Convex Polytopes, Algebraic Geometry, and Combinatorics

Laura Escobar and Kiumars Kaveh

In the last several decades, convex geometry methods have The first observation is related to describing a polytope as proven very useful in algebraic geometry specifically to un- an intersection of finitely many half spaces. It appears in derstand discrete invariants of algebraic varieties. An ap- the notion of a Newton polygon introduced in Section 1 proach to study algebraic varieties is to assign to a family and is important in tropical geometry. The second observa- of varieties a corresponding family of combinatorial ob- tion is related to the construction of a polytope as a convex jects which encode geometric information about the vari- hull of finitely many points. It is central to the proof ofthe eties. Often, the combinatorial objects that arise are con- BKK theorem (see Theorem 1.2) and its generalization to vex polytopes, and convex geometry has been an essential Newton–Okounkov bodies described in Section 3. tool for this strategy. The emergence of convexity in algebraic geometry is 1. Newton Polytopes and Toric Varieties rooted in the following geometric observations. Let 풜 ⊂ The origin of appearances of convex polyhedra in algebraic ℤ푛 be a finite set, then: geometry goes back to Sir Isaac Newton and Ferdinand 1. For any vector 휉 ∈ ℝ푛, the maximum/minimum of the Minding. Newton introduced what we now know as the dot products 휉 ⋅ 푥, 푥 ∈ 풜, is attained on the boundary Newton diagram of a polynomial 푓(푥, 푡) in two variables 푥 of the convex hull of 풜. and 푡. Given an equation 푓(푥, 푡) = 0 regarded as implicitly 1 defining 푥 in terms of 푡, Newton was interested in express- 2. As 푘 → ∞ the rescaled 푘-fold sums { (푥1 + ⋯ + 푥푘) ∣ 푘 ing 푥 as an infinite series in 푡. He knew that in general 푥 ∈ 풜} converge to the convex hull of 풜. 푖 푥 = 푥(푡) may not be a power series but a series of the form ∑ 푐 푡푖/푘 for some fixed integer 푘 > 0, a series with frac- Laura Escobar is an assistant professor of mathematics at Washington Univer- 푖≥0 푖 sity in St. Louis. Her email address is [email protected]. She was supported tional exponents. In general one expects to have as many in part by NSF grant DMS-1855598. solutions as the degree of 푓 in 푥. 푎 푏 Kiumars Kaveh is an associate professor of mathematics at the University of Let 푓(푥, 푡) = ∑푎,푏 푐푎,푏푥 푡 . Consider the Newton dia- Pittsburgh. His email address is [email protected]. He was partially supported gram (or Newton polygon) of 푓(푥, 푡) to be the lower convex 2 by NSF grant DMS-1601303, a Simons Foundation Collaboration Grant for hull of {(푎, 푏) ∣ 푐푎,푏 ≠ 0} ⊂ ℝ . This is the lower part of the Mathematicians, and a Simons Fellowship. boundary of the convex hull stretching from the leftmost Communicated by Notices Associate Editor Stephan Ramon Garcia. to the rightmost point. Suppose the slopes of the (nonver- For permission to reprint this article, please contact: tical) line segments in the Newton diagram are 휇1, … , 휇푠. [email protected]. Newton showed that the fractional exponents for the first DOI: https://doi.org/10.1090/noti2137 terms in the series representations of solutions for 푥 are

1116 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 8 푠 훼푖 finite-dimensional vector space 퐿풜 = {푓(푥) = ∑푖=0 푐푖푥 ∣ ∀푖, 푐푖 ∈ ℂ} consisting of Laurent polynomials with exponents from 풜. The convex hull of 풜, which we denote Δ풜, is the Newton polytope of a generic element of 퐿풜.

