VVARIATION OF STRATIFICATIONS FROM TORIC GIT
CHI-YU CHENG
Abstract.
When a reductive group acts on an algebraic variety, a linearizedample line bundle induces a stratification on the variety where the strata areordered by the degrees of instability. In this paper, we study variation ofstratifications coming from the group actions in the GIT quotient constructionfor projective toric varieties. Cox showed that each projective toric variety isa GIT quotient of an affine space by a diagonalizable group with respect tolinearizations that come from ample divisors on the toric variety. We providea sufficient conditions for two ample divisors to induce the same stratificationand formulate two types of walls in the ample cone that completely capturetwo kinds of variations. We also prove that the variation is intrinsic to theprimitive collections and the relations among ray generators of the fans.
Contents
1. Introduction 21.1. The main results 41.2. VSIT vs. VGIT 61.3. Outline of the paper 71.4. Acknowledgements 72. Some linear algebra 72.1. Notations 72.2. Linear programming 82.3. Distance to linear subspaces 93. Instability in invariant theory 113.1. Notations and conventions 123.2. Definition and a numerical criterion for instability 133.3. Numerical analysis of instability 143.4. Stratification induced by a character 153.5. The structure of a stratum 174. Recap on toric varieties 184.1. Set up 194.2. The ample cone 214.3. GIT quotient construction of projective toric varieties 215. Toric VSIT 245.1. Notations and conventions 255.2. Extensions to the ample cone 255.3. Wall and semi-chamber 275.4. The main results 315.5. Variation of stratification - an elementary example 375.6. Relations to the structure of the fan 416. The computer program 466.1. Computing the Picard group 47 a r X i v : . [ m a t h . AG ] F e b CHI-YU CHENG P at two points 65References 651. Introduction
Geometric Invariant Theory (abbreviated as GIT) was developed in [MFK94]by Mumford to construct quotients for group actions on algebraic varieties. Whilethere are standardized quotient constructions in various mathematical categoriesincluding the category of differentiable manifolds, forming quotients within thecategory of algebraic varieties is less immediate.The local pieces of an algebraic variety are affine varieties. An affine varietyis the spectrum of a ring, consisting of algebraic functions defined on the affinevariety. Constructing varieties that parametrize orbits is not immediate even forgroup actions on affine varieties. While it is natural to consider the spectrum ofthe subring of functions that are constant on the orbits, namely, the invariantfunctions, there might not be enough invariants to separate the orbits.Mumford realized some orbits are so exceptional, that they must be left out ofthe quotient. The notion of stability was then introduced by Mumford to specifyan invariant open subset, known as the semistable locus that allows for a goodquotient [Ses72]. Since then GIT has led to fruitful results in algebraic geometry,especially constructions of various moduli spaces. Examples include sequence oflinear subspaces [Mum63], representations of quivers [Kin94], vector bundles ona curve [Mum63], and more generally coherent sheaves on a projective variety[Ses67].GIT stability however, depends on the choice of a linearized line bundle.Variation of Geometric Invariant Theory (abbreviated as VGIT) quotients comingfrom different choices of linearized line bundles were well studied by [DH98] and[Tha96] in the 90’s. The main result is that when a reductive group G actson a normal projective variety over the field of complex numbers, the space of G -linearized ample line bundles has a finite wall and chamber decomposition suchthat(1) stability and the GIT quotient do not change inside each chamber,(2) the variation of GIT quotients after wall crossing is described by a flip.It is worthwhile pointing out that VGIT has applications to birational geometry.It not only provides examples of flips, but also realizes Mori theory as an instanceof VGIT. In [HK00] it was shown that if a projective variety is a Mori dreamspace, it is a GIT quotient of an affine variety by a torus. In this case the Morichambers correspond to VGIT chambers.Our work takes on the theme of variations. We study the variation of stratifi-cations in invariant theory (abbreviated as VSIT) that occurs in the GIT quotient ARIATION OF STRATIFICATIONS FROM TORIC GIT 3 construction of projective toric varieties. This is where instability in invarianttheory (abbreviated as IIT) and toric varieties meet.The field of toric varieties is famous for being a good testing ground in algebraicgeometry and toric varieties can be built from very concrete objects that are knownas fans. Moreover, there is a well understood dictionary translating combinatorialproperties of a fan to the geometric properties of its associated toric variety, givingthis field a rich amount of computability. The main driving force behind thispaper is the generous number of examples toric varieties offer (Section 7) and theinteresting connections many results in this paper have to the combinatorics ofthe fans (Section 5.6).IIT on the other hand, studies the points that are not semistable, namelythe unstable points. Although the unstable locus is discarded when one takesa GIT quotient, it possesses interesting and useful properties. For instance, let X be an affine variety equipped with an action by a reductive group G over thefield C of complex numbers. A character χ : G → C × then gives a linearizationof the trivial line bundle O X . The pioneering IIT work [Kem78] due to Kempfstates that every χ -unstable point is maximally destabilized by a one parametersubgroup of G in a numerical sense that comes from the Hilbert-Mumford criterion([MFK94],[Kin94]). In addition, such a one parameter subgroup when taken to beindivisible, is unique up to conjugacy by some parabolic subgroup of G .Therefore, a character χ of G induces the following decomposition(1.1) X = X ss ( χ ) ∪ (cid:0) (cid:91) λ S χ [ λ ] (cid:1) where X ss ( χ ) is the semistable locus with respect to χ and S χ [ λ ] consists of pointsin X that are maximally destabilized by some element in the conjugacy class [ λ ]of the indivisible one parameter subgroup λ in G . This decomposition actuallyinduces a stratification of X which we now define. Definition 1.1.
Let Y be a topological space. A finite collection of locally closedsubspaces { Y a | a ∈ A } forms a stratification of Y if Y is a disjoint union ofthe strata Y a and there is a strict partial order on the index set A such that ∂Y a (cid:48) ∩ Y a (cid:54) = ∅ only if a > a (cid:48) .In [Hes79], Hesselink prescribes to each piece in (1.1) a degree of instability andshowed that the collection of pieces, when strictly partially ordered by degrees ofinstability, forms a stratification of X . We shall refer to this stratification as thestratification of X induced by χ .We would like to point out that the stratification just introduced is not solelyaesthetic but has actual applications to obtaining certain topological invariants ofthe quotients. For instance, in [Kir84] Kirwan used Hesselink’s stratification toproduce an inductive formula for the equivariant Betti numbers of the semistableloci, which in good cases will be the Betti numbers of the quotients of nonsingularcomplex projective varieties.The stratifications that we are concerned with in this paper are those thatoccur from the GIT quotient construction of projective toric varieties. To get apicture of the stratifications under this setting, let Σ be a fan. We let X Σ be thetoric variety built from the fan, Cl( X Σ ) be its divisor class group, and Σ(1) bethe collection of rays in Σ. We also let C Σ(1) = Spec C [ x ρ | ρ ∈ Σ(1)]
CHI-YU CHENG be the affine space whose coordinates are indexed by rays in Σ. In [Cox95] and[CLS11], Cox showed that every projective toric variety X Σ is a GIT quotient of C Σ(1) by the diagonalizable group G := Hom Z (Cl( X Σ ) , C × )with respect to characters χ D that come from ample divisors D on X Σ . Moreover,the χ D -unstable locus ( C Σ(1) ) us ( χ D ) can be described by the vanishing loci ofcertain collections x ρ ’s, known as the primitive collections of the fan Σ (SeeDefinition 4.5). Specifically,(1.2) ( C Σ(1) ) us ( χ D ) = (cid:91) C V ( { X ρ | ρ ∈ C } )where the union is taken over primtive collections C of the fan.With these data we can think about stratifications of C Σ(1) and the varia-tion induced by different ample divisors on X Σ . Here is how we decide if thestratifications induced by two ample divisors are equivalent. Definition 1.2.
We say two stratifications { Y a | a ∈ A } and { Y b | b ∈ B } of atopological space Y are equivalent if there is a bijection Φ : A → B such that(1) Φ preserves strata. That is, Y Φ( a ) = Y a for all a ∈ A , and(2) Φ preserves order. That is, Φ( a ) > Φ( a (cid:48) ) if and only if a > a (cid:48) for all a, a (cid:48) ∈ A .With this notion we then say there is a type one variation between the strat-ifications of C Σ(1) induced by ample divisors D and D (cid:48) if there is no bijectionthat satisfies condition (1). We say there is a type two variation between thestratifications of C Σ(1) induced by ample divisors D and D (cid:48) if there is a bijectionthat satisfies condition (1), but not condition (2). Namely, there is a change oforder among the strata.It is important to point out that the variation studied in this paper is finerthan VGIT in the following sense. Equality (1.2) indicates that the semistablelocus stays constant among all ample divisors. Hence by choosing different ampledivisors, variation of stratifications only occurs in the fixed unstable locus. Weare now ready to state the main results of this paper.1.1. The main results.
The major task of this paper is to nail down at whichample divisors the stratification undergoes variations, and provide a sufficientcondition for two ample divisors to induce equivalent stratifications.
