We define and study the vanishing sequence along a real valuation of sections of a line bundle on a projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure. In particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.
We generalize the theory of Newton-Okounkov bodies of big divisors to the case of graded linear series. One of the results is the generalization of slice formulas and the existence of generic Newton-Okounkov bodies for birational graded linear series. We also give a characterization of graded linear series which have full volume in terms of their base locus.
This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies.
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.
Okounkov bodies, which are closed convex sets defined for big line bundles, have rich information on the line bundles. On the other hand, Seshadri constants are invariants which measure the positivity of line bundles. In this paper, we prove that Okounkov bodies give lower bounds of Seshadri constants.
The aim of this paper is to study the Okounkov bodies associated to abundant divisors. As a main result, we prove that the valuative Okounkov bodies of an abundant divisor encode all the numerical properties. We apply this result to recover the asymptotic base loci of an abundant divisor from the valuative Okounkov bodies. We also give a criterion of when the valuative and limiting Okounkov bodies of an abundant divisor coincide by comparing their Euclidean volumes. To obtain these results, we prove some variants of Fujita's approximations for Okounkov bodies using Iitaka fibrations.
We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, 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…
Towards the boundary of the big cone, Newton-Okounkov bodies do not vary continuously and in fact the body of a boundary class is not well defined. Using the global Okounkov body one can nonetheless define a numerical invariant, the numerical Newton-Okounkov body. We show that if a normal projective variety has a rational polyhedral global Okounkov body, it admits a Minkowski basis provided one includes numerical Newton-Okounkov bodies above non-big classes. Under the same assumption, we also show that the dimension of the numerical Newton-Okounkov body is the numerical Kodaira dimension.
An Okounkov body is a convex subset of Euclidean space associated to a divisor on a smooth projective variety with respect to an admissible flag. In this paper, we recover the asymptotic base loci from the Okounkov bodies by studying various asymptotic invariants such as the asymptotic valuations and the moving Seshadri constants. Consequently, we obtain the nefness and ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we compute the divisorial Zariski decomposition by the Okounkov bodies, and find upper and lower bounds for moving Seshadri constants given by the siz…
We prove that the Okounkov body of a big divisor with respect to a general flag on a smooth projective surface whose pseudo-effective cone is rational polyhedral decomposes as the Minkowski sum of finitely many simplices and line segments arising as Okounkov bodies of nef divisors.
The purpose of this paper is to investigate the close relation between Okounkov bodies and Zariski decompositions of pseudoeffective divisors on smooth projective surfaces. Firstly, we completely determine the limiting Okounkov bodies on such surfaces, and give applications to Nakayama constants and Seshadri constants. Secondly, we study how the shapes of Okounkov bodies change as we vary the divisors in the big cone.
We prove that the existence of a finite Minkowski base for Okounkov bodies on a smooth projective variety with respect to an admissible flag implies rational polyhedrality of the global Okounkov body. As an application of this general result, we deduce that the global Okounkov body of a surface with finitely generated pseudo-effective cone with respect to a general flag is rational polyhedral. We give an alternative proof for this fact which recovers the generators more explicitly. We also prove the rational polyhedrality of global Okounkov bodies in the case of certain homogeneous 3-folds usi…
We first prove some basic properties of Okounkov bodies, and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Küronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.
We compute Okounkov bodies of projective complexity-one T-varieties with respect to two types of invariant flags. In particular, we show that the latter are rational polytopes. Moreover, using results of Dave Anderson and Nathan Ilten, we briefly exhibit explicit links to degenerations and T-deformations. Finally, we prove that the global Okounkov body of a rational projective complexity-one T-variety is rational polyhedral.
Based on the work of Okounkov (\cite{Ok96}, \cite{Ok03}), Lazarsfeld and Musta\c t\u a (\cite{LM08}) and Kaveh and Khovanskii (\cite{KK08}) have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then…
An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld-Mustaţă and Kaveh-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieti…
We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.
We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors which is defined in terms of refined divisorial Zariski decompositions. Our results extend Roé's work on surfaces to higher dimensional varieties although our proof is essentially different in nature.
The purpose of this paper is to study the slices of the Okounkov bodies of Mori dream spaces. First, we analyze all the slices of the Okounkov bodies of big divisors on Mori dream spaces associated to some admissible flags. As a byproduct, we obtain their descriptions on Mori dream threefolds. Finally, we consider its application to the rational polyhedrality of them.
Given a smooth Mori dream space $X$ we construct a model dominating all the small $\mathbb{Q}$-factorial modifications via tropicalization. This construction allows us to recover a Minkowski basis for the Newton-Okounkov bodies of divisors on $X$ and hence the movable cone of $X$. The existence of such basis allows us to prove the polyhedrality of the global Newton-Okounkov body and the existence of toric degenerations.