Theorem 1.2 (BKK). For a generic choice of 푓1, … , 푓푛 ∈ 퐿풜, ∗ 푛 the number of solutions 푥 in (ℂ ) of the system 푓1(푥) = ⋯ = 푓푛(푥) = 0 is the same and is equal to 푛! vol(Δ풜), where vol denotes the standard Lebesgue measure in ℝ푛. Figure 1. Example from Newton’s letter to Oldenburg dated October 24, 1676. Remark 1.3. In fact, the above form of this theorem is due to Kushnirenko. Khovanskii has found many differ- the −휇푖. This observation is a key step in proving the well- ent proofs for this theorem (worthy of Guinness Book of known Newton–Puiseux Theorem, which describes the al- Records?). Extending the BKK theorem, he also found for- gebraic closure of the field of formal power series ℂ((푡)) mulas in terms of the Newton polytope Δ풜 for genera and (see [MS15, Theorem 2.1.5]). Euler characteristics of subvarieties defined by 푓1(푥) = ⋯ = Motivated by Newton’s classic work, Vladimir Arnold 푓푘(푥) = 0, for 푘 ≤ 푛, and, as before, generic elements asked his students to work on generalizations and ana- 푓푖 ∈ 퐿풜 (see [Hov78]). logues of the notion of Newton diagram/polygon in sev- 푠 eral variables. This resulted in the modern definition of Consider the subset 푇풜 ⊂ ℂℙ defined by the Newton polytope of a polynomial. Unlike the Newton 훼0 훼푠 ∗ 푛 푇풜 = {(푥 ∶ ⋯ ∶ 푥 ) ∣ 푥 ∈ 푇 = (ℂ ) }. diagram, which is a local concept related to multiplicities of roots of a polynomial, the Newton polytope (Defini- The subset 푇풜 is isomorphic to an algebraic torus. One can 푛 tion 1.1) is a global concept which can be thought of as a show that if the differences of elements in 풜 generate ℤ , ∗ 푛 푠 refinement of the notion of degree of a polynomial. then 푇풜 is isomorphic to 푇 = (ℂ ) . Let 푋풜 ⊂ ℂℙ be the ∗ 푛 푠 Throughout we denote the multiplicative group of closure of 푇풜. Note that the torus 푇 = (ℂ ) acts on ℂℙ nonzero complex numbers by ℂ∗. The product (ℂ∗)푛 of by ∗ 훼0 훼푠 푛 copies of ℂ is an algebraic group often called an alge- 푥 ⋅ (푧0 ∶ ⋯ ∶ 푧푠) = (푥 푧0 ∶ ⋯ ∶ 푥 푧푠), 1 푛 braic torus (it contains the usual topological torus (푆 ) and the variety 푋풜 is the closure of the orbit of (1 ∶ ⋯ ∶ 1). as a maximal compact subgroup). The algebra of regu- Recall that the degree of an 푛-dimensional projective vari- ∗ 푛 lar functions on (ℂ ) is the algebra of Laurent polynomials ety 푋 ⊂ ℂℙ푠 is equal to the number of intersection points ± ± ℂ[푥1 , … , 푥푛 ]. We will use the multi-index notation, and for of 푋 with a generic plane of codimension 푛. This notion 푛 훼 훼 = (푎1, … , 푎푛) ∈ ℤ we write 푥 to denote the monomial generalizes the degree of a polynomial and provides a mea- 푎 푎 1 푛 푠 푥1 ⋯ 푥푛 . surement for how complex the embedding of 푋 in ℂℙ is. The BKK theorem can be restated as giving a formula for Definition 1.1. Let 푓(푥) = ∑ 푐 푥훼 be a Laurent polyno- 훼 훼 the degree of 푋 ⊂ ℂℙ푠. mial. The Newton polytope of 푓 is the convex hull of the 풜 finite set {훼 ∣ 푐훼 ≠ 0}. It is a convex polytope with vertices Theorem 1.4 (Alternative statement). The degree of 푋풜 ⊂ 푛 푠 in ℤ . ℂℙ is equal to 푛! vol(Δ풜). Remark 1.5. One of the simplest proofs of the BKK theorem, due to Khovanskii, uses the above restatement and • the classical Hilbert theorem on the relation between the • degree of a projective variety and the leading term of its Hilbert polynomial. This proof enables a generalization • • of this theorem to arbitrary systems of equations and is the basis of the theory of Newton–Okounkov bodies (Section 3). Figure 2. Newton polytope of 푓(푥, 푦) = 푥3 + 3푥푦 + 푦2 + 1. Since 푋풜 is the closure of an orbit, it is invariant under the 푇-action. It is a 푇-toric variety in the sense of the fol- One of the major discoveries of the Moscow Newton lowing definition. polyhedra school is the celebrated BKK theorem named after David Bernstein (younger brother of Joseph Bernstein), Definition 1.6. An (abstract) 푇-toric variety is an irre- Askold Khovanskii, and Anatoli Kushnirenko. Fix a finite ducible variety with an algebraic 푇-action that has a finite 푛 set of lattice points 풜 = {훼0, … , 훼푠} ⊂ ℤ and consider the number of 푇-orbits.

SEPTEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1117 푠 푛 For 푋 a 푇-toric variety there is an open dense orbit 푈0 ⊂ 휇풜 ∶ ℂℙ → ℝ is defined by 푋. After replacing 푇 with 푇/푇0, where 푇0 is the 푇-stabilizer 푠 2 of 푈0, without loss of generality we can assume that 푈0 is |푧푖| 휇풜 ∶ (푧0 ∶ ⋯ ∶ 푧푠) ↦ ∑ ( 푠 ) 훼푖 ∈ Δ풜. isomorphic to 푇 itself. 2 푖=0 ∑푗=0 |푧푗| The degree of 푋풜 is not the only geometric information we can obtain from the polytope Δ풜. In general, there is One easily verifies that 휇 is invariant under the action a beautiful correspondence between the algebro-geometry of the topological torus (푆1)푛 ⊂ 푇 = (ℂ∗)푛. Moreover, 푠 of toric varieties and the combinatorics of convex poly- 휇풜(ℂℙ ) = 휇풜(푋풜) = Δ풜. topes. The following is an example of this correspondence. Remark 1.10. The variety 푋풜 inherits a volume form/ Theorem 1.7. There is a one-to-one correspondence between measure, called the Liouville measure, from the standard 푠 the faces of the polytope Δ풜 and the torus orbits in 푋풜. Fubini–Study metric on ℂℙ . One can directly compute the pushforward of the Liouville measure on 푋풜 to Δ풜 and To make the connection between toric varieties and con- show that the pushforward measure is (a constant multi- vex polytopes more tight, one usually assumes 푋 is a nor- ple of) the Lebesgue measure on Δ풜. From this one can mal variety. Some authors include this in the definition of give an elegant proof of the BKK theorem. This proof is a toric variety (see [Ful93]). The connection between the due to Khovanskii. For a beautiful account of these ideas theory of toric varieties and Newton polytopes was discov- and more details, see [Ati83]. ered by Khovanskii in now classic papers [Hov77,Hov78]. Example 1.11 (Baby example). Let 푛 = 1 and 풜 = {0, 1} ⊂ 1 Remark 1.8. Using the rich correspondence between geom- ℤ. One sees that 푋풜 = ℂℙ . The moment map 휇풜 ∶ etry of toric varieties and combinatorics of convex poly- ℂℙ1 → [0, 1] is the height function illustrated in Figure 3. topes, Victor Batyrev famously proved the mirror symmetry The Fubini–Study metric on ℂℙ1 is the usual metric on the conjecture for smooth toric varieties. This is a deep conjec- sphere, and the Liouville measure is just the surface area on ture in algebraic geometry inspired by high-energy physics, the sphere. and in particular string theory. Batyrev’s work is based on the Khovanskii–Danilov computation of Hodge–Deligne Remark 1.12. In the above example, the fact that the push- 1 numbers using Newton polytopes. forward of the surface area on the sphere ℂℙ (the Fubini– Study form in this case) is equal to the Lebesgue measure Remark 1.9. The correspondence between toric varieties was apparently known to Archimedes! It is directly related and convex polytopes has also proven to be an extremely to Archimedes’s theorem on surface area of a cylinder vs. powerful tool in attacking some deep combinatorial prob- surface area of a sphere. Cicero describes visiting the tomb lems. Given a sequence 햿 = (햿0, … , 햿푛−1) of nonnegative of Archimedes, on top of which there were a sphere and a integers, one asks whether there is an 푛-dimensional poly- cylinder that Archimedes had requested be placed on his tope 푃 all of whose faces are simplices so that 햿푖 is the tomb to represent his mathematical discoveries. number of 푖-dimensional faces of 푃 for all 푖 = 0, … , 푛 − 1. The Dehn–Sommerville equations give necessary and sufficient conditions on the sequence 햿 for this to hold. Richard Stanley gave a proof of (the necessity of) Dehn– Sommerville equations by interpreting the vector 햿 in terms μ of the (intersection) cohomology of projective toric varieties (see [Ful93, Section 5.6]). Alternatively Khovanskii found a proof of the Dehn–Sommerville equations based on Morse theory on polytopes/toric varieties.

Interestingly there is a natural (continuous and almost everywhere differentiable) map 휇풜 ∶ 푋풜 → Δ풜. It is Figure 3. Moment map of the sphere. called the moment map of the variety 푋풜. It is a special case of the notion of moment map of a Hamiltonian torus action from symplectic geometry and classical mechanics. Here we regard (the smooth locus of) 푋풜 as a symplectic 2. The Permutohedron and Associahedron manifold with respect to the restriction of the standard Now suppose we have a linear action of a torus 푇 = (ℂ∗)푛 Fubini–Study form on ℂℙ푠. This map can be written ex- on ℂℙ푠 and a 푇-invariant subvariety 푋 such that the 푇- plicitly (without any knowledge of symplectic geometry re- stabilizer of 푋 is trivial. This yields a moment map 휇 ∶ 푋 → 푛 푛 quired). Given a finite set 풜 = {훼0, … , 훼푠} ⊂ ℤ the map ℝ , and its image is a polytope Δ푋 . (As mentioned before,

