Okounkov Bodies Research Articles

Newton–Okounkov bodies of exceptional curve valuations

We prove that the Newton–Okounkov body associated to the flag E_{\bullet}:= \{ X=X_r \supset E_r \supset \{q\} \} , defined by the surface X and the exceptional divisor E_r given by any divisorial valuation of the complex projective plane \mathbb{P}^2 , with respect to the pull-back of the line-bundle \mathcal{O}_{\mathbb{P}^2} (1) is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton–Okounkov bodies which turn out to be triangular.

Probabilistic Saturations and Alt’s Problem

Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of solutions outside of the base locus to a system arising as the general linear combination of polynomials. In particular, we derive effective symbolic and numeric methods for studying such systems using probabilistic saturations that can be employed using both finite fields and floating-point computations. We give bounds on the size of finite field required to achieve a desired level of certainty. These methods can also be applied to many other problems where similar systems arise such as computing the volumes of Newton-Okounkov bodies and computing intersection theoretic invariants including Euler characteristics, Chern classes, and Segre classes.

Divisorial instability and Vojta’s main conjecture for $\mathbb{Q}$-Fano varieties

We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $\mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.

Generalized GCD for toric Fano varieties

We study the greatest common divisor problem for torus invariant blowing-up morphisms of nonsingular toric Fano varieties. Our main result applies the theory of Okounkov bodies together with an arithmetic form of Cartan's Second Main theorem, which has been established by Ru and Vojta. It also builds on Silverman's geometric concept of greatest common divisor. As a special case of our results, we deduce a bound for the generalized greatest common divisor of pairs of nonzero algebraic numbers.

Polyhedral realizations of crystal bases and convex-geometric Demazure operators

The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano.

A Reider-type theorem for higher syzygies on abelian surfaces

Building on the theory of infinitesimal Newton--Okounkov bodies and previous work of Lazarsfeld--Pareschi--Popa, we present a Reider-type theorem for higher syzygies of ample line bundles on abelian surfaces. As an application of our methods we confirm a conjecture of Gross and Popescu on abelian surfaces with a very ample primitive polarization of type $(1,d)$, whenever $d\geq 23$.

Relating transfinite diameters using an Okounkov body

We derive relations between the transfinite diameter of a locally circled subset K of the complexified sphere in $$\mathbb {C}^3$$ and notions of weighted transfinite diameter of the projection of K to $$\mathbb {C}^2$$ . Our method is based on connecting a Chebyshev transform of K on an Okounkov body of V to the classical Chebyshev transform of the projection of K to $$\mathbb {C}^2$$ .

A generalization of Δ-genus for big divisors on projective varieties

Let X be a complex projective variety, and let D be a big Cartier divisor on X. For this data, we define an invariant that refines the Δ-genus introduced by Fujita for nef and big divisors. We relate to the Okounkov body of D, and give some examples. In particular, we show that can be arbitrarily negative already on surfaces.

Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians

We use cluster structures and mirror symmetry to explicitly describe a natural class of Newton-Okounkov bodies for Grassmannians. We consider the Grassmannian $X=Gr_{n-k}(\mathbb C^n)$, as well as the mirror dual Landau-Ginzburg model $(\check{X}^\circ, W_q:\check{X}^\circ \to \mathbb C)$, where $\check{X}^\circ$ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian $\check{X} = Gr_k((\mathbb C^n)^*)$, and the superpotential W_q has a simple expression in terms of Pl\"ucker coordinates. Grassmannians simultaneously have the structure of an $\mathcal{A}$-cluster variety and an $\mathcal{X}$-cluster variety. Given a cluster seed G, we consider two associated coordinate systems: a $\mathcal X$-cluster chart $\Phi_G:(\mathbb C^*)^{k(n-k)}\to X^{\circ}$ and a $\mathcal A$-cluster chart $\Phi_G^{\vee}:(\mathbb C^*)^{k(n-k)}\to \check{X}^\circ$. To each $\mathcal X$-cluster chart $\Phi_G$ and ample `boundary divisor' $D$ in $X\setminus X^{\circ}$, we associate a Newton-Okounkov body $\Delta_G(D)$ in $\mathbb R^{k(n-k)}$, which is defined as the convex hull of rational points. On the other hand using the $\mathcal A$-cluster chart $\Phi_G^{\vee}$ on the mirror side, we obtain a set of rational polytopes, described by inequalities, by writing the superpotential $W_q$ in the $\mathcal A$-cluster coordinates, and then "tropicalising". Our main result is that the Newton-Okounkov bodies $\Delta_G(D)$ and the polytopes obtained by tropicalisation coincide. As an application, we construct degenerations of the Grassmannian to toric varieties corresponding to these Newton-Okounkov bodies. Additionally, when $G$ corresponds to a plabic graph, we give a formula for the lattice points of the Newton-Okounkov bodies, which has an interpretation in terms of quantum Schubert calculus.

Intersection theory of toric 𝑏-divisors in toric varieties

We introduce toric b b -divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric b b -divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b b -divisor is equal to the number of lattice points in this convex set and we give a Hilbert–Samuel-type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to b b -divisors with Newton–Okounkov bodies. The main motivation for studying toric b b -divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.