Theorem A (Proposition 5.12, Proposition 5.14, Theorem 5.18, Theorem 5.21,and Theorem 5.22) . Let X Σ be a projective toric variety. There are two typesof walls in the cone of ample divisors, called type one walls and type two wallsrespectively. The following properties hold for walls:(1) There are finitely many walls.(2) A type one (resp. type two) wall is a rational hyperplane (resp. homoge-neous quadratic hypersurface).(3) Type one walls capture type one variations in the following sense: Let D, D (cid:48) be two ample divisors on X Σ . Then the stratifications induced by D and D (cid:48) undergo a type one variation only if the line segment DD (cid:48) intersectsa type one wall properly in the ample cone. Namely, DD (cid:48) crosses a typeone wall. ARIATION OF STRATIFICATIONS FROM TORIC GIT 5 (4) Type two walls capture type two variations in the following sense: Let
D, D (cid:48) be two ample divisors on X Σ . If the stratifications induced by D and D (cid:48) undergo a type two variation and if the line segment DD (cid:48) intersectsno type one walls properly, then DD (cid:48) intersects a type two wall properlyin the ample cone. Namely, DD (cid:48) crosses a type two wall.Away from the walls, the ample cone is decomposed into semi-chambers such thatthe following properties hold:(1) There are finitely many semi-chambers.(2) A semi-chamber is a cone, possibly not convex.(3) The stratification stays constant in a semi-chamber in the following sense:If two ample divisors are in the same semi-chamber, then they induceequivalent stratifications of C Σ(1) . Next, we provide two results that have connections to the combinatorics ofthe fans. The first one is related to primitive relations for simplicial completetoric varieties (Definition 4.9). The significance of primitive relations is that forsimplicial projective toric varieties, its Mori cone can be generated by primitiverelations (see [Bat91]).The notion of primitive relations occurred to us in the context of Cox’s quotientconstruction for projective toric varieties. Equality (1.2) that describes the GITunstable locus was established in [CLS11] by considering the invariants withrespect to characters induced by ample divisors. We supply an alternative proofof (1.2) in Proposition 4.8, using the Hilbert-Mumford criterion, Theorem 3.3.Below is the theorem describing the connections between primitive relationsand the destabilizing one parameter subgroups we used in Proposition 4.8. Onefact to have in mind is that the one parameter subgroups of the group in Cox’squotient construction correspond to relations among the ray generators of the fan(Proposition 4.1).
Theorem B.
Let X Σ be a projective toric variety. For each primitive col-lection C of the fan, there is a one parameter subgroup λ ( C ) of the group G = Hom Z (Cl( X Σ ) , C × ) depending only on C , that fails the Hilbert-Mumfordcriterion for all points in V ( { x ρ | ρ ∈ C } ) and for all ample divisors. Moreover,if Σ is simplicial, − λ ( C ) is an integral positive multiple of the primitive relationof C . As an important application, the above theorem allows us to consider stratifi-cations induced by R -ample divisors as well in Section 5.2 .Finally, we proved that the variation of stratifications is intrinsic to the relationsamong the ray generators and the primitive collections of the fan. More precisely,we consider two complete fans Σ and Σ of the same dimension that admit abijection Ψ : Σ (1) → Σ (1)between the rays such that the following two properties hold:(1) Ψ( C ) is a primitive collection for Σ if and only if C is a primitive collectionfor Σ , and(2) for any { a ρ | ρ ∈ Σ (1) } ⊂ Z , (cid:80) ρ a ρ u ρ = 0 ⇔ (cid:80) ρ a ρ u Ψ( ρ ) = 0 where u ρ (resp. u Ψ( ρ ) ) stands for the ray generator of ρ (resp. Ψ( ρ )).In Section 5.6.1, we say such two complete fans Σ and Σ are amply equivalent . CHI-YU CHENG
Warning 1.3.
We note that ample equivalence between two fans does not resultin an isomorphism between the toric varieties associated to them. A quick exampleis to consider the complete fan Σ ⊂ R with ray generators (1 , , (0 , − , −
1) and the complete fan Σ ⊂ R with ray generators (2 , , (1 , − − , − and Σ are amply equivalent. However, Σ is a smooth fanbut Σ is not.If Σ and Σ are amply equivalent, then by Cox’s quotient construction and thefact that Σ (1) (cid:39) Σ (1), the affine spaces for the quotient constructions of X Σ and X Σ can be identified. Write C Σ(1) as the common affine space. Based onthe description (1.2) using primitive collections and the assumption that Σ andΣ are amply equivalent, the unstable loci discarded in the quotient constructionsfor both X Σ and X Σ can be identified in C Σ(1) . Write Z (Σ) ⊂ C Σ(1) as thecommon unstable locus.For i = 1 ,
2, let G i be the group Hom Z (Cl( X Σ i ) , C × ) that realizes X Σ i asthe quotient of C Σ(1) and ΓΓΓ( G i ) be the set of one parameter subgroups of G i .Since each G i is abelian, each ΓΓΓ( G i ) has a natural abelian group structure. Wethen prove in Proposition 5.25 and Proposition 5.28 that there is a Q -linearisomorphism ϕ : ΓΓΓ( G ) Q (cid:39) ΓΓΓ( G ) Q whose dual ϕ ∗ : Cl( X Σ ) Q → Cl( X Σ ) Q between the Q -divisor class groups restricts to a bijection between their Q -ampledivisors. The stratifications induced by Q -ample divisors on X Σ and X Σ arerelated by the following adjunction: Theorem C (Theorem 5.29) . For every Q -ample divisor D on X Σ , the assign-ment of the strata S ϕ ∗ ( χ D ) λ S χ D ϕ ( λ ) defined for all λ ∈ ΓΓΓ( G ) Q is an equivalence of stratifications of Z (Σ) induced by D and ϕ ∗ ( D ) . The adjunction also holds for R -ample divisors D on X Σ and λ ∈ ΓΓΓ( G ) R . VSIT vs. VGIT.
Suppose X is a variety endowed with an action by areductive group G over C . It is defined in VGIT that two G -linearized ample linebundles are GIT-equivalent if their semistable loci in X are the same. We canthink about a finer equivalence than GIT-equivalence. We define two G -linearizedample line bundles to be SIT-equivalent if they induce equivalent stratificationsof X . Since the stratifications induced by two SIT-equivalent line bundles musthave a common stratum of the least instability, which is the semistable locus,SIT-equivalence is finer than GIT-equivalence among G -linearized ample linebundles.While GIT-equivalence classes are well-understood by VGIT, very little isknown about the geometry of SIT-equivalence classes. For example, it is knownthat GIT-equivalence classes on X corresponds to relative interiors of a collectionof convex rational polyhedral cones that is named by [Res00] as a GIT fan forthe action of G on X . In Section 7.2, we will see that a semi-chamber does haveto be convex and its closure may not be a polyhedral cone. Moreover, in VGITdifferent chambers correspond to different GIT-equivalence classes. We will seean example in Section 7.1 where two different semi-chambers are contained in asingle SIT-equivalence class. This phenomenon also shows that wall crossing isnot sufficient for variations of stratifications but only necessary (Theorem A). ARIATION OF STRATIFICATIONS FROM TORIC GIT 7
Outline of the paper.
This paper is notation heavy. In view of this, westart Section 2, Section 3, Section 4, and Section 5 with conventions and notations.If at any point of a section the reader forgets what certain symbols stand for, thereader may consult the beginning of the section.Most of the analysis and computations in this paper come down to basic linearalgebra, especially linear programming. We therefore introduce results from linearprogramming that will be used later at the very beginning, Section 2.In Section 3, we then recall various important IIT theorems, including theHilbert Mumford criterion (Theorem 3.3), Kempf’s result (Theorem 3.6) aboutmaximally destablizing one parameter subgroups, and Hesselink’s stratificationinduced by a character (Theorem 3.9).In Section 4 we recall Cox’s GIT quotient construction of projective toricvarieties, a numerical criterion for a divisor to be ample (Theorem 4.2), and thethe description of the unstable locus using primitive collections (Proposition 4.8).Section 5 is the heart of the paper. We first extend the notions of stability andstratfications induced by ample divisors to those induced by R -ample divisorsin Section 5.2, allowing us to analyze VSIT in the entire ample cone. We thenintroduce the notion of walls (Definition 5.9) and semi-chambers (Definition 5.13) inthe ample cone at Section 5.3 and prove the main result Theorem A at Section 5.4.In Section 5.5 we supply an example where two types of walls and two typesof variations are present in the ample cone and where walls and semi-chamberscorresponds to SIT-equivalence classes. We then spend the rest of Section 5exploring the connections of VSIT to the combinatorics of the fan in Section 5.6where another main result Theorem C is established.Due to the computability offered by toric varieties, we wrote a computerprogram to generate a number of examples. Section 6 explains the functions andlogic of the program and in Section 7, we present some examples computed bythe program to address certain questions we have.1.4. Acknowledgements.
This paper is a part of my PhD thesis. I would liketo thank my PhD advisor Jarod Alper for many insightful directions and hispatience for guiding me through the revision of my PhD thesis. Many results inthis paper were also inspired by discussions with Dan Edidin, Matthew Satriano,and encouragements from Zinovy Reichstein and Jack Hall.2.
Some linear algebra
This section records all results from linear programming and linear algebranecessary for later discussions. Section 2.3 will be used when we discuss type twowalls at Section 5.3.2.1.
Notations.
Let V be a finite dimensional real vector space endowed withan inner product ( − , − ) : V × V → R . Let || − || : V → R be the induced normon V . The space V and any subspace W ⊂ V have the induced metric topology.Obviously the Pythagorean law holds: If ( v, w ) = 0, then(2.1) || v + w || = || v || + || w || . For any vector v ∈ V and any subspace W ⊂ V , we set Proj W v to be theprojection of v onto W along the orthogonal complement W ⊥ . Namely,Proj W v ∈ W and ( v − Proj W v, w ) = 0 for all w ∈ W. If f : V → R is a linear functional, we let CHI-YU CHENG • f ∗ ∈ V be the unique vector in V such that(2.2) f ( v ) = ( v, f ∗ ) for all v ∈ V. • H f be the hyperplane { v ∈ V | f ( v ) = 0 } , and • H + f be the half-space { v ∈ V | f ( v ) ≥ } . A polyhedral cone in V is an intersection of finitely many half-spaces. Let σ bea polyhedral cone. The hyperplane H f is a supporting hyperplane for σ if σ ⊂ H + f .A face F of σ is H f ∩ σ for some supporting hyperplane H f of σ . Notation-wisewe write F (cid:22) σ. Obviously any face of a polyhedral cone is a polyhedral cone. The relative interior of a polyhedral cone σ is the interior of σ in the subspace spanned by σ and iswritten as Relint( σ ) . We note without proof that for a polyhedral cone σ , its collection of faces is finiteand(2.3) σ = (cid:71) F (cid:22) σ Relint( F ) . Finally, for a linear functional f : V → R and a polyhedral cone σ ⊂ V , wedefine the following finite set of vectorsΛ fσ = {− Proj
Sp( F ) f ∗ (cid:12)(cid:12) − Proj
Sp( F ) f ∗ ∈ σ, F (cid:22) σ } where Sp( F ) is the linear subspace spanned by F .2.2. Linear programming.