1118 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 8 Furthermore, 퐹푛 is a 풟-invariant subvariety and therefore has a moment map. The image of this map is the permutohedron 햯푛. The permutohedron 햯푛 is the convex hull of the 푛! per- mutations of (0, 1, … , 푛 − 1). Actually all of these points are vertices of 햯푛. This polytope is the Newton polytope of the Vandermonde determinant 1 1 ⋯ 1 ⎡ ⎤ 푥 푥 ⋯ 푥 det ⎢ 1 2 푛 ⎥ = ∏ (푥 − 푥 ). ⎢ ⋮ ⋮ ⋮ ⎥ 푗 푖 ⎢ ⎥ 1≤푖<푗≤푛 푛−1 푛−1 푛−1 ⎣푥1 푥2 ⋯ 푥푛 ⎦ Figure 4. Surface area of the sphere vs. the cylinder. The combinatorics of 햯푛 reflects some of the geometry of the flag variety. For example, the vertices and edge direc- this is a special case of the moment map from symplec- tions of 햯 encode the 푇-equivariant cohomology (and tic geometry.) In the previous section we further assumed 푛 hence the usual cohomology) of 퐹 . that 푋 was a toric variety. This implied the existence of 푛 an open dense orbit 푈0 ⊂ 푋 isomorphic to 푇 which in Remark 2.1. More generally, there is a large class of vari- turn forced dim(푇) = dim(푋). In general we may have eties with a torus action for which one can give an elegant that dim(푇) < dim(푋), which takes us out of the setting of description of their cohomology rings in terms of the com- toric varieties. The polytope Δ푋 encodes some geometric binatorial data of vertices and edges of their moment graphs. information of 푋, but unlike the toric case, the moment This is the class of GKM varieties named after Mark Goresky, map does not yield a tight connection between Δ푋 and 푋. Robert Kottwitz, and Robert MacPherson. A 푇-variety 푋 is Regardless, the moment map construction gives rise to im- a GKM variety if it has a finite number of 푇-fixed points, a portant combinatorics associated to 푋. In this section we finite number of 푇-invariant curves, and the 푇-action is so- give two examples of moment polytopes arising from the called equivariantly formal. Examples include flag varieties study of flag varieties and see they are combinatorially in- and toric varieties. teresting. The first one will be associated to a projective variety that is not toric. There is a classical formula for the volume of the permutohedron. Let 퐻푛 be the affine hyperplane given by 2.1. The permutohedron. The flag variety 퐹푛 consists of 푛(푛−1) 푛 푥1 + ⋯ + 푥푛 = ; note that 햯푛 lies on the affine hy- the nested sequences {0} = 푉0 ⊊ 푉1 ⊊ ⋯ ⊊ 푉푛 = ℂ of 2 푛 vector subspaces of ℂ . One can realize this space as a perplane 퐻푛. The volume of 햯푛 as a polytope in 퐻푛 (nor- 푛 subvariety of a product of projective spaces as follows. An malized so that every primitive parallelepiped in 퐻푛 ∩ ℤ 푛−2 element of 퐹푛 can be represented by an invertible 푛×푛 ma- has volume 1) equals 푛 . This is equal to the number of trix 푀 by setting 푉푖 to be the row-span of the top 푖 rows of trees on 푛 labeled vertices. However, since 퐹푛 is not a toric 푀. Given an invertible matrix 푀 and a set 퐼 ⊂ {1, 2, … , 푛}, variety, the BKK theorem does not apply and 푛푛−2 is not let 푝퐼 (푀) be the minor of 푀 given by the top |퐼| rows and the degree of 퐹푛. The volume of 햯푛 equals the degree of the columns in 퐼. For 1 ≤ 푖 ≤ 푛, the homogeneous coor- the largest 푇-toric subvariety in 퐹푛. (푛)−1 dinates (푝퐼 (푀) ∣ |퐼| = 푖) ∈ ℂℙ 푖 are called the Plücker coordinates of the subspace 푉푖. Sending 푉1, … , 푉푛−1 to their Plücker coordinates yields an embedding

(푛)−1 ( 푛 )−1 퐹푛 ↪ ℂℙ 1 × ⋯ × ℂℙ 푛−1 , and the image is a closed subvariety. To realize 퐹푛 as a pro- (푛)×⋯×( 푛 )−1 jective variety in ℂℙ 1 푛−1 , we can apply the Segre Figure 5. From left to right, the permutohedra 햯3 and 햯4. embedding multiple times. Define a torus action on the Plücker coordinates asfol- 2.2. The associahedron. Consider the ways to put paren- lows. Realize the torus (ℂ∗)푛 as the group 풟 of invertible theses on 푡1 ⋯ 푡푛 so that we end up with binary prod- diagonal 푛 × 푛 matrices. Given 퐷 ∈ 풟 and an invertible ucts. These products can be arranged into a polyhedral matrix 푀, the action of 퐷 on the Plücker coordinates 푝 (푀) 퐼 complex, called the associahedron 햠푛. Any balanced place- sends them to the Plücker coordinates of 푀퐷. This ac- ment of parentheses yields a face with vertices the binary (푛)×⋯×(푛)−1 tion comes from a linear action of 풟 on ℂℙ 1 푛 . products that contain the given parentheses. For example,

SEPTEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1119 Schubert varieties are subvarieties of the 퐹 defined by (푡1(푡2푡3))푡4 푛 imposing conditions on how the flags intersect the coordi- 푡1((푡2푡3)푡4) nate subspaces of ℂ푛. Varieties generalizing the one above

((푡1푡2)푡3)푡4 can be used to resolve singularities of transverse intersections of Schubert varieties. 푡 (푡 (푡 푡 )) 1 2 3 4 Returning to the case at hand, the action of 풟 = (ℂ∗)3

