Toric Varieties Research Articles

Toric configuration spaces: the bipermutahedron and beyond

We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical configuration spaces. The corresponding toric varieties provide modular compactifications of the algebraic configuration spaces, with boundary parametrising transverse configurations on tropical expansions. The rubber torus, used to identify equivalent configurations, plays a key role. As an application, we obtain a modular interpretation for the bipermutahedral variety.

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

Quadratic enrichment of the logarithmic derivative of the zeta function

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure.

Generalized toric varieties, LVMB manifolds and Lie groupoids

Generalized toric varieties, LVMB manifolds and Lie groupoids

Investigating Secant Varieties

A variety is a geometric object that locally looks like the set of points that are the solutions of some polynomial equations. Any given variety gives rise to a sequence of secant varieties indexed by the nonnegative integers. The kth secant variety is the union of all planes spanned by k+1 points on the original variety. Like vector spaces in linear algebra, varieties have an intrinsic dimension. A naive parameter count produces an upper bound on the dimension of the kth secant variety called its expected dimension. A secant variety is called defective if its actual dimension is less than its expected dimension. Despite being well understood in a few interesting cases, the classification of varieties with defective secant varieties in general remains far from complete. For a specific class of varieties called smooth toric varieties, our goal is to determine which ones admit defective secant varieties. Using the computer algebra system Macaulay2, we conducted experiments on this class of varieties and collected data on defectiveness, focusing on varieties of low dimension and Picard rank. Toric varieties have strong connections to combinatorics and convex polytopes; for any toric variety, there is a corresponding polytope that contains data about the variety. From the polytope, we are able to construct a particular kind of matrix that encodes information about the variety and its secants, which allowed us to prove defectivity for various Picard rank 2 toric varieties. Conversely, applying results from a paper by Laface–Massarenti–Rischter (2022) to the polytopes, we were able to prove non-defectivity of all secant varieties of certain toric surfaces with Picard rank 2. These statements together yield a classification of defective secant varieties for a subset of 2-dimensional smooth toric varieties with Picard rank 2.

The structure of exceptional sequences on toric varieties of Picard rank two

The structure of exceptional sequences on toric varieties of Picard rank two

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

Tropical Mirror

We describe the tropical curves in toric varieties and define the tropical Gromov-Witten invariants. We introduce amplitudes for the higher topological quantum mechanics (HTQM) on special trees and show that the amplitudes are equal to the tropical Gromov-Witten invariants. We show that the sum over the amplitudes in $A$-model HTQM equals the total amplitude in B-model HTQM, defined as a deformation of the $A$-model HTQM by the mirror superpotential. We derived the mirror superpotentials for the toric varieties and showed that they coincide with the superpotentials in the mirror Landau-Ginzburg theory. We construct the mirror dual states to the evaluation observables in the tropical Gromov-Witten theory.

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.

The Manin–Peyre conjecture for smooth spherical Fano threefolds

The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.

Surface counterexamples to the Eisenbud-Goto conjecture

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in P 5 \mathbb {P}^5 , and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in P 4 \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.

Period Integrals of Hypersurfaces via Tropical Geometry

Abstract Let $\left \{ Z_{t} \right \}_{t}$ be a one-parameter family of complex hypersurfaces of dimension $d \geq 1$ in a toric variety. We compute asymptotics of period integrals for $\left \{ Z_{t} \right \}_{t}$ by applying the method of Abouzaid–Ganatra–Iritani–Sheridan, which uses tropical geometry. As integrands, we consider Poincaré residues of meromorphic $(d+1)$-forms on the ambient toric variety, which have poles along the hypersurface $Z_{t}$. The cycles over which we integrate them are spheres and tori, which correspond to tropical $(0, d)$-cycles and $(d, 0)$-cycles on the tropicalization of $\left \{ Z_{t} \right \}_{t}$, respectively. In the case of $d=1$, we explicitly write down the polarized logarithmic Hodge structure of Kato–Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.

On orbits of automorphism groups on horospherical varieties

AbstractIn this paper, we describe orbits of the automorphism group on an affine horospherical variety in terms of degrees of homogeneous with respect to natural grading locally nilpotent derivations. In case of (possibly nonnormal) toric varieties, a description of orbits of the automorphism group in terms of corresponding weight monoid is obtained.

Tate resolutions on toric varieties

We develop an analogue of Eisenbud–Fløystad–Schreyer’s Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier–Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinson’s resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

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.

Solving equations using Khovanskii bases

We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.

Normal Completions of Toric Varieties Over Rank One Valuation Rings and Completions of Γ-Admissible Fans

Abstract We show that any normal toric variety over a rank one valuation ring admits an equivariant open embedding in a normal toric variety which is proper over the valuation ring, after a base-change by a finite extension of valuation rings. If the value group $\Gamma $ is discrete or divisible then no base-change is needed. We give explicit examples that show that existing methods do not produce such normal equivariant completions. Our approach is combinatorial and proceeds by showing that $\Gamma $-admissible fans admit $\Gamma $-admissible completions. In order to show this we prove a combinatorial analog of Noetherian reduction, which we believe will be of independent interest.

The characterization of (𝑛 − 1)-spheres with 𝑛 + 4 vertices having maximal Buchstaber number

Abstract We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of ( n − 1 ) (n-1) -dimensional PL spheres with n + 4 n+4 vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number 4. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number 4. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.

On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.