This section presents results from linear program-ming that will be used extensively throughout the paper, especially Corollary 2.2.Specifically Kempf’s theorem Theorem 3.6, the structure of strata described byTheorem 3.17, and the computation of stratifications, the formulation of wallslater at Section 5 all depend on results from this section.The story begins with the following theorem whose proof is a direct modificationof the proof presented in [Kem78] and is omitted.
Theorem 2.1.
Let S ⊂ V be the unit sphere and σ ⊂ V be a polyhedral cone.Suppose f : V → R is a linear functional that assumes a negative value at somepoint v ∈ σ . Then f attains the relative minimum on S ∩ σ at a unique point s .In this case, if F (cid:22) σ is the face of σ such that s ∈ Relint( F ) , then s = − Proj
Sp( F ) f ∗ || Proj
Sp( F ) f ∗ || . Corollary 2.2.
Let S ⊂ V be the unit sphere and σ ⊂ V be a polyhedral cone.Suppose f : V → R is a linear functional that attains a negative value at somepoint v ∈ σ . Let s ∈ S ∩ σ be the unique point where f achieves the relativeminimum on S ∩ σ . Then there is a unique vector v ∈ Λ fσ such that s = v || v || . In this case || v || ≥ || v (cid:48) || for all v (cid:48) ∈ Λ fσ and f ( s ) = −|| v || . ARIATION OF STRATIFICATIONS FROM TORIC GIT 9
Proof.
For any F (cid:22) σ , we have f (cid:18) − Proj
Sp( F ) f ∗ || Proj
Sp( F ) f ∗ || (cid:19) = ( f ∗ , − Proj
Sp( F ) f ∗ ) || Proj
Sp( F ) f ∗ || = (Proj Sp( F ) f ∗ , − Proj
Sp( F ) f ∗ ) || Proj
Sp( F ) f ∗ || = − || Proj
Sp( F ) f ∗ || || Proj
Sp( F ) f ∗ || = −|| Proj
Sp( F ) f ∗ || . Hence any v ∈ Λ fσ that has the longest length will achieve the relative minimumof f on σ by Theorem 2.1. The statement that there is a unique longest v in Λ fσ follows from the uniqueness of s . (cid:3) Let f : V → R be a linear functional that takes a negative value on σ . Wesee that Corollary 2.2 reduces the search range for s ∈ σ ∩ S where f attains therelative minimum to the finite set Λ fσ . We will use Corollary 2.2 repeatedly in thecontext of toric GIT later at Section 5.We now specialize Theorem 2.1 with some rationality assumptions. Corollary 2.3below is the basis of Kempf’s theorem Theorem 3.6. For this we fix a lattice M ⊂ V such that M ⊗ Z R = V . A lattice point m ∈ M is indivisible if m = N · m (cid:48) for some N ∈ N and m (cid:48) ∈ M implies N = 1. A rational polyhedral cone (withrespect to M) is a polyhedral cone whose supporting hyperplanes are define bylinear functionals that take integral values on M . Corollary 2.3. M ⊂ V be a lattice with M ⊗ Z R = V , σ ⊂ V be a rationalpolyhedral cone and S ⊂ V be the unit sphere. Suppose the inner product ( − , − ) takes integral values on M × M and f : V → R is a linear functional that takesintegral values on M . If f assumes a negative value on σ and f attains the relativeminimum at s on S ∩ σ , then the ray R > · s contains a unique nonzero indivisiblelattice point in M .Proof. It is easy to check that the rationality assumptions imposed here implythat the ray R > · ( − Proj
Sp( F ) f ∗ ) intersects M away from 0 for each face F (cid:22) σ with Proj Sp( F ) f ∗ (cid:54) = 0. Now apply the description of s in Theorem 2.1. (cid:3) Below is an elementary observation that will be used later in Section 5.2 toformulate stratifications induced by R -ample divisors on a projective toric variety. Lemma 2.4.
Let v ∈ V and W , W be two subspaces of V . Then the followingtwo conditions for Proj W v and Proj W v are equivalent:(1) Proj W v and Proj W v are linearly dependent.(2) Proj W v = Proj W v ,In this case, we actually have Proj W v = Proj W ∩ W v = Proj W v Proof.
Condition (2) obviously implies condition (1). Let us show (1) implies (2).For this, condition (1) implies both Proj W v and Proj W v are in W ∩ W . SinceProj W v ∈ W ∩ W , we haveProj W v = Proj W ∩ W (Proj W v ) = Proj W ∩ W v. Similarly, we have Proj W v = Proj W ∩ W v . The lemma is proved. (cid:3) Distance to linear subspaces.
This section is intended to understand thestructure of type two walls described by Proposition 5.12. However, readers donot need the knowledge from this section to understand the definition of typetwo walls given at Section 5.3. Accordingly readers may return to this section ifneeded.
It turns out that type two walls defined in Section 5.3 corresponds to collectionsof vectors that solve an equi-distance problem which we describe in this section.The main result Proposition 2.6 of this section will be translated into Proposi-tion 5.12 to describe the structure of type two walls for simplicial projective toricvarieties.Let v be a vector in V and W ⊂ V be a subspace. We define the distance from v to W to be dist( v, W ) = || v − Proj W v || . Our interest is to describe the set of vectors { v ∈ V | dist( v, W ) = dist( v, W ) } that are equi-distant to a pair of subspaces W , W . First note that the definitionof dist( v, W ) makes sense for if w ∈ W is a vector, then || v − w || = || v − Proj W v + Proj W v − w || = (cid:112) || v − Proj W v || + || Proj W v − w || ≥ || v − Proj W v || by the Pythagorean law Equation (2.1). Namely, Proj W v is closest to v amongall vectors in W .There is an explicit way to compute dist( v, W ). Let B = { f ∗ , . . . , f ∗ (cid:15) } be anorthonormal basis for W ⊥ . Equivalently, W is cut out by the linear functionals { f , . . . , f (cid:15) } corresponding to B . Then v − Proj W v = Proj W ⊥ v = (cid:15) (cid:88) i =1 f i ( v ) · f ∗ i . Therefore, dist( v, W ) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:15) (cid:88) i =1 f i ( v ) . Now if W , W are two subspaces of V , let • { f ∗ , . . . , f ∗ (cid:15) } be a basis of W ⊥ ∩ W ⊥ , • A = { f ∗ , . . . , f ∗ (cid:15) , g ∗ , . . . , g ∗ m } be a basis of W ⊥ , and • B = { f ∗ , . . . , f ∗ (cid:15) , h ∗ , . . . , h ∗ n } be a basis of W ⊥ .Applying the Gram Schmidt process, we may assume both A , B are orthonormal.With these it follows that(2.4) dist( v, W ) = dist( v, W ) ⇔ m (cid:88) i =1 g i ( v ) = n (cid:88) j =1 h j ( v ) Moreover, it follows from the construction that A ∪ B is a basis of W ⊥ + W ⊥ soit can be extended to a full basis C of V . With the coordinate induced by thebasis C , Equation (2.4) looks like(2.5) x + · · · + x m = y + · · · + y n where m and n are codimensions of W and W in W + W respectively.We now introduce some terminologies from differential geometry to describe thestructure of the set of equi-distant vectors. Let F : N → M be a C ∞ map betweenreal manifolds. We say c ∈ M is a regular value of F if either c is not in theimage of F or at every point p ∈ F − ( c ), the differential (d F ) p : T p N → T F ( p ) M is surjective. The inverse image F − ( c ) of a regular value c is called a regular levelset . We now recall the regular level set theorem from differential geometry: ARIATION OF STRATIFICATIONS FROM TORIC GIT 11
Theorem 2.5 (Regular level set theorem) . Let F : N → M be a C ∞ map ofmanifolds, with dim N = n and dim M = m . Then a nonempty regular level set F − ( c ) , where c ∈ M , is a regular submanifold of N of dimension equal to n − m . We may now describe the equi-distant collection in V . Proposition 2.6.
Let W and W be two sub-spaces of V and let Z be the collec-tion of points v ∈ V that are equi-distant to W and W . If there is a containment,say W ⊂ W , then Z is the linear subspace W . If there is no containmentbetween W and W , then Z is a regular submanifold of V of codimension 1 awayfrom a subspace of codimension at least 2.Proof. The case that there is a containment between W and W is clear. If thereare no containments between W and W , then W + W properly contains W and W . By Equation (2.5), Z is defined by x + · · · + x m − y − · · · − y n = 0 withboth m, n >
0. Let F : V → R be the C ∞ function x + · · · + x m − y − · · · − y n .Let V (cid:48) ⊂ V be the subspace V ( x , · · · , x m , y , · · · , y n ). Then 0 ∈ R is a regularvalue for the restriction F : V − V (cid:48) → R and F − (0) is obviously a nonemptyregular level set. By the regular level set theorem, Theorem 2.5, Z − V (cid:48) is aregular submanifold of codimension 1. (cid:3) Instability in invariant theory
In this section we recall some fundamental results from IIT. This field isrelatively old and has a well-established literature. A list of standard referencesto this subject may include [Kem78], [Nes79], [Hes79], and [Kir84].IIT originates from Mumford’s numerical criterion presented in [MFK94] thatreduces testing stability for arbitrary reductive group actions to one dimensionaltorus actions given by one parameter subgroups. Mumford’s conjecture on theexistence of the maximally destabilizing one parameter subgroups that fail thenumerical criterion was proved by Kempf in his famous paper [Kem78]. Beyondexistence Kempf actually established the uniqueness of maximally destabilizingone parameter subgroups up to conjugacy by some parabolic subgroup. Hesselinkthen showed in [Hes79] that the unstable locus is stratified by the conjugacyclasses of Kempf’s maximally destabilizing one parameter subgroups.We will first recall Mumford’s definition of stability (Definition 3.2) and hisnumerical criterion (Theorem 3.3) in Section 3.2. We then make precise themeaning of maxiamally destabilizing one parameter subgroups (Definition 3.5),and present Kempf’s theorem (Theorem 3.6) at Section 3.3. With these we wouldbe able to recall Hesselink’s stratification induced by a character (Theorem 3.9) anddefine variations of stratfications induced by different characters (Definition 3.10,Definition 3.11). Finally, we describe the structure of a stratum (Theorem 3.17)at Section 3.5.Instead of supplying IIT results in full generality, we will focus on finitedimensional representations of a diagonalizable group G as this is the setting fortoric GIT. In this case the set of one parameter subgroups of G has a group structureof a lattice and because G is abelian, the conjugacy class of a one parametersubgroup λ is just the singleton { λ } . Hence there would be no ambiguity givenby conjugacy about the uniqueness of the maximally destabilizing one parametersubgroups. Notations and conventions.