(푡1푡2)(푡3푡4) on 퐹3 induces an action of 풟 on 퐵 and a moment map. The moment polytope of the image is Loday’s realization Figure 6. A realization of 햠4. of 햠4. It turns out that the dimension of 퐵 is equal to the dimension of 햠4, which implies that this variety is actually 푡1(푡2푡3푡4) corresponds to the edge with vertices 푡1(푡2(푡3푡4)) the toric variety of Loday’s realization of 햠4. and 푡1((푡2푡3)푡4). See Figure 6 for an example. An alternative description of the associahedron 햠푛 is given by considering 3. Newton–Okounkov Bodies the vertices to be triangulations of an (푛+1)-gon where two The success of toric methods encouraged algebraic geome- triangulations are adjacent if you can obtain one by flipping ters to try to extend the scope of convex geometric meth- a diagonal of the other; see Figure 7 for an example. ods in algebraic geometry. Many of the results about toric varieties have been extended to varieties with actions of so-called reductive groups. These are the complex algebraic counterparts of compact Lie groups and include familiar examples from linear algebra such as GL(푛, ℂ) and SL(푛, ℂ). A large class of varieties with reductive group actions that extends that of toric varieties is the class of spherical varieties. (We caution that the adjective spherical here is not directly related to sphere, but rather to spherical functions from representation theory.) The BKK theorem has been generalized to spherical varieties by Michel Brion and Boris Kazarnovskii and to more general reductive group actions by Kaveh and Khovanskii (see [KK12b] and references therein). Far more generally, the theory of Newton–Okounkov bodies extends the BKK theorem to arbitrary projective va- Figure 7. The polyhedral complex with vertices triangulations rieties. Let 푋 ⊂ ℂℙ푠 be an 푛-dimensional projective vari- of a regular hexagon. ety. Generalizing the BKK formula for degree of 푋풜, we would like to construct a convex body (i.e., a convex com- 푛 A realization of 햠푛 is an (푛 − 2)-dimensional polytope pact subset) Δ ⊂ ℝ such that its volume gives the degree whose face structure equals the face structure of 햠푛. There of 푋. In this full generality Δ may not be a polytope but are many different polytopes that can arise in this way; the only a convex body. To do this we need an extra choice of paper [CSZ15] has a survey about the different realizations. a function 푣 ∶ ℂ(푋) ⧵ {0} → ℤ푛 satisfying the properties The Newton polytope of ∏1≤푖<푗≤푛−1(푥푖 + 푥푖+1 + ⋯ + 푥푗) of a valuation on the field of rational functions ℂ(푋). Instead is Jean-Louis Loday’s realization of 햠푛. The toric variety of of giving the abstract definition of a valuation (from com- Loday’s realization of 햠푛 can be constructed using concepts mutative algebra), we explain the geometric construction from the flag variety; see [Esc16]. For brevity we describe of a typical valuation on the field of rational functions. It only the case 푛 = 4. Let 푒1, 푒2, 푒3 be the standard basis vec- is a generalization of the familiar notion of leading term of tors of ℂ3. Consider the variety 퐵 consisting of the tuples a polynomial. 푛 (푉1,1, 푉2,1, 푉2,2) of vector spaces such that the following inci- First we equip the additive group ℤ with the lexi- dences hold: cographic order. Pick a smooth point 푝 in 푋 and let (푢 , … , 푢 ) be a system of parameters at 푝. That is, the

⟨푒1, 푒2, 푒3⟩ 1 푛 ⊂ ⊂ 푢푖 are rational functions that are regular at 푝 such that

푉2,1 푉2,2 푢1(푝) = ⋯ = 푢푛(푝) = 0, and their differentials at 푝 are lin- ⊂

⊂ ⊂ ⊂ early independent (in other words, they generate the max- ⟨푒 ⟩ 푉1,1 ⟨푒 ⟩ 1 3 imal ideal of 푝). It is well known that every rational func- One can then use the Plücker embedding and multiple tion 푓 ∈ ℂ(푋) can be uniquely expressed as a formal Lau- 푎 푎푛 3 3 3 1 ( )×( )×( )−1 rent series in the 푢푖, that is, 푓 = ∑훼=(푎 ,…,푎 ) 푐훼푢1 ⋯ 푢푛 . Segre embeddings to embed 퐵 into ℙ 1 2 2 . 1 푛

