Projective Toric Varieties Research Articles

On Mori cone of Bott towers

A Bott tower of height r is a sequence of projective bundlesXr⟶πrXr−1⟶πr−1⋯⟶π2X1=P1⟶π1X0={pt}, where Xi=P(OXi−1⊕Li−1) for a line bundle Li−1 over Xi−1 for all 1≤i≤r and P(−) denotes the projectivization. These are smooth projective toric varieties and we refer to the top object Xr also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.

Nearest points on toric varieties

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.

Positive geometries and canonical forms

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects — the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra — which have been loosely referred to as “positive geometries”. The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of “positive geometries” and their associated “canonical forms” as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones — via “triangulation” on the one hand, and “push-forward” maps between geometries on the other. We present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties, both for the simplest “simplex-like” geometries and the richer “polytope-like” ones. We also illustrate a number of strategies for computing canonical forms for large classes of positive geometries, ranging from a direct determination exploiting knowledge of zeros and poles, to the use of the general triangulation and push-forward methods, to the representation of the form as volume integrals over dual geometries and contour integrals over auxiliary spaces. These methods yield interesting representations for the canonical forms of wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

Gauss maps of toric varieties

We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

On the n-th row of the graded Betti table of an n-dimensional toric variety

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of projective space where we prove a special case of a conjecture of Ein and Lazarsfeld. We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.

Toric chordality

We study the geometric change of Chow cohomology classes in projective toric varieties under the Weil–McMullen dual of the intersection product with a Lefschetz element. Based on this, we introduce toric chordality, a generalization of graph chordality to higher skeleta of simplicial complexes with a coordinatization over characteristic 0, leading us to a far-reaching generalization of Kalai's work on applications of rigidity of frameworks to polytope theory. In contrast to “homological” chordality, the notion that is usually studied as a higher-dimensional analogue of graph chordality, we will show that toric chordality has several advantageous properties and applications.∘Most strikingly, we will see that toric chordality allows us to introduce a higher version of Dirac's propagation principle.∘Aside from the propagation theorem, we also study the interplay with the geometric properties of the simplicial chain complex of the underlying simplicial complex, culminating in a quantified version of the Stanley–Murai–Nevo generalized lower bound theorem.∘Finally, we apply our technique to give a simple proof of the generalized lower bound theorem in polytope theory and∘prove the balanced generalized lower bound conjecture of Klee and Novik.

Quantum 𝒟-modules for toric nef complete intersections

Let [Formula: see text] be a smooth projective toric variety with [Formula: see text] ample line bundles. Let [Formula: see text] be the zero locus of [Formula: see text] generic sections. It is well known that the ambient quantum [Formula: see text]-module of [Formula: see text] is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of [Formula: see text] and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum [Formula: see text]-module.

SIMPLICIAL WEDGE COMPLEXES AND PROJECTIVE TORIC VARIETIES

Let K be a fan-like simplicial sphere of dimension n-1 such that its associated complete fan is strongly polytopal, and let v be a vertex of K. Let K(v) be the simplicial wedge complex obtained by applying the simplicial wedge operation to K at v, and let <TEX>$v_0$</TEX> and <TEX>$v_1$</TEX> denote two newly created vertices of K(v). In this paper, we show that there are infinitely many strongly polytopal fans <TEX>${\Sigma}$</TEX> over such K(v)'s, different from the canonical extensions, whose projected fans <TEX>${Proj_v}_i{\Sigma}$</TEX> (i = 0, 1) are also strongly polytopal. As a consequence, it can be also shown that there are infinitely many projective toric varieties over such K(v)'s such that toric varieties over the underlying projected complexes <TEX>$K_{{Proj_v}_i{\Sigma}}$</TEX> (i = 0, 1) are also projective.

Computing characteristic classes of subschemes of smooth toric varieties

Let XΣ be a smooth complete toric variety defined by a fan Σ and let V=V(I) be a subscheme of XΣ defined by an ideal I homogeneous with respect to the grading on the total coordinate ring of XΣ. We show a new expression for the Segre class s(V,XΣ) in terms of the projective degrees of a rational map specified by the generators of I when each generator corresponds to a numerically effective (nef) divisor. Restricting to the case where XΣ is a smooth projective toric variety and dehomogenizing the total homogeneous coordinate ring of XΣ via a dehomogenizing ideal we also give an expression for the projective degrees of this rational map in terms of the dimension of an explicit quotient ring. Under an additional technical assumption we construct what we call a general dehomogenizing ideal and apply this construction to give effective algorithms to compute the Segre class s(V,XΣ), the Chern–Schwartz–MacPherson class cSM(V) and the topological Euler characteristic χ(V) of V. These algorithms can, in particular, be used for subschemes of any product of projective spaces Pn1×⋯×Pnj or for subschemes of many other projective toric varieties. Running time bounds for several of the algorithms are given and the algorithms are tested on a variety of examples. In all applicable cases our algorithms to compute these characteristic classes are found to offer significantly increased performance over other known algorithms.