Throughout the section G is a diagonalizablealgebraic group over the field C of complex numbers and X = A n C is a repre-sentation of G . We write C [ G ] and C [ X ] as the coordinate rings of G and X respectively.Letting G m = Spec C [ t ] t be the one dimensional torus over C , we set χχχ ( G ) := { χ : G → G m } to be the group of characters of G andΓΓΓ( G ) := { λ : G m → G } to be the set of one parameter subgroups of G . Since G is abelian, the set ΓΓΓ( G )has a natural abelian group structure. We set for χχχ ( G ) the vector spaces χχχ ( G ) Q := χχχ ( G ) ⊗ Z Q , χχχ ( G ) R := χχχ ( G ) ⊗ Z R and similarly for ΓΓΓ( G ).Let (cid:104)− , −(cid:105) : χχχ ( G ) × ΓΓΓ( G ) → Z be the pairing defined by the formula χ ( λ ( t )) = t (cid:104) χ,λ (cid:105) for all t ∈ G m . Remark 3.1. If G = ( G m ) r = { ( t , . . . , t r ) | t i ∈ G m } is a torus, ΓΓΓ( G ) can beidentified as Z r where each ( b , . . . , b r ) ∈ Z r induces a one parameter subgroup λ : G m → G given by t (cid:55)→ ( t b , . . . , t b r ) . Likewise χχχ ( G ) is identified as Z r where each ( a , . . . , a r ) ∈ Z r defines a character χ : G → G m given by ( t , . . . , t r ) (cid:55)→ t a · · · t a r r . One easily computes that (cid:104) χ, λ (cid:105) = (cid:80) i a i b i . Hence the natural pairing (cid:104)− , −(cid:105) is aperfect pairing.If G is diagonalizable, then G (cid:39) ( G m ) r × F for some finite group F . In this case χχχ ( G ) (cid:39) Z r ⊕ P where P is a finite abeliangroup and ΓΓΓ( G ) (cid:39) Z r . The pairing (cid:104)− , −(cid:105) when extended over Q or R , is aperfect pairing.Since G is diagonalizable, we assume the action of G on X is diagonalized.Namely, we fix n characters χ , . . . , χ n of G so that for every g ∈ G and x =( x , . . . , x n ) ∈ X , we have g · x = ( χ ( g ) · x , . . . , χ n ( g ) · x n ) . Letting [ n ] = { , . . . , n } , we define for any subset S ⊂ [ n ] the following polyhe-dral cone in ΓΓΓ( G ) R that is rational with respect to the lattice ΓΓΓ( G ): σ S = { v ∈ ΓΓΓ( G ) R |(cid:104) χ i , v (cid:105) ≥ i ∈ S } . For x = ( x , . . . , x n ) ∈ X , we define S x = { i ∈ [ n ] | x i (cid:54) = 0 } as the states of x and the following polyhedral cone σ x := σ S x . ARIATION OF STRATIFICATIONS FROM TORIC GIT 13
Given a one parameter subgroup λ ∈ ΓΓΓ( G ), we define the subset X λ = { x ∈ X | lim t → λ ( t ) · x exists } , which is the following linear subspace of X : { ( x , . . . , x n ) ∈ X | x i = 0 if (cid:104) χ i , λ (cid:105) < } . Finally, a norm on ΓΓΓ( G ) is a real valued function || − || : ΓΓΓ( G ) → R such thatthere is an inner product ( − , − ) on ΓΓΓ( G ) R × ΓΓΓ( G ) R → R that takes integralvalues on ΓΓΓ( G ) × ΓΓΓ( G ), and for any λ ∈ ΓΓΓ( G ), we have || λ || = ( λ, λ ) / . Note that a norm always exists. One can either choose an isomorphism ΓΓΓ( G ) (cid:39) Z r and take the standard norm, or choose an embedding G (cid:44) → ( C × ) s , and restrictthe standard norm to ΓΓΓ( G ) via the inclusion ΓΓΓ( G ) (cid:44) → ΓΓΓ(( C × ) s ).We fix a norm || − || on ΓΓΓ( G ). Moreover, abusing the notations, we also write (cid:104)− , −(cid:105) and || − || for their extensions over Q or R .3.2. Definition and a numerical criterion for instability.
Mumford’s sta-bility condition depends on the choice of a linearized line bundle. In the case ofgroup action on an affine space, linearizations of the trivial line bundle correspondto characters of the group.
Definition 3.2.
Let χ : G → G m be a character. Let(1) ˆ σ : C [ X ] → C [ G ] ⊗ C C [ X ] be the co-action, and(2) χ (cid:93) : C [ t ] t → C [ G ] be the map that corresponds to χ .An element f ∈ C [ X ] is χ - invariant of weight d if ˆ σ ( f ) = χ (cid:93) ( t ) d ⊗ f . We saya point x ∈ X is χ - semistable if there is a χ -invariant f of positive weight suchthat f ( x ) (cid:54) = 0. We say a point x ∈ X is χ - unstable if x is not χ -semistable. Wewrite X ss ( χ ) as the set of χ -semistable points and X us ( χ ) as the complement X − X ss ( χ ) . It follows from the definition that X ss ( χ ) is a G -invariant open subvariety andthat X us ( χ ) is a G -invariant closed subvariety. Moreover, X ss ( χ ) = X ss ( χ d ) forany d > X by the group G with respect to χ : Let C [ X ] χ,d be the space of χ -invariant elements of weight d . The space ⊕ d ≥ C [ X ] χ,d has a natural graded ring structure. We define X// χ G := Proj( ⊕ d ≥ C [ X ] χ,d ) . Then there is a map X ss ( χ ) → X// χ G that is constant on G -orbits, submersiveand induces a bijection between points in X// χ G and closed orbits in X ss ( χ ) (see[MFK94] for more detail). Moreover, X// χ G is a quasi-projective variety that isknown as the GIT quotient of X by G with respect to χ .An alternative way to test stability is to restrict the group action to oneparameter subgroups. Let λ be a one parameter subgroup of G . We say lim t → λ ( t ) · x exists if the domain of the map λ x : G m → X defined by t (cid:55)→ λ ( t ) · x can beextended to A C . Theorem 3.3. (Hilbert-Mumford criterion. [MFK94] , [Kin94] ) A point x ∈ X is χ -semistable if and only if for each one parameter subgroup λ : G m → G suchthat the limit lim t → λ ( t ) · x exists, we have (cid:104) χ, λ (cid:105) ≥ . Example 3.4.
Let the one dimensional torus G m acts on A C by t · ( x, y ) = ( tx, t − y ) for all t ∈ G m and for all ( x, y ) ∈ A C . Let χ be the identity id : G m → G m . We will check χ -semistability with χ -invariants and then with the numerical criterion Theorem 3.3.To begin with, an element f ∈ C [ x, y ] = C [ A C ] is χ -invariant of weight d if andonly if f ( tx, t − y ) = t d · f ( x, y ) . This is equivalent to saying that f ∈ x d · C [ xy ].In particular, f is divisible by x d . This shows that ( A C ) ss ( χ ) ⊂ D ( x ). Conversely, x ∈ C [ x, y ] is χ -invariant of weight 1. It follows that ( A C ) ss ( χ ) ⊃ D ( x ). Inconclusion, we get ( A C ) ss ( χ ) = D ( x ).To test stability with Theorem 3.3, let λ : G m → G m be a one parametersubgroup given by t (cid:55)→ t a and p = ( x, y ) be a point in A C . Suppose lim t → λ ( t ) · p =( t a x, t − a y ) exists. If x (cid:54) = 0, we have a = (cid:104) χ, λ (cid:105) ≥
0. We see that D ( x ) ⊂ ( A C ) ss ( χ ).Conversely, if x = 0, then the one parameter subgroup given by a = − p but (cid:104) χ, λ (cid:105) = − <
0. This shows that V ( x ) ⊂ ( A C ) us ( χ ). In conclusion, wehave ( A C ) ss ( χ ) = D ( x ) as was shown earlier.One also easily checks that ( A C ) ss ( − χ ) = D ( y ), indicating the dependence ofstability of a point on the choice of linearizations.3.3. Numerical analysis of instability.
In this section we are going to makeprecise in Definition 3.5 on what it means for a one parameter subgroup to be amaximally destabilizing one, then recall Kempf’s main theorem Theorem 3.6.For a point x ∈ X , set C x = { λ ∈ ΓΓΓ( G ) | lim t → λ ( t ) · x exits } . According to the numerical criterion, Theorem 3.3, x ∈ X us ( χ ) if and only if thereis a one parameter subgroup λ such that(1) λ ∈ C x , and(2) (cid:104) χ, λ (cid:105) < x ∈ X us ( χ ), it is natural to ask if there is a one parametersubgroup that contributes to the highest instability measured by the negativequantities (cid:104) χ, λ (cid:105) among all λ ∈ C x . An immediate problem is that (cid:104) χ, λ N (cid:105) = N · (cid:104) χ, λ (cid:105) for any λ ∈ ΓΓΓ( G ) and N ∈ N . To get rid of the dependency on multiplesof one parameter subgroups, we divide the function (cid:104) χ, −(cid:105) : ΓΓΓ( G ) → Z by thenorm || − || on ΓΓΓ( G ). For x ∈ X us ( χ ), we set M χ ( x ) = inf λ ∈ C x \{ } (cid:104) χ, λ (cid:105)|| λ || . Definition 3.5.