1120 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 8 Then we define 푣 ∶ ℂ(푋) ⧵ {0} → ℤ푛 by Example 3.4. With notation as before, let 푋 ⊂ ℂℙ푠 be a projective variety with degree 푑. Let 푣 be the valuation 푣(푓) = min{훼 ∣ 푐 ≠ 0}. 훼 obtained from a system of parameters corresponding to Here the minimum is taken with respect to the lexico- 푛 = dim(푋) hyperplane sections in general position. Then 푛 푛 graphic order on ℤ (one shows that the minimum always Δ(푋, 푣) is the simplex in ℝ with vertices 0, 푒1, … , 푒푛−1, and exists in this case). Note that the choice of 푣 is independent 푑푒푛, where {푒1, … , 푒푛} is the standard basis. of the choice of embedding of 푋 into projective space. Δ(푋, 푣) Let ℂ[푋] be the homogeneous coordinate ring of 푋. If It is possible that is not a polytope (see [LM09, Section 6.3]). For random choices of 푋 and 푣 one expects 퐼 ⊂ ℂ[푥0, … , 푥푠] is the homogeneous ideal defining 푋 ⊂ 푠 that the convex body Δ(푋, 푣) is not a polytope, cf. work of ℂℙ , then ℂ[푋] = ℂ[푥0, … , 푥푠]/퐼 and it is a ℤ≥0-graded al- Küronya–Lozovanu–Maclean. gebra. Denote by ℂ[푋]푚 the 푚th graded piece of ℂ[푋] so that Remark 3.5. One can also define local versions of Newton– ℂ[푋] = ℂ[푋] . ⨁ 푚 Okounkov bodies. In commutative algebra terms, to a pri- 푚∈ℤ≥0 mary ideal 퐼 in a local algebra 푅 one can associate a convex Using the valuation 푣 we would like to associate a convex set Γ(퐼) inscribed in a cone 퐶(푅) such that the volume of its body Δ to the graded algebra ℂ[푋]. Fix a nonzero element complement gives the Samuel multiplicity 푒(퐼) (see [KK14] ℎ ∈ ℂ[푋]1. First, we construct a semigroup 푆 = 푆(푋, 푣) ⊂ as well as work of Dale Cutkosky). ℤ × ℤ푛 by >0 The notion of a toric degeneration provides a geometric 푆 = {(푚, 푣(푓/ℎ푚)) ∣ 0 ≠ 푓 ∈ ℂ[푋] }. explanation for why Theorem 3.2 holds. ⋃ 푚 푚>0 Definition 3.6. A toric degeneration of an embedded pro- 푛+1 Let 퐶 ⊂ ℝ be the closure of the convex hull of 푆 ∪ {0}. jective variety 푋 ⊂ ℂℙ푠 is a family 휋 ∶ 픛 → ℂ where 픛 ⊂ ℂℙ푠 × ℂ and 휋 is the projection on the second fac- Definition 3.1. The Newton–Okounkov body Δ = Δ(푋, 푣) of tor, such that the following hold: 푋 ⊂ ℂℙ푠 is the intersection of the convex cone 퐶 with the 푛+1 ∗ hyperplane 푥1 = 1 in ℝ . One shows that Δ is bounded (1) The family is trivial over ℂ with fiber isomorphic and hence a convex body. to 푋. That is, 휋−1(ℂ∗) ≅ 푋 × ℂ∗. −1 (2) The fibers 푋푡 ∶= 휋 (푡), 푡 ∈ ℂ, are all reduced and The construction of Δ(푋, 푣) appears (in passing) in irreducible (by (1) it suffices that 푋0 is reduced [Oko96, Oko03]. It was defined in a more general setting and irreducible). and systematically studied in [LM09,KK12a]. −1 (3) The fiber 푋0 = 휋 (0) (special fiber) is a toric vari- The main theorem regarding Newton–Okounkov bod- ety with respect to an action of 푇 = (ℂ∗)푛 induced ies is a far generalization of the BKK theorem (see from a linear action of 푇 on ℂℙ푠. [Oko03,LM09,KK12a]). A toric degeneration is a “deformation” of a given vari- Theorem 3.2 (Okounkov, Lazarsfeld–Mustat¸ˇa,Kaveh– ety to a toric variety such that some useful intersection the- Khovanskii). With notation as above, oretic data, in particular the degree, are preserved under deg(푋) = 푛! vol(Δ). the deformation, hence enabling us to obtain some geometric information about the embedding 푋 ⊂ ℂℙ푠 from its degenerated toric variety. An important application of realizing degree as vol- We should point out that degenerations of curves is a ume of a convex body is that one can apply the cel- classical and very well-studied subject. ebrated Brunn–Minkowski inequality (which is an inequality about volumes of subsets in Euclidean space) to Newton–Okounkov bodies to obtain a simple proof of a deep fact, known as the Hodge inequality, about intersection numbers of hypersurfaces on varieties (see [Oko03,LM09,KK12a]). Figure 8. The family {((푥 ∶ 푦 ∶ 푧), 푡) ∣ 푦2푧 = 푥3 + 푡3푧3} ⊂ ℂℙ2 × ℂ Example 3.3. Let 푋 ⊂ ℂℙ2 be a plane algebraic curve de- gives a toric degeneration of the elliptic curve 푦2푧 = 푥3 + 푧3 to 2 3 fined by a homogeneous polynomial of degree 푑. Let 푣 be the singular toric variety 푦 푧 = 푥 . Topologically, a donut the order of vanishing at a smooth point in 푋. One shows, shape degenerates to a pinched sphere. The two circles around the arm and the hole of the donut contract to a point by direct computation or using Theorem 3.2, that Δ(푋, 푣) on the pinched sphere. is the line segment [0, 푑].

SEPTEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1121 Using standard methods from commutative algebra (namely, Rees algebra associated to a valuation) it can be shown that whenever the semigroup 푆 = 푆(푋, 푣) is finitely generated 푋 ⊂ ℂℙ푠 admits a toric degeneration to the toric 푠 variety 푋풜 ⊂ ℂℙ associated to a finite set 풜 of generators of 푆 [AND13]. To be precise we want 푆 to be generated in level 1. Remark 3.7. These toric degenerations can be used in symplectic geometry for constructions of full-dimensional Hamiltonian torus actions as well as to obtain general re- Figure 9. The Gröbner fan of the ideal (푦2푧 − 푥3 + 7푥푧2 − 2푧3). sults and constructions about symplectic ball embeddings The ray colored red is the only (full-dimensional) prime cone. from symplectic topology [HK15, Kav19]. a well-known result that the equivalence classes partition Remark 3.8. The Newton–Okounkov bodies of the flag ℚ푠+1 into relatively open rational polyhedral cones. This manifold have been computed by various mathemati- partition is usually referred to as the Gröbner fan of 퐼, and cians with respect to some geometric valuations. For we denote it by Σ(퐼). example, in [Kav15] the author gives valuations for We will say that a cone 퐶 ∈ Σ(퐼) is a prime cone if the which the resulting Newton–Okounkov bodies are the corresponding initial ideal in퐶(퐼) = in푤(퐼), ∀푤 ∈ 퐶, is string polytopes of Peter Littelmann, Arkady Berenstein, a prime ideal in ℂ[푥0, … , 푥푠]. One of the main results in and Andrei Zelevinsky, a generalization of the Gelfand– [KM19] establishes a correspondence between (full rank) Zetlin polytopes. Gelfand–Zetlin polytopes are in one- valuations on ℂ[푋] whose corresponding value semigroup to-one correspondence with irreducible representations is finitely generated and (full-dimensional) prime cones of GL(푛, ℂ). Moreover, the number of integral points in the Gröbner fan of 퐼. Thus to each such cone one can in a given Gelfand–Zetlin polytope is equal to the di- naturally associate a Newton–Okounkov body (which in mension of the irreducible representation it corresponds this case is in fact a polytope) Δ퐶. The following example to. Valentina Kiritchenko gave alternative valuations such is from [KM19, p. 300]. that the resulting Newton–Okounkov bodies are Feigin– Fourier–Littelmann–Vinberg polytopes. Also, polytopes Example 3.9. Consider the ideal 퐼 = (푦2푧−푥3 +7푥푧2 −2푧3) of Nakashima–Zelevinsky were realized as Newton– defining an elliptic curve 퐸 ⊂ ℂℙ2. The Gröbner fan of 퐼 Okounkov bodies of flag varieties by Naoki Fujita and lives in ℝ3. Since 퐼 is a homogeneous ideal, one sees that Hironori Oya. There are also recent interesting connec- every cone in the Gröbner fan is invariant under adding tions between Newton–Okounkov bodies and cluster alge- scalar multiples of the vector (1, 1, 1). Thus we can think of bras (see [RW19] as well as [KM19, p. 298] and references the Gröbner fan as living in ℝ3/⟨(1, 1, 1)⟩ ≅ ℝ2. It consists therein). of the seven cones (three 2-dimensional cones, three rays, and the origin) in Figure 9. One computes that only the 푋 ⊂ ℂℙ푠 Given an embedded projective variety , one ray colored red is a (full-dimensional) prime cone. is interested to know when there is a valuation 푣 on ℂ(푋) such that the corresponding value semigroup 푆 is finitely The above correspondence leads to the following ques- generated. In [KM19] the authors provide a criterion for tion: How does the Newton–Okounkov polytope change if we this in terms of tropical geometry and Gröbner theory. cross from one prime cone to an adjacent prime cone in the Gröb- With notation as before let 퐼 ⊂ ℂ[푥0, … , 푥푠] be the ho- ner fan? The preprint [EH19] gives the following answer: mogeneous ideal defining 푋 ⊂ ℂℙ푠. Let us recall the basic • notion of initial form of a polynomial from the Gröbner 푠+1 훼 basis theory. Take 푤 ∈ ℚ and let 푓(푥) = ∑ 푠+1 푐훼푥 훼∈ℤ≥0 (푓) be a polynomial. The initial form in푤 is defined to be 헉1 헉2 훼 the sum of terms 푐훼푥 where the dot product ⟨푤, 훼⟩ is minimum. More precisely, let 푚 = min{⟨푤, 훼⟩ ∣ 푐 ≠ 0}. Then 0 훼 •휉 훼 in푤(푓) = ∑ 푐훼푥 . ⟨푤,훼⟩=푚0 • The initial ideal of 퐼 with respect to the weight 푤 is the ideal Figure 10. Two Newton–Okounkov polytopes that project onto generated by the in (푓) for all 0 ≠ 푓 ∈ 퐼. the same polytope and such that the ﬁbers under the 푤 projection maps are the same length. Given a homogeneous ideal 퐼, let us say that two vec- 푠+1 tors 푤1, 푤2 ∈ ℚ are equivalent if in푤1 (퐼) = in푤2 (퐼). It is