Khovanskii Bases, Higher Rank Valuations, and Tropical Geometry

Given a finitely generated algebra $A$, it is a fundamental question whether $A$ has a full rank discrete (Krull) valuation $\mathfrak{v}$ with finitely generated value semigroup. We give a necessary and sufficient condition for this in terms of tropical geometry of $A$. In the course of this we introduce the notion of a Khovanskii basis for $(A,\mathfrak{v})$ which provides a framework for far extending Gröbner theory on polynomial algebras to general finitely generated algebras. In particular, this makes a direct connection between the theory of Newton--Okounkov bodies and tropical geometry, and toric degenerations arising in both contexts. We also construct an associated compactification of ${Spec}\,(A)$. Our approach includes many familiar examples such as the Gel'fand--Zetlin degenerations of coordinate rings of flag varieties as well as wonderful compactifications of reductive groups. We expect that many examples coming from cluster algebras naturally fit into our framework.

Newton–Okounkov bodies on projective bundles over curves

In this article, we study Newton–Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe’s estimates used to compute the volume function on these varieties, we compute all Newton–Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton–Okounkov bodies.

A Product Formula for Volumes of Divisors Via Okounkov Bodies

AbstractWe generalize Kawamata’s product formula for volumes of canonical divisors to arbitrary divisors using Okounkov bodies.

Toric degenerations and symplectic geometry of smooth projective varieties

Let $X$ be an $n$-dimensional smooth complex projective variety embedded in $\mathbb{C}\mathbb{P}^{N}$. We construct a smooth family $\mathcal{X}$ over $\mathbb{C}$ with an embedding in $\mathbb{C}\mathbb{P}^{N} \times \mathbb{C}$ whose generic fiber is $X$ and the special fiber is the torus $(\mathbb{C}^*)^n$ sitting in $\mathbb{C}\mathbb{P}^{N}$ via a monomial embedding. We use this to show that if $\omega$ is an integral K\"ahler form on $X$ then for any $\epsilon > 0$ there is an open subset $U_\epsilon \subset X$ such that $vol(X \setminus U_\epsilon) < \epsilon$ and $U_\epsilon$ is symplectomorphic to $(\mathbb{C}^*)^n$ equipped with a (rational) toric K\"ahler form. As an application we obtain lower bounds for the Gromov width of $(X, \omega)$ in terms of its associated Newton-Okounkov bodies. We also show that if $\omega$ lies in the class $c_1(L)$ of a very ample line bundle $L$ then $(X, \omega)$ has a full symplectic packing with $d$ equal balls where $d$ is the degree of $(X, L)$.

Newton–Okounkov bodies for Bott–Samelson varieties and string polytopes for generalized Demazure modules

A Newton–Okounkov body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this generalizes a Newton polytope for a toric variety. This convex body inherits information about algebraic, geometric, and combinatorial properties of the original projective variety; for instance, Kaveh showed that string polytopes in the representation theory of algebraic groups are examples of Newton–Okounkov bodies for Schubert varieties. In this paper, we extend the notion of string polytopes for Demazure modules to generalized Demazure modules, and prove that the resulting generalized string polytopes are identical to the Newton–Okounkov bodies for Bott–Samelson varieties with respect to a specific valuation. As an application of this result, we show that these are indeed polytopes.

Mini-Workshop: Positivity in Higher-dimensional Geometry: Higher-codimensional Cycles and Newton–Okounkov Bodies

There are several flavors of positivity in Algebraic Geometry. They range from conditions that determine vanishing of cohomology, to intersection theoretic properties, and to convex geometry. They offer excellent invariants that have been shown to govern the classification and the parameterization programs in Algebraic Geometry, and are finer than the classical topological ones. This mini-workshop aims to facilitate research collaboration in the area, strengthening the relationship between various positivity notions, beyond the now classical case of divisors/line bundles.

Local positivity of linear series on surfaces

We study asymptotic invariants of linear series on surfaces with the help of Newton-Okounkov polygons. Our primary aim is to understand local positivity of line bundles in terms of convex geometry. We work out characterizations of ample and nef line bundles in terms of their Newton-Okounkov bodies, treating the infinitesimal case as well. One of the main results is a description of moving Seshadri constants via infinitesimal Newton-Okounkov polygons. As an illustration of our ideas we reprove results of Ein-Lazarsfeld on Seshadri constants on surfaces.

Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies

In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computations of Newton-Okounkov bodies of Hessenberg varieties. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, and then show that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and the special fiber is a regular nilpotent Hessenberg variety, are flat and have reduced fibres. This result further allows us to give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Pl\"ucker embedding. Furthermore, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, which are suitable for computing Newton-Okounkov bodies. As an application of our results, we explicitly compute many Newton-Okounkov bodies of the two-dimensional Peterson variety with respect to Pl\"ucker embeddings.

Okounkov bodies associated to pseudoeffective divisors

Okounkov bodies associated to pseudoeffective divisors

Local Okounkov bodies and limits in prime characteristic

This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call p-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert–Kunz multiplicity, and F-signature). We associate to each p-family of ideals an object in Euclidean space that is analogous to the Newton–Okounkov body of a graded family of ideals, which we call a p-body. Generalizing the methods used to establish volume formulas for the Hilbert–Kunz multiplicity and F-signature of semigroup rings, we relate the volume of a p-body to a certain asymptotic invariant determined by the corresponding p-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn–Minkowski theorem for Hilbert–Kunz multiplicity, and a uniformity result concerning the positivity of a p-family.