We say a one parameter subgroup λ is χ - adapted to x if (cid:104) χ,λ (cid:105)|| λ || = M χ ( x ). Moreover, we say λ is χ - adapted to a subset S ⊂ X us ( χ ) if λ is χ -adaptedto every point in S. Note that it is not immediate that M χ ( x ) is finite. Even so it is not immediatethat the infimum is attained by a one parameter subgroup from C x . The followingTheorem 3.6, a special case of a theorem due to Kempf, resolves these issues. Theorem 3.6 (Kempf, [Kem78]) . Let χ : G → G m be a character. For an x ∈ X us ( χ ) , we have(1) the value M χ ( x ) is finite,(2) the function M χ ( − ) : X us ( χ ) → R assumes finitely many values, ARIATION OF STRATIFICATIONS FROM TORIC GIT 15 (3) there is a unique indivisible one parameter subgroup λ χ,x that is χ -adaptedto x ,(4) for any g ∈ G , λ χ,g · x = λ χ,x , and in particular,(5) M χ ( − ) is constant on G -orbits. Namely, for each g ∈ G and x ∈ X us ( χ ) ,we have M χ ( x ) = M χ ( g · x ) . Theorem 3.6 is a consequence of Corollary 2.3. The connection is made by the
Lemma 3.7.
For each x ∈ X , C x ⊗ Z R ⊂ ΓΓΓ( G ) R is the rational polyhedral cone σ x . Equivalently, the set of lattice points of σ x is exactly C x .Proof. Recall that the action is given by g · ( x , . . . , x n ) = ( χ ( g ) · x , . . . , χ n ( g ) · x n )for all g ∈ G and ( x , . . . , x n ) ∈ X . If λ : G m → G is a one parameter subgroupand if t ∈ G m , we get λ ( t ) · ( x , . . . , x n ) = ( t (cid:104) χ ,λ (cid:105) x , . . . , t (cid:104) χ n ,λ (cid:105) x n ) . Hence lim t → λ ( t ) · x exists ⇔ (cid:104) χ i , λ (cid:105) ≥ i with x i (cid:54) = 0 ⇔ λ ∈ σ x . (cid:3) We now demonstrate how Theorem 3.6 can be proved.
Proof.
Let f ( − ) := (cid:104) χ, −(cid:105) : ΓΓΓ( G ) R → R be the linear functional. The assumption that x is χ -unstable implies that f takesa negative value on C x and therefore on σ x by Lemma 3.7. Corollary 2.3 thenimplies that M χ ( x ) = inf λ ∈ σ x \{ } f ( λ ) || λ || , and that M χ ( χ ) is attained at a unique indivisible one parameter subgroup λ χ,x .This proves statement (1) and (3). Statement (2) follows from the fact that thereare only finitely many σ x as x runs through X us ( χ ). For statements (4), simplynote that σ x = σ g · x for all g ∈ G . (cid:3) Stratification induced by a character.
In this section we recall Hes-selink’s theorem Theorem 3.9 and define stratifications (Definition 3.10), theequivalences and variations among them on a topological space (Definition 3.11)in preparation of VSIT at Section 5.Let χ ∈ χχχ ( G ) and λ ∈ ΓΓΓ( G ). We define the following subset of XS χλ = { x ∈ X us ( χ ) | λ χ,x = λ } and the subset of ΓΓΓ( G ) Λ χ = { λ χ,x | x ∈ X us ( χ ) } . We point out that Λ χ is a finite set. This follows from the discussion we had atthe end of the previous section that each λ χ,x comes from minimizing the function (cid:104) χ, −(cid:105)||−|| : ΓΓΓ( G ) R → R on σ x and that there are only finitely many σ x as x runsthrough X us ( χ ) . With these notations and Theorem 3.6, we have a decomposition X = X ss ( χ ) ∪ ( (cid:91) λ ∈ Λ χ S χλ ) . Let us define a strict partial ordering on the set Λ χ by setting λ > λ (cid:48) if (cid:104) χ, λ (cid:105)|| λ || < (cid:104) χ, λ (cid:48) (cid:105)|| λ (cid:48) || . For the convenience of this paper, we also define the same strict partial orderingon the collection { S χλ } λ ∈ Λ χ by S χλ > S χλ (cid:48) ⇔ λ > λ (cid:48) . Remark 3.8.
The reason why there is a flip of inequalities defining the strictpartial ordering on Λ χ is that (cid:104) χ,λ (cid:105)|| λ || < (cid:104) χ,λ (cid:48) (cid:105)|| λ (cid:48) || implies the left hand side is morenegative. Namely, the stratum S χλ contains points that are higher in instabilitymeasured by the function (cid:104) χ, −(cid:105)||−|| . The following theorem due to Hesselink states:
Theorem 3.9. (Hesselink, [Hes79] ) Let χ be a character of G. Then X = X ss ( χ ) ∪ ( (cid:91) λ S χλ ) is a finite disjoint union of G -invariant, locally closed subvarieties of X . Moreover, S χλ ∩ ∂S χλ (cid:48) (cid:54) = ∅ only if λ > λ (cid:48) . Proof.
That each S χλ is G -invariant follows from statement (4) of Theorem 3.6.Note that the last statement of the theorem we are proving implies for any λ ∈ Λ χ ,we have S χλ = S χλ \ (cid:91) λ (cid:48)(cid:48) >λ S χλ (cid:48)(cid:48) , giving us the locally closedness for each stratum. Hence it remains to show thelast statement. To do this, let z ∈ S χλ ∩ ∂S χλ (cid:48) ⊂ S χλ (cid:48) . Since X λ (cid:48) is closed andcontains S χλ (cid:48) , we have z ∈ X λ (cid:48) . Namely, lim t → λ (cid:48) ( t ) · z exists. Since λ (cid:54) = λ (cid:48) , it mustbe the case that (cid:104) χ,λ (cid:105)|| λ || < (cid:104) χ,λ (cid:48) (cid:105)|| λ (cid:48) || . (cid:3) The decomposition in Theorem 3.9 induces a stratification of X , which we nowdefine. Definition 3.10.
Let Y be a topological space. A finite collection of locallyclosed subspaces { Y a | a ∈ A } forms a stratification of Y if Y is a disjoint unionof the strata Y a and there is a strict partial order on the index set A such that ∂Y a (cid:48) ∩ Y a (cid:54) = ∅ only if a > a (cid:48) .For each character χ , we index the semistable locus X ss ( χ ) by the trivial oneparameter subgroup e and assign e the new lowest order 0 (as in zero instability)in the set Λ χ (cid:116) { e } . Doing so makes the decomposition of X in Theorem 3.9 astratification. We refer to the stratification as the stratification induced by χ andeach stratum as a χ - stratum . Obviously the stratification depends on the choiceof a character. To compare stratifications, we make the following Definition 3.11.
We say two stratifications { Y a | a ∈ A } and { Y b | b ∈ B } of atopological space Y are equivalent if there is a bijection Φ : A → B such that(1) Φ preserves strata. That is, Y Φ( a ) = Y a for all a ∈ A , and(2) Φ preserves order. That is, Φ( a ) > Φ( a (cid:48) ) if and only if a > a (cid:48) for all a, a (cid:48) ∈ A . ARIATION OF STRATIFICATIONS FROM TORIC GIT 17
Otherwise, we say(1) there is a type one variation between { Y a | a ∈ A } and { Y b | b ∈ B } ifthere is no bijection between A and B that satisfies condition (1), or(2) there is a type two variation between { Y a | a ∈ A } and { Y b | b ∈ B } ifthere is a bijection between A and B that satisfies condition (1), but notcondition (2). Definition 3.12.
We say two characters χ and χ of G are SIT - equivalent withrespect to the action of G on X if χ and χ induce equivalent stratifications of X .3.5. The structure of a stratum.
In this section, we prove in Theorem 3.17that for a character χ ∈ χχχ ( G ) and for each λ ∈ Λ χ , the closure S χλ of the stratum S χλ is the subspace X λ , and S χλ is obtained by removing a union of subspaces from S χλ . This implies each stratum is irreducible and smooth. Theorem 3.17 is alsogoing to be applied to Theorem 5.31 for the description of strata in the toric GITsetting. We first introduce some notations. • If S ⊂ [ n ], we let – V ( S ) = V ( { x i | i ∈ S } ) ⊂ X ; – D ( S ) = D ( (cid:81) i ∈ S x i ) ⊂ X ; – L ( S ) = D ( S ) ∩ V ([ n ] − S );and if A ⊂ C ⊂ [ n ], we set • L ( A ; C ) = ∪ B L ( B ) where the union is over all B with A ⊂ B ⊂ C .Note that L ( A ; C ) = V ([ n ] − C ) ∩ D ( A ) so all subsets of X listed above areirreducible locally closed G -invariant subvarieties. With these notations, we havethe immediate observation: Proposition 3.13.
Let χ ∈ χχχ ( G ) be a character and λ ∈ Λ χ . The followingstatements for a point x ∈ X are equivalent:(1) x ∈ S χλ ,(2) the function (cid:104) χ, −(cid:105)||−|| attains negative relative minimum on σ x at λ , and(3) L ( S x ) ⊂ S χλ .Proof. That statement (1) is equivalent to statement (2) follows from the proofof Theorem 3.6. To prove statement (1) implies statement (3), simply note that S y = S x so that σ y = σ x for all y ∈ L ( S x ). Now the equivalence of statement (1)and statement (2) implies y ∈ S χλ for all y ∈ L ( S x ) if x ∈ S χλ . Finally statement (3)implies statement (1) because x ∈ L ( S x ). Therefore, statement (1) is equivalentto statement (3). (cid:3) We obtain the first preliminary observation of a stratum:
Corollary 3.14.
Let χ ∈ χχχ ( G ) be a character. Then for each λ ∈ Λ χ , the stratum S χλ is a finite disjoint union of G -invariant locally closed subvarieties of the form L ( S ) for some S ⊂ [ n ] . Next, we prove that among all subsets S ⊂ [ n ] such that L ( S ) ⊂ S χλ , there is aunique maximal one. Lemma 3.15.
Let χ ∈ χχχ ( G ) be a character. For every λ ∈ Λ χ , there is a uniquemaximal subset M ⊂ [ n ] with respect to the property that L ( M ) ⊂ S χλ . Moreover, M = { i ∈ [ n ] |(cid:104) χ i , λ (cid:105) ≥ } . In particular, V ([ n ] − M ) = X λ . Proof.