1122 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 8 Theorem 3.10 (Wall-crossing, Escobar–Harada). Suppose [Kav19] Kiumars Kaveh, Toric degenerations and symplectic ge- that 퐶1, 퐶2 are two prime cones in Σ(퐼) that share a ometry of smooth projective varieties, J. Lond. Math. Soc. (2) codimension-1 face. There exist a polytope Δ of dim(Δ) + 1 = 99 (2019), no. 2, 377–402. MR3939260 dim(Δ ) = dim(Δ ) and two natural surjective projections [KK12a] Kiumars Kaveh and A. G. Khovanskii, Newton- 퐶1 퐶2 Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2) 176 (2012), no. 2, 헉1 헉2 Δ퐶1 ⟶ Δ ⟵ Δ퐶2 925–978. MR2950767 [KK12b] Kiumars Kaveh and Askold G. Khovanskii, Convex such that the fibers of 헉1 and 헉2 of any point 휉 ∈ Δ are 1- bodies associated to actions of reductive groups, Mosc. Math. J. dimensional polytopes of the same Euclidean length (up to a 12 (2012), no. 2, 369–396, 461. MR2978761 global constant). Moreover, there exists a piecewise linear bi- [KK14] Kiumars Kaveh and Askold Khovanskii, Convex bod- jection 향 ∶ Δ → Δ which makes the following diagram ies and multiplicities of ideals, Proc. Steklov Inst. Math. 286 퐶1 퐶2 (2014), no. 1, 268–284. Reprint of Tr. Mat. Inst. Steklova commute: 286 (2014), 291–307. MR3482603 Δ 향 Δ [KM19] Kiumars Kaveh and Christopher Manon, Khovan- 퐶1 퐶2 skii bases, higher rank valuations, and tropical geometry, SIAM J. Appl. Algebra Geom. 3 (2019), no. 2, 292–336. 헉1 헉2 MR3949692 [LM09] Robert Lazarsfeld and Mircea Mustat¸˘a, Convex bodies Δ associated to linear series, Ann. Sci. Ec.´ Norm. Sup´er. (4) 42 (2009), no. 5, 783–835. MR2571958 It was observed by Nathan Ilten and Christopher [MS15] Diane Maclagan and Bernd Sturmfels, Introduction Manon in 2017 that the geometric wall-crossing phenom- to tropical geometry, Graduate Studies in Mathematics, enon for Newton–Okounkov bodies, as described above, vol. 161, American Mathematical Society, Providence, RI, can also be derived from the theory of complexity-one T- 2015. MR3287221 varieties. [Oko03] Andrei Okounkov, Why would multiplicities be log- concave?, The orbit method in geometry and physics (Mar- References seille, 2000), 2003, pp. 329–347. MR1995384 [AND13] Dave Anderson, Okounkov bodies and toric de- [Oko96] Andrei Okounkov, Brunn-Minkowski inequality for generations, Math. Ann. 356 (2013), no. 3, 1183–1202. multiplicities, Invent. Math. 125 (1996), no. 3, 405–411. MR3063911 MR1400312 [Ati83] M. F. Atiyah, Angular momentum, convex polyhedra [RW19] K. Rietsch and L. Williams, Newton-Okounkov bodies, and algebraic geometry, Proc. Edinburgh Math. Soc. (2) 26 cluster duality, and mirror symmetry for Grassmannians, Duke (1983), no. 2, 121–133. MR705256 Math. J. 168 (2019), no. 18, 3437–3527. MR4034891 [CSZ15] Cesar Ceballos, Francisco Santos, and Günter M. Ziegler, Many nonequivalent realizations of the associahedron, Combinatorica 35 (2015), no. 5, 513–551. MR3437894 [Esc16] Laura Escobar, Brick manifolds and toric varieties of brick polytopes, Electron. J. Combin. 23 (2016), no. 2, Pa- per 2.25, 18. MR3512647 [EH19] Laura Escobar and Megumi Harada, Wall-crossing for Newton-Okounkov bodies and the tropical Grassmannian, arXiv:1912.04809, 2019. [Ful93] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures Laura Escobar Kiumars Kaveh in Geometry. MR1234037 Credits [HK15] Megumi Harada and Kiumars Kaveh, Integrable systems, toric degenerations and Okounkov bodies, Invent. Math. Figure 1 was taken from https://cudl.lib.cam.ac.uk 202 (2015), no. 3, 927–985. MR3425384 /view/MS-ADD-03977/18. Reproduced by kind permis- [Hov77] A. G. Hovanski˘ı, Newton polyhedra, and toroidal vari- sion of the Syndics of Cambridge University Library. eties, Funkcional. Anal. i Priloˇzen. 11 (1977), no. 4, 56–64, All other figures were created by the authors. 96. MR0476733 Photo of Laura Escobar is by Sean Garcia and is courtesy of [Hov78] A. G. Hovanski˘ı, Newton polyhedra, and the genus Washington University in St. Louis. of complete intersections, Funktsional. Anal. i Prilozhen. 12 Photo of Kiumars Kaveh is courtesy of the University of Pitts- (1978), no. 1, 51–61. MR487230 burgh. [Kav15] Kiumars Kaveh, Crystal bases and Newton-Okounkov bodies, Duke Math. J. 164 (2015), no. 13, 2461–2506. MR3405591

SEPTEMBER 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1123