In this paper we explore the connection between asymptotic base loci and Newton-Okounkov bodies associated to infinitesimal flags. Analogously to the surface case, we obtain complete characterizations of augmented and restricted base loci. Interestingly enough, an integral part of the argument is a study of the relationship between certain simplices contained in Newton-Okoukov bodies and jet separation; our results also lead to a convex geometric description of moving Seshadri constants.
In this note we focus on three independent problems on Okounkov bodies for projective varieties. The main goal is to present a geometric version of the classical Fujita Approximation Theorem, a Jow-type theorem and a cardinality formulae for Minkowski bases on a certain class of smooth projective surfaces.
The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new app…
In recent years, the interaction between the local positivity of divisors and Okounkov bodies has attracted considerable attention, and there have been attempts to find a satisfactory theory of positivity of divisors in terms of convex geometry of Okounkov bodies. Many interesting results in this direction have been established by Choi--Hyun--Park--Won and Küronya--Lozovanu separately. The first aim of this paper is to give uniform proofs of these results. Our approach provides not only a simple new outlook on the theory but also proofs for positive characteristic in the most important cases. …
Given a homogeneous ideal in a polynomial ring over C, we adapt the construction of Newton-Okounkov bodies to obtain a convex subset of Euclidean space such that a suitable integral over this set computes the Segre zeta function of the ideal. That is, we extract the numerical information of the Segre class of a subscheme of projective space from an associated (unbounded) Newton-Okounkov convex set. The result generalizes to arbitrary subschemes of projective space the numerical form of a previously known result for monomial schemes.
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 mult…
We define and study the global Okounkov moment cone of a projective spherical variety X, generalizing both the global Okounkov body and the moment body of X defined by Kaveh and Khovanskii. Under mild assumptions on X we show that the global Okounkov moment cone of X is rational polyhedral. As a consequence, also the global Okounkov body of X, with respect to a particular valuation, is rational polyhedral.
Up to a factor 1/n!, the volume of a big line bundle agrees with the Euclidean volume of its Okounkov body. The latter is the convex hull of top rank valuation vectors of sections, all with respect to a single flag. In this text we give a different volume formula, valid in the ample cone, also based on top rank valuation vectors, but mixing data along several different flags.
Let $π: X \rightarrow \mathbb{P}^2$ be the blow-up of $\mathbb{CP}^2$ in $n$ points $x_i$ in very general position, and let $E_i$ be the exceptional divisor over $x_i$. For $0 \leq n \leq 9$ we calculate Okounkov bodies of graded linear series given by sections of multiples of line bundles $π^\ast \mathcal{O}_{\mathbb{P}^2}(d) \otimes \mathcal{O}_X(-m\sum_{i=1}^n E_i)$ with respect to a flag consisting of a line on $\mathbb{CP}^2$ and a point on the line in general position. Furthermore, we show what Nagata's Conjecture predicts on these Okounkov bodies when $n > 9$.
The theory of Newton-Okounkov bodies attaches a convex body to a line bundle on a variety equipped with flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this paper, we study Newton-Okounkov bodies for schemes defined over discrete valuation rings. We give the basic properties and then focus on the case of toric schemes and semistable curves. We provide a description of the Newton-Okounkov bodies for semistable curves in terms of the Baker--Norine theory of linear systems on graphs, finding a connection with tropical geometry.…
We study topological properties of functions on Okounkov bodies as introduced by Boucksom-Chen and Witt-Nyström. We note that they are continuous over the whole Okounkov body whenever the body is polyhedral, on the other hand, we exhibit an example that shows that continuity along the boundary does not hold in general.
Given a smooth projective algebraic surface X, a point O in X and a big divisor D on X, we consider the set of all Newton-Okounkov bodies of D with respect to valuations of the field of rational functions of X centred at O, or, equivalently, with respect to a flag (E,p) which is infinitely near to O, in the sense that there is a sequence of blowups mapping the smooth, irreducible rational curve E to O. The main objective of this paper is to start a systematic study of the variation of these infinitesimal Newton-Okounkov bodies as (E, p) varies, focusing on the case X = P2.
We show that the Okounkov body of a big divisor with finitely generated section ring is a rational simplex, for an appropriate choice of flag; furthermore, when the ambient variety is a surface, the same holds for every big divisor. Under somewhat more restrictive hypotheses, we also show that the corresponding semigroup is finitely generated.
Based on the work of Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata independently associated a convex body, called the Okounkov body, to a big divisor on a normal projective variety with respect to an admissible flag. Although the Okounkov bodies carry rich positivity data of big divisors, they only provide information near a single point. The purpose of this paper is to introduce a convex body of a big divisor that is effective in handling the positivity theory associated with multi-point settings. These convex bodies open the door to approach the local positivity theory at multiple points…
Given a big divisor $D$ on a normal complex projective variety $X$, we show that the restricted volume of $D$ along a very general complete-intersection curve $C\subset X$ can be read off from the Okounkov body of $D$ with respect to an admissible flag containing $C$. From this we deduce that if two big divisors $D_1$ and $D_2$ on $X$ have the same Okounkov body with respect to every admissible flag, then $D_1$ and $D_2$ are numerically equivalent.