Let M = { i ∈ [ n ] |(cid:104) χ i , λ (cid:105) ≥ } . Let S ⊂ [ n ] be a subset such that L ( S ) ⊂ S χλ .We need to show that(1) S ⊂ M , and(2) L ( M ) ⊂ S χλ .For statement (1), since S χλ ⊂ X λ , we have L ( S ) ⊂ X λ = V ([ n ] − M ). This implies S ⊂ M , justifying statement (1). This in turn implies σ M ⊂ σ S . For statement(2), note that λ is by definition in σ M . In sum, we obtain λ ∈ σ M ⊂ σ S . By Proposition 3.13, the function (cid:104) χ, −(cid:105)||−|| attains relative minimum on σ S at λ .Therefore, the function (cid:104) χ, −(cid:105)||−|| also attains relative minimum on σ M at λ . Thisproves statement (2). (cid:3) We also note that a stratum has the following absorption property:
Lemma 3.16.
Let χ ∈ χχχ ( G ) be a character and λ ∈ Λ χ . Then if C ⊂ A ⊂ [ n ] is a pair of subsets such that both L ( C ) and L ( A ) are contained in S χλ , we have L ( B ) ⊂ S χλ for all B with C ⊂ B ⊂ A .Proof. Suppose we have C ⊂ B ⊂ A and both L ( C ) and L ( A ) are contained in S χλ . Then we have λ ∈ σ A ⊂ σ B ⊂ σ C . The lemma is now a direct result of Proposition 3.13. (cid:3)
Finally, we are ready to describe a stratum:
Theorem 3.17.
Let χ ∈ χχχ ( G ) be a character and λ ∈ Λ χ . Let M be the maximalsubset of [ n ] as in Lemma 3.15. If each A j for j = 1 , . . . , N is a subset of [ n ] thatis minimal with respect to the property that L ( A j ) ⊂ S χλ , then S χλ = V ([ n ] − M ) (cid:92) (cid:18) N (cid:91) j =1 D ( A j ) (cid:19) = X λ (cid:92) (cid:18) N (cid:91) j =1 D ( A j ) (cid:19) . In particular, S χλ as an open subvariety of X λ , is irreducible, connected and smooth.Moreover, S χλ is obtained by removing a finite union of linear subspaces from X λ and the zariski closure S χλ is the linear subspace X λ .Proof. By Corollary 3.14 and Lemma 3.16, we have S χλ = N (cid:91) j =1 L ( A j ; M ) = N (cid:91) j =1 (cid:18) V ([ n ] − M ) ∩ D ( A j ) (cid:19) = V ([ n ] − M ) (cid:92) (cid:18) N (cid:91) j =1 D ( A j ) (cid:19) . The theorem is a direct result of the above computation. (cid:3) Recap on toric varieties
The main reference for this section is [CLS11]. In this section we recall thefact due to Cox that every projective toric variety X is the GIT quotient of anaffine space by a diagonalizable group with respect to characters induced by ampledivisors on X . Our toric varieties are assumed to be normal. In this case a toricvariety is constructed from a fan. We refer the reader for the definition of a fanand how to construct a toric variety from a fan to chapter 3 in [CLS11].Starting at Section 4.1, we adopt notations and conventions from [CLS11] andrecall the finite presentation (4.2) of the divisor class group of a toric variety. With ARIATION OF STRATIFICATIONS FROM TORIC GIT 19 (4.2) we derive at Proposition 4.1 several properties of the diagonalizable groupthat occurs in the quotient construction of toric varieties. We then briefly recall anumerical criterion, Theorem 4.2 for a torus invariant divisor to be ample anddescribe in Theorem 4.4 the cone of ample divisors on a projective toric variety.Finally, in Section 4.3 we recall Cox’s GIT quotient construction for projectivetoric varieties at Theorem 4.7 and Proposition 4.8.4.1.
Set up.
Let N be a lattice and M = Hom Z ( N, Z ) be the dual. Let (cid:104)− , −(cid:105) : M × N → Z be the perfect pairing where (cid:104) m, n (cid:105) = m ( n ). Set N R and M R to be N ⊗ Z R and M ⊗ Z R respectively.Let Σ ⊂ N R be a fan. We write X Σ as the toric variety constructed from thefan Σ. In this case T N := N ⊗ Z C × is the torus embedded in X Σ . The lattice N is realized as the group of one parameter subgroups ΓΓΓ( T N ) of T N in the followingway: given an element n ∈ N , the assignment C × → T N defined by t (cid:55)→ n ⊗ t is a one parameter subgroup of T N . The lattice M is realized as the group ofcharacters χχχ ( T N ) of T N in the following way: given an element m ∈ M and anelement n ⊗ t ∈ T N , the assignment T N → C × defined by n ⊗ t (cid:55)→ t (cid:104) m,n (cid:105) is a character of T N . Under these identifications the pairing (cid:104)− , −(cid:105) : M × N → Z agrees with the pairing defined for χχχ ( T N ) and ΓΓΓ( T N ) in Section 3.1.By Σ( i ) we mean the collection of i -dimensional cones in Σ and Σ max meansthe collection of maximal cones in Σ. For each ray ρ ∈ Σ(1), we write u ρ ∈ N as the indivisible lattice point that spans ρ over R . If σ is a polyhedral cone, σ (1) means the collection of extremal rays of σ . Namely, σ (1) consists of onedimensional faces of σ .By the orbit-cone correspondence (theorem 3.2.6 in [CLS11]), each ray ρ ∈ Σ(1)corresponds to a T N -invariant divisor D ρ ⊂ X Σ . Let Z Σ(1) = (cid:77) ρ ∈ Σ(1) Z · D ρ be the free abelian group generated by the divisors D ρ for ρ ∈ Σ(1). Each m ∈ M ,viewed as a character χ m of T N , is a rational function on X Σ . The principaldivisor div( χ m ) satisfies the formula(4.1) div( χ m ) = (cid:88) ρ ∈ Σ(1) (cid:104) m, u ρ (cid:105) · D ρ . It is proved in theorem 4.1.3 in [CLS11] that the natural map Z Σ(1) → Cl( X Σ ) issurjective with kernel the image of the map M → Z Σ(1) defined by m (cid:55)→ div( χ m ).Namely, we have an exact sequence(4.2) M Z Σ(1)
Cl( X Σ ) 0 . Moreover, it is exact on the left if Σ is complete. Let CDiv T N ( X Σ ) be the groupof torus invariant Cartier divisors on X Σ . Then there is the exact sequence(4.3) M CDiv T N ( X Σ ) Pic( X Σ ) 0 that fits into the commutative diagram of short exact sequences: M CDiv T N ( X Σ ) Pic( X Σ ) 0 M Z Σ(1)
Cl( X Σ ) 0 . When Σ is smooth, there is no distinction between sequence (4 .
2) and sequence(4 . . When Σ is complete, as C × is divisible, we get another short exact sequenceby applying Hom Z ( − , C × ) to sequence (4 . G ( C × ) Σ(1) T N G = Hom Z (Cl( X Σ ) , C × ). Facts about the group G are summarized in the Proposition 4.1.
Let Σ be a complete fan and G = Hom Z (Cl( X Σ ) , C × ) . Then(1) G is diagonalizable.(2) Cl( X Σ ) (cid:39) χχχ ( G ) .(3) An element t ∈ ( C × ) Σ(1) is in G if and only if (cid:89) ρ ∈ Σ(1) t (cid:104) m,u ρ (cid:105) ρ = 1 for all m ∈ M. (4) There is an exact sequence (4.5) 0 ΓΓΓ( G ) Z Σ(1) N ϕ dual to (4.2) where if { e ρ } is the standard basis for Z Σ(1) , ϕ is defined by e ρ (cid:55)→ u ρ . In particular, ΓΓΓ( G ) records the relation among ray generatorsof Σ(1) . Proof.
Statement (1) follows from the inclusion G ⊂ ( C × ) Σ(1) . Statement (2)follows from the construction of G . For statement (3), see lemma 5.1.1 in [CLS11].For statement(4), the inclusion G (cid:44) → ( C × ) Σ(1) induces an inclusion ΓΓΓ( G ) (cid:44) → Γ(( C × ) Σ(1) ) (cid:39) Z Σ(1) . By statement (3), a one parameter subgroup b ∈ Z Σ(1) isin ΓΓΓ( G ) if and only if (cid:89) ρ ∈ Σ(1) t b ρ ·(cid:104) m,u ρ (cid:105) = t (cid:104) m, (cid:80) ρ b ρ u ρ (cid:105) = 1 for all t ∈ C × and for all m ∈ M. Hence (cid:104) m, (cid:80) ρ ∈ Σ(1) b ρ u ρ (cid:105) = 0 for all m ∈ M. Since the pairing between M and N is perfect, we have (cid:88) ρ ∈ Σ(1) b ρ u ρ = 0 . Namely, ΓΓΓ( G ) is exactly the relation among ray generators. Moreover, the dual of M → Z Σ(1) defined in (4.2) is ϕ . Hence (4.5) is the dual of (4.2). (cid:3) Notation
We will also write the pairing (and its extension over R ) between χχχ ( G )and ΓΓΓ( G ) as (cid:104)− , −(cid:105) when the context is clear. Moreover, for a divisor D ∈ Cl( X Σ ),we write χ D as the corresponding character of G . ARIATION OF STRATIFICATIONS FROM TORIC GIT 21
The ample cone.
We supply a numerical criterion to test if a divisor on atoric variety is ample.
Theorem 4.2.
Let Σ be a fan and D = (cid:80) ρ a ρ D ρ be a divisor on X Σ . Then D is Cartier if and only if for all cone σ ∈ Σ max , there is an m σ ∈ M such that (cid:104) m, u ρ (cid:105) = − a ρ for all ρ ∈ σ (1) . Such m σ is unique modulo σ ⊥ ∩ M . Moreover, if Σ is complete, and D = (cid:80) ρ a ρ D ρ is Cartier with m σ ∈ M for each σ ∈ Σ max as in the first part of the theorem,then D is ample if and only if (cid:104) m σ , u ρ (cid:105) > − a ρ for all σ ∈ Σ max and for all ρ / ∈ σ (1) . Corollary 4.3.
