {\bf m}}: [0, \infty)\longrightarrow \mathbb{R}$ for all monomial prime ideals $I$ of $R$ of height one where $(R, {\bf m})$ is the homogeneous coordinate ring associated to a projectively normal toric pair $(X, D)$, such that $$\int_{0}^{\infty}g_{I, {\bf m}}(\lambda)d\lambda=\beta(I, {\bf m}),$$ where $\beta(I, {\bf m})$ is the second coefficient of the Hilbert-Kunz function of $I$ with respect to the maximal ideal ${\bf m}$, as proved by Huneke-McDermott-Monsky \cite{HMM2004}. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map $\tau_{{\bf m}}:\text{Cl}(R)\longrightarrow \mathbb{R}$ introduced in \cite{HMM2004} to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function $g_{I, {\bf m}}$ is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert polynomial of the rings and the ideals. \keywords{coefficients of Hilbert-Kunz function\and projective toric variety\and Hilbert-Kunz density function\and $\beta$-density function\and monomial prime ideal of height one.}