The main goal of this article is to construct "arithmetic Okounkov bodies" for an arbitrary pseudo-effective (1,1)-class $α$ on a Kähler manifold. Firstly, using Boucksom's divisorial Zariski decompositions for pseudo-effective (1,1)-classes on compact Kähler manifolds, we prove the differentiability of volumes of big classes for Kähler manifolds on which modified nef cones and nef cones coincide; this includes Kähler surfaces. We then apply our differentiability results to prove Demailly's transcendental Morse inequality for these particular classes of Kähler manifolds. In the second part, we…
The purpose of the paper is to initiate the development of the theory of Newton Okounkov bodies of curve classes. Our denition is based on making a fundamental property of NewtonOkounkov bodies hold also in the curve case: the volume of the NewtonOkounkov body of a curve is a volume-type function of the original curve. This construction allows us to conjecture a new relation between NewtonOkounkov bodies, we prove it in certain cases.
Let $\mathscr{L} \rightarrow X$ be an ample line bundle over a nonsingular complex projective variety $X$. We construct an admissable flag $X_0 \subseteq X_1 \subseteq...\subseteq X_n=X$ of subvarieties for which the associated Okounkov body for $\mathscr{L}$ is a rational polytope.
We give a uniform proof for the conjectured Gromov width of coadjoint orbits of all compact connected simple Lie groups, by analyzing simplices in Newton-Okounkov bodies.
In recent years, the study of Newton-Okounkov bodies on normal varieties has become a central subject in the asymptotic theory of linear series, after its introduction by Lazarsfeld-Mustata and Kaveh-Khovanskii. One reason for this is that they encode all numerical equivalence information of divisor classes (by work of Jow). At the same time, they can be seen as local positivity invariants, and Küronya-Lozovanu have studied them in depth from this point of view. We determine what information is encoded by the set of all Newton-Okounkov bodies of a big divisor with respect to flags centered at …
A Newton-Okounkov convex 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 has various kinds of information about the original projective variety; for instance, Kaveh showed that the string polytopes from representation theory 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 id…
The Hodge algebra structures on the homogeneous coordinate rings of Grassmann varieties provide semi-toric degenerations of these varieties. In this paper we construct these semi-toric degenerations using quasi-valuations and triangulations of Newton-Okounkov bodies.
A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this is deeply connected with representation theory. For instance, the Littelmann string polytopes and the Feigin-Fourier-Littelmann-Vinberg polytopes are examples of Newton-Okounkov convex bodies. In this paper, we prove that the Newton-Okounkov convex body of a Schubert variety with respect to a specific valuation is identical to the Nakashima-Zelevinsky polyhedral realization of a Demazure crystal. As an application of this result, we show that Kashiwara'…
Let Δbe the Okounkov body of a divisor D on a projective variety X. We describe a geometric criterion for Δto be a lattice polytope, and show that in this situation X admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.
The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation $ν_{max}$ c…
We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme of points on $\mathbb C^2$. We obtain an upper bound for the Newton-Okounkov body of the Hilbert scheme of points on any smooth toric surface. We conjecture that this upper bound coincides with the exact Newton-Okounkov body for the Hilbert schemes of points on $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, and Hirzebruch surfaces. These results imply upper bounds for the effective cones of these Hilbert schemes, which are also conjecturally sharp in the above cases.
Despite their noted potential in polynomial-system solving, there are few concrete examples of Newton-Okounkov bodies arising from applications. Accordingly, in this paper, we introduce a new application of Newton-Okounkov body theory to the study of chemical reaction networks, and compute several examples. An important invariant of a chemical reaction network is its maximum number of positive steady states, which is realized as the maximum number of positive real roots of a parametrized polynomial system. Here, we introduce a new upper bound on this number, namely the `Newton-Okounkov body bo…
We prove that on a Bott-Samelson variety $X$ every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from $X$. Applying this result, we prove the rational polyhedrality of the global Newton-Okounkov body of a Bott-Samelson variety with respect to the so called `horizontal' flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.
The global Okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety. In the case of a rank two toric vector bundle E on a smooth projective toric variety, we use its Klyachko filtrations to give an explicit description of the global Okounkov body of P(E). In particular, we show that this is a rational polyhedral cone and that P(E) is a Mori dream space.
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of California Davis（加州大学戴维斯分校）；University of Massachusetts（马萨诸塞大学）；University of British Columbia（不列颠哥伦比亚大学）