Suppose there is full dimensional cone in Σ . Then Pic( X Σ ) isfree of finite rank. In particular, this holds when Σ is complete.Proof. Because of the short exact sequence (4.3), we know that Pic( X Σ ) is finitelygenerated. Hence it is sufficient to prove that Pic( X Σ ) is torsion free. Let σ be afull dimensional cone in Σ. Let D = (cid:80) ρ a ρ D ρ be a torus invariant Cartier divisorand K · D is linearly equivalent to 0 for some K >
0. We need to prove that D is linearly equivalent to 0. By Theorem 4.2, there exists an m σ ∈ M such that (cid:104) m σ , u ρ (cid:105) = − a ρ for all ρ ∈ σ (1). By the short exact sequence (4.3), there exists an m ∈ M such that (cid:104) m, u ρ (cid:105) = − K · a ρ for all ρ ∈ Σ(1). Hence (cid:104) K · m σ , u ρ (cid:105) = (cid:104) m, u ρ (cid:105) for all ρ ∈ σ (1). Since σ is full dimensional, K · m σ = m . As Z Σ(1) is torsion free,we also have div( χ m σ ) = D . Hence D is zero in Pic( X Σ ) . (cid:3) We write Amp( X Σ )as the collection of ample divisors on X Σ . The ample cone for X Σ is the real coneover Amp( X Σ ) in Pic( X Σ ) R and is denoted byAmp( X Σ ) R . The ample cone of projective toric varieties are well understood. It is theinterior of the nef cone. We recall theorem 6.3.22 from [CLS11]:
Theorem 4.4.
Let X Σ be a projective toric variety. Then the cone generated bynef divisors, called the nef cone is a full dimensional, strongly convex rationalpolyhedral cone in Pic ( X Σ ) R . Moreover, a Cartier divisor is ample if and only ifit is in the interior of the nef cone. GIT quotient construction of projective toric varieties.
In this sec-tion we recall Cox’s GIT quotient construction for projective toric varieties. Themain result of this section is Proposition 4.8 where we supply an alternativecalculation of the unstable locus by cooking up a one parameter subgroup to desta-bilize points in the linear subspace defined by each primitive collection. The oneparameter subgroups we used in Proposition 4.8 are related to primitive relationson simplicial projective toric varieties (Definition 4.9). The primitive relationswere known to generate the Mori cone on a simplicial projective toric variety[Bat91]. In addition, the one parameter subgroups introduced in Proposition 4.8will allow us to extend the notion of stratifications induced by ample divisors to R -ample divisors in the ample cone later at Section 5.2.To begin with, for each fan Σ, we define the affine space C Σ(1) = Spec C [ x ρ | ρ ∈ Σ(1)] . For each cone σ ∈ Σ max , define the monomial x ˆ σ := (cid:89) ρ/ ∈ σ (1) x ρ . Then define the ideal B (Σ) = ( x ˆ σ | σ ∈ Σ max ) ⊂ C [ x ρ | ρ ∈ Σ(1)] . We then have the vanishing locus Z (Σ) = V ( B (Σ)) ⊂ C Σ(1) . There is a cleaner description of B (Σ) and Z (Σ), using the notion of primitivecollections . Definition 4.5.
A subset C ⊂ Σ(1) is a primitive collection if:(1) C (cid:54)⊂ σ (1) for all σ ∈ Σ, and(2) for any proper subset C (cid:48) of C , there is a cone σ ∈ Σ such that C (cid:48) ⊂ σ (1). Proposition 4.6.
The ideal B (Σ) has the primary decomposition B (Σ) = (cid:92) C ( { x ρ | ρ ∈ C } ) which induces Z (Σ) = (cid:91) C V ( { x ρ | ρ ∈ C } ) . Here both the union and the intersection are taken over all primitive collections C of Σ(1) .Proof.
See proposition 5.1.6 from [CLS11]. (cid:3)
The closed subset Z (Σ) is used for Cox’s quotient construction for toric varieties.Below is Cox’s theorem from [Cox95]. We refer the reader to chapter 5 in [CLS11]for the definition of an almost geometric quotient of a variety by an algebraicgroup. Theorem 4.7.
Let X Σ be a toric variety and G = Hom Z (Cl( X Σ ) , C × ) act on C Σ(1) via the inclusion G ⊂ ( C × ) Σ(1) . Then there is a map π : C Σ(1) \ Z (Σ) → X Σ that realizes X Σ as an almost geometric quotient of C Σ(1) \ Z (Σ) by G . Since G acts on C Σ(1) and Cl( X Σ ) (cid:39) χχχ ( G ), we can associate the GIT quo-tient C Σ(1) // χ D G with any divisor D ∈ Cl( X Σ ). It is natural to ask if there is adivisor D ∈ Cl( X Σ ) such that C Σ(1) // χ D G (cid:39) X Σ . This amounts to ask if thereis a divisor D such that ( C Σ(1) ) ss ( χ D ) = C Σ(1) \ Z (Σ). The answer is affirma-tive if X Σ is projective and if D ∈ Amp( X Σ ). In [CLS11] Cox proved this bycomputing the χ D -invariants. We give an alternative proof using the numericalcriterion for semistability (Theorem 3.3), and the numerical criterion for ampleness(Theorem 4.2). Proposition 4.8.
Let X Σ be a projective toric variety and D ∈ Cl( X Σ ) be anample divisor. Then ( C Σ(1) ) ss ( χ D ) = C Σ(1) \ Z (Σ) . ARIATION OF STRATIFICATIONS FROM TORIC GIT 23
Proof.
We first show that a point x / ∈ Z (Σ) is χ D -semistable for any ample divisor D = (cid:80) ρ a ρ D ρ . Let λ ∈ Z Σ(1) be a one parameter subgroup of G such thatlim t → λ ( t ) · x exists. For each ρ ∈ Σ(1), we have( λ ( t ) · x ) ρ = t λ ρ · x ρ . By definition of Z (Σ), there is a maximal cone σ ∈ Σ max such that x ˆ σ ( x ) (cid:54) = 0.Hence lim t → λ ( t ) · x exists only if λ ρ ≥ ρ / ∈ σ (1). Recall we have m σ ∈ M from Theorem 4.2. Now (cid:104) χ D , λ (cid:105) = (cid:88) ρ a ρ λ ρ = (cid:88) ρ ∈ σ (1) a ρ λ ρ + (cid:88) ρ/ ∈ σ (1) a ρ λ ρ = − (cid:88) ρ ∈ σ (1) (cid:104) m σ , u ρ (cid:105) λ ρ + (cid:88) ρ/ ∈ σ (1) a ρ λ ρ ≥ − (cid:88) ρ ∈ σ (1) (cid:104) m σ , u ρ (cid:105) λ ρ − (cid:88) ρ/ ∈ σ (1) (cid:104) m σ , u ρ (cid:105) λ ρ = − (cid:104) m σ , (cid:88) ρ λ ρ u ρ (cid:105) = 0by Theorem 4.2 and the fact that ΓΓΓ( G ) is the relation among ray generators.Conversely, let x ∈ Z (Σ), we need to establish a one parameter subgroup λ of G such that • lim t → λ ( t ) · x exists, and • (cid:104) χ D , λ (cid:105) < C such that x ∈ V ( { x ρ | ρ ∈ C } ). Let u = (cid:88) ρ ∈ C u ρ . Then since Σ is complete, there is a maximal cone σ ∈ Σ max such that u ∈ σ .By proposition 11.1.7 in [CLS11], there is a subcollection S ⊂ σ (1) such that u ∈ Cone( S ) and such that Cone( S ) is a simplicial cone. Hence there are b ρ ∈ Q ≥ for ρ ∈ σ (1) such that u = (cid:80) ρ ∈ σ (1) b ρ u ρ . Let K be a positive integer large enoughso that K · b ρ ∈ N for all ρ ∈ σ (1). Then we have a relation K · u = (cid:88) ρ ∈ σ (1) ( K · b ρ ) u ρ . Let(4.6) c ρ = − K if ρ ∈ C \ σ (1) ,K · ( b ρ −
1) if ρ ∈ C ∩ σ (1) ,K · b ρ if ρ ∈ σ (1) \ C, . Since each c ρ is an integer and (cid:80) ρ c ρ u ρ = 0, we see that the collection { c ρ } ρ ∈ Σ(1) induces a one parameter subgroup λ of G . Furthermore c ρ ≥ ρ / ∈ C .Hence lim t → λ ( t ) · x exists. It remains to show that (cid:104) χ D , λ (cid:105) = (cid:80) ρ a ρ c ρ <
0. Since C is a primitive collection, there is at least one ρ (cid:48) ∈ C \ σ (1) and c ρ (cid:48) = − K < Therefore, we have (cid:88) ρ a ρ c ρ = (cid:88) ρ ∈ σ (1) a ρ c ρ + (cid:88) ρ/ ∈ σ (1) a ρ c ρ < − (cid:88) ρ ∈ σ (1) (cid:104) m σ , u ρ (cid:105) c ρ − (cid:88) ρ/ ∈ σ (1) (cid:104) m σ , u ρ (cid:105) c ρ = − (cid:104) m σ , (cid:88) ρ c ρ u ρ (cid:105) = 0 , where the m σ comes from Theorem 4.2. (cid:3) It is worthwhile pointing out that the one parameter subgroup we cooked upin Equation (4.6) does not depend on the divisor D , but only on the primitivecollection C . Namely, the one parameter subgroup defined by Equation (4.6) failsthe Hilbert-Mumford criterion for every point x ∈ V ( { x ρ | ρ ∈ C } ) and for everyample divisor D .The one parameter subgroup introduced in Equation (4.6) is related to primitiverelations for simplicial complete fans. Let Σ be a complete simplicial fan and C ⊂ Σ(1) be a primitive collection. Then the sum u = (cid:80) ρ ∈ C u ρ is in the relativeinterior of a unique cone σ ∈ Σ. Since Σ is simplicial, there exists unique q ρ ∈ Q ≥ for each ρ ∈ σ (1) such that u = (cid:80) ρ ∈ σ (1) q ρ u ρ . Definition 4.9.