On Fano Schemes of Toric Varieties

Given a finite set of lattice points $\mathcal{A}$, we consider the associated homogeneous binomial ideal $I_\mathcal{A}$ and projective toric variety $X_\mathcal{A}$. We give a concise combinatorial description of all linear subspaces contained in the variety $X_\mathcal{A}$, or, equivalently, all solutions in linear forms to the system of binomial equations determined by $I_\mathcal{A}$. More precisely, we study the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ whose closed points correspond to $k$-dimensional linear spaces contained in $X_\mathcal{A}$. We show that the irreducible components of $\mathbf{F}_k(X_\mathcal{A})$ are in bijection to maximal Cayley structures for $\mathcal{A}$ of length at least $k$. We explicitly describe these irreducible components and their intersection behavior, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore, in the special case $k=\dim X_\mathcal{A}-1$, we describe the nonreduced structure of $\mathbf{F}_k(X_\mathcal{A})$.

Density of quasismooth hypersurfaces in simplicial toric varieties

This paper investigates the density of hypersurfaces in a projective normal simplicial toric variety over a finite field having a quasismooth intersection with a given quasismooth subscheme. The result generalizes the formula found by B. Poonen for smooth projective varieties. As an application, we further analyze the density of hypersurfaces with bounds on their number of singularities and on the length of their singular schemes.

Stringy Chern classes of singular toric varieties and their applications

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional reflexive polytopes to the number 12 in dimension d>3.

The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces

We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for non-trivial weights) is isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant $K$-theory and complex cobordism rings of more general singular toric varieties, namely those whose integral cohomology is concentrated in even dimensions, are isomorphic to rings of appropriate piecewise formal power series. Finally, we confirm the corresponding descriptions for any smooth, compact, projective toric variety, and rewrite them in a face ring context. In many cases our results agree with those of Vezzosi and Vistoli for algebraic $K$-theory, Anderson and Payne for operational $K$-theory, Krishna and Uma for algebraic cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we summarize the details of these coincidences.

Projective toric polynomial generators in the unitary cobordism ring

According to Milnor and Novikov's classical result, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: , . In this paper we solve the well-known problem of constructing geometric representatives for the among smooth projective toric varieties, , . Our proof uses a family of equivariant modifications (birational isomorphisms) of an arbitrary complex manifold of complex dimension (, ). The key fact is that the change of the Milnor number under these modifications depends only on the dimension and the number and does not depend on the manifold itself. Bibliography: 22 titles.

Facets of secondary polytopes and chow stability of toric varieties

Chow stability is one notion of Mumford’s Geometric Invariant Theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope [7]. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is H-semistable if and only if it is H-polystable with respect to the standard complex torus action H . This essentially means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.

Toric polynomial generators of complex cobordism

Although it is well-known that the complex cobordism ring is a polynomial ring $\Omega_{*}^{U}\cong\mathbb{Z}\left[\alpha_{1},\alpha_{2},\ldots\right]$, an explicit description for convenient generators $\alpha_{1},\alpha_{2},\ldots$ has proven to be quite elusive. The focus of the following is to construct complex cobordism polynomial generators in many dimensions using smooth projective toric varieties. These generators are very convenient objects since they are smooth connected algebraic varieties with an underlying combinatorial structure that aids in various computations. By applying certain torus-equivariant blow-ups to a special class of smooth projective toric varieties, such generators can be constructed in every complex dimension that is odd or one less than a prime power. A large amount of evidence suggests that smooth projective toric varieties can serve as polynomial generators in the remaining dimensions as well.

Logarithmic stable toric varieties and their moduli

The Chow quotient of a toric variety by a subtorus, as dened by Kapranov{Sturmfels{ Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a xed projective toric variety, as constructed by Alexeev and Brion. We show that, after we endow both spaces with the structure of a logarithmic stack, the spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties it satises.

Growth of balls of holomorphic sections on projective toric varieties

Let [Formula: see text] be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety [Formula: see text]. Given a continuous toric metric [Formula: see text] on [Formula: see text], we define the energy at equilibrium of [Formula: see text] where [Formula: see text] is the weight of the metrized toric divisor [Formula: see text]. We show that this energy describes the asymptotic behavior as [Formula: see text] of the volume of the [Formula: see text]-norm unit ball induced by [Formula: see text] on the space of global holomorphic sections [Formula: see text].

Projective bundles over toric surfaces

Let [Formula: see text] be the Whitney sum of complex line bundles over a topological space [Formula: see text]. Then, the projectivization [Formula: see text] of [Formula: see text] is called a projective bundle over [Formula: see text]. If [Formula: see text] is a nonsingular complete toric variety, then so is [Formula: see text]. In this paper, we show that the cohomology ring of a nonsingular projective toric variety [Formula: see text] determines whether it admits a projective bundle structure over a nonsingular complete toric surface. In addition, we show that two [Formula: see text]-dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds.

Equivariant principal bundles and logarithmic connections on toric varieties

Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.