The tuple a ∈ Q Σ(1) where a ρ = , if ρ ∈ C ∩ \ σ (1)1 − b ρ , if ρ ∈ C ∩ σ (1) b ρ , if ρ ∈ σ (1) \ C , otherwise . is called the primitive relation of C .5. Toric VSIT
Let X Σ be a projective toric variety. The tasks in this section break mainlydown to the following:(1) Extend the notions of stability and stratification induced by an ampledivisor to those induced by any R -ample divisor in the ample cone andexplain the strategy to compute stratifications induced by R -ample divisorsat Section 5.2.(2) Identify certain collections of R -ample divisors on X Σ that may cause thestratification to change as walls at Section 5.3.(3) Provide a sufficient condition for two R -ample divisors to induce equivalentstratifications and prove that walls capture variations at Section 5.4.They are formulated as Theorem 5.18, Theorem 5.21 and Theorem 5.22respectively.(4) Relate VSIT (Theorem 5.29) and several topological properties of thestrata to the combinatorial properties of the fan Σ at Section 5.6.We also supply an example at Section 5.5 where we compute stratificationsinduced by all R -ample divisors. The picture is simple enough for us to summarizestratifications and their variations at Table 1. ARIATION OF STRATIFICATIONS FROM TORIC GIT 25
Notations and conventions.
Let X Σ be a projective toric variety and G = Hom Z (Cl( X Σ ) , C × ). We equip ΓΓΓ( G ) R with the inner product( − , − ) : ΓΓΓ( G ) R × ΓΓΓ( G ) R → R obtained by restricting the standard inner product on R Σ(1) via the R -extensionof the inclusion ΓΓΓ( G ) (cid:44) → Z Σ(1) in sequence (4.5). The induced norm is written as || − || : ΓΓΓ( G ) R → R . If D is an element in Pic( X Σ ) R , we write χ ∗ D as the unique vector in ΓΓΓ( G ) R such that ( χ ∗ D , v ) = (cid:104) χ D , v (cid:105) for all v ∈ ΓΓΓ( G ) R . We let L be the set of subsets of Σ(1) where S ∈ L if and only if there is aprimitive collection C such that S ⊂ Σ(1) − C . With this we see that a point x ∈ C Σ(1) is in Z (Σ) if and only if S x ∈ L where S x was defined as the state of x in Section 3.1. If S ∈ L , we define(1) the polyhedral cone σ S = { v ∈ ΓΓΓ( G ) R |(cid:104) χ D ρ , v (cid:105) ≥ ρ ∈ S } , and(2) the subspace W S = { v ∈ ΓΓΓ( G ) R |(cid:104) χ D ρ , v (cid:105) = 0 for all ρ ∈ S } .Moreover, if D ∈ Amp( X Σ ) R , we define the finite subset of ΓΓΓ( G ) R (3) Λ DS = {− Proj W Z χ ∗ D (cid:12)(cid:12) − Proj W Z χ ∗ D ∈ σ S , Z ⊂ S } . If x is a point in Z (Σ) and D ∈ Amp( X Σ ) R , we letΛ Dx := Λ DS x . Obviously the set Λ Dx depends only on the states of x . We will see in Theorem 5.5that for every S ∈ L , there is a unique longest vector λ DS ∈ Λ DS . Letting λ Dx = λ DS x ,we define Λ D := { λ DS | S ∈ L } = { λ Dx | x ∈ Z (Σ) } . We note without proof of the
Fact 5.1.
Each face of σ S is of the form W Z ∩ σ S for some Z ⊂ S and thesubspace spanned by a face of σ S is of the form W Z for some Z ⊂ S .5.2. Extensions to the ample cone.
We need a dedicated analysis of the amplecone to understand variation of stratifications. The first step is to extend thenotions of stability and stratifications induced by ample divisors to the entireample cone. For stability, the extension adopts numerical conditions imposed bythe Hilbert-Mumford criterion.
Definition 5.2.
Let X Σ be a projective toric variety and D ∈ Amp( X Σ ) R . Wesay a point x ∈ C Σ(1) is χ D - semistable if the function (cid:104) χ D , −(cid:105) : ΓΓΓ( G ) R → R takesnon-negative values at the cone σ x . Else, we say x is χ D - unstable .We now show that the χ D -semistable locus stays constant throughout the entireample cone. Proposition 5.3.
Let X Σ be a projective toric variety, D ∈ Amp( X Σ ) R , and x ∈ C Σ(1) . Then the linear function (cid:104) χ D , −(cid:105) : ΓΓΓ( G ) R → R assumes a negativevalue at the cone σ x if and only if x ∈ Z (Σ) .Proof. Suppose x ∈ Z (Σ). Then x ∈ V ( { x ρ | ρ ∈ C } ) for some primitive collection C . By Proposition 4.8, there is a one parameter subgroup λ ( C ) ∈ σ y such that (cid:104) χ D (cid:48) , λ ( C ) (cid:105) < D (cid:48) ∈ Amp( X Σ ) and for all y ∈ V ( { x ρ | ρ ∈ C } ). Writing D = (cid:80) i r i · D i for D i ∈ Amp( X Σ ) and r i >
0, we see that (cid:104) χ D , λ ( C ) (cid:105) <
0. Since λ ( C ) ∈ σ x , the if part is proved. Conversely, suppose x / ∈ Z (Σ). Then by Proposition 4.8, (cid:104) χ D (cid:48) , λ (cid:105) ≥ λ ∈ σ x and for all D (cid:48) ∈ Amp( X Σ ). Again writing D = (cid:80) i r i D i for r i > D i ∈ Amp( X Σ ), we get (cid:104) χ D , λ (cid:105) ≥ λ ∈ σ x . The proposition is proved. (cid:3) Hence ( C Σ(1) ) ss ( χ D ) = C Σ(1) − Z (Σ)= C Σ(1) − (cid:91) C V ( x ρ | ρ ∈ C ) for all D ∈ Amp( X Σ ) R . (5.1)We can also perform numerical analysis of instability as in Section 3.3 on theentire ample cone. Definition 5.4.
Let X Σ be a projective toric variety and let x ∈ Z (Σ). For any D ∈ Amp( X Σ ) R , we set M D ( x ) = inf λ ∈ σ x \{ } (cid:104) χ D , λ (cid:105)|| λ || . We say an element λ ∈ ΓΓΓ( G ) R is χ D - adapted to x if (cid:104) χ D ,λ (cid:105)|| λ || = M D ( x ). We say λ is χ D - adapted to a subset Y ⊂ Z (Σ) if λ is χ D -adapted to all points in Y .Kempf’s theorem, Theorem 3.6 can now be formulated in the ample cone. Theorem 5.5.
Let X Σ be a projective toric variety. For any x ∈ Z (Σ) and D ∈ Amp( X Σ ) R , there is a unique vector λ Dx ∈ Λ Dx that is χ D -adapted to x . Inthis case, M D ( x ) = −|| λ Dx || . Proof.
By Proposition 5.3 and Theorem 2.1 and, the function (cid:104) χ D , −(cid:105)|| − || : ΓΓΓ( G ) R \{ } → R achieves the relative minimum at a unique ray in σ x . The existence and theuniqueness of λ Dx in Λ Dx is a direct consequence of Fact 5.1 and Corollary 2.2.Finally, the equality M D ( x ) = −|| λ Dx || is derived by the same calculation done inCorollary 2.2. (cid:3) Next, we formulate Hesselink’s stratification of C Σ(1) for any D ∈ Amp( X Σ ) R .For each λ ∈ Λ D , we define the subset S χ D λ = { x ∈ Z (Σ) | λ Dx = λ } . By Lemma 2.4, there is a unique λ ∈ Λ D that is χ D -adapted to x for each point x ∈ Z (Σ). Therefore, we have the decomposition C Σ(1) = ( C Σ(1) ) ss ( χ D ) ∪ (cid:0) (cid:91) λ ∈ Λ D S χ D λ (cid:1) . We define a strict partial order on Λ D by setting λ > λ (cid:48) in Λ D if || λ || > || λ (cid:48) || . Using the same argument from Theorem 3.9, one then shows that theabove decomposition induces a stratification of C Σ(1) and that it is equivalent toHesselink’s stratification whenever D is an ample divisor.We refer to the stratification from the above decomposition as the stratificationof C Σ(1) induced by D ∈ Amp( X Σ ) R . To compute the stratification induced by D ∈ Amp( X Σ ) R , we first note that the vector λ Dx only depends on the states S x of x so λ DS is χ D -adapted to L ( S ) for each S . Hence computing stratificationinduced by D is simply grouping { L ( S ) } S ∈ L by { λ DS } S ∈ L . To compute λ DS , one ARIATION OF STRATIFICATIONS FROM TORIC GIT 27 simply computes the finite set Λ DS , and finds the longest vector in it. For S whosesize is small, Λ DS can easily be computed by hand. The complexity of enumeratingΛ DS grows exponentially with respect to the size of S because it involves the powerset of S . For larger S , we would rely on the computer program we wrote. SeeSection 6.5 for more information.We now define SIT-equivalence for R -ample divisors. Definition 5.6.
Let X Σ be a projective toric variety. We say two R -ampledivisors on X Σ are SIT-equivalent if they induce equivalent stratifications of C Σ(1) .We end this section with a continuity result, Corollary 5.8 that is needed later.In short, we prove that for every point x ∈ Z (Σ), choosing the longest vector λ Dx ∈ Λ Dx is continuous with respect to D in the ample cone. Proposition 5.7.
Let X Σ be a projective toric variety. For any x ∈ Z (Σ) , theassignment ξ x : Amp( X Σ ) R → ΓΓΓ( G ) R defined by (5.2) D (cid:55)→ λ Dx is continuous.Proof. Let S x ⊂ Σ(1) be the states of x . For each Z ⊂ S x , define the subset ofthe ample coneAmp( X Σ ) Z = { D ∈ Amp( X Σ ) R | − Proj W Z χ ∗ D = λ Dx } . By Theorem 2.1, the union ∪ Z ⊂ S x Amp( X Σ ) Z is the entire ample cone. Theassignment ξ x on Amp( X ) Z is given by D (cid:55)→ − Proj W Z χ ∗ D , which is obviously continuous. (cid:3) Corollary 5.8.
Let X Σ be a projective toric variety. For any x ∈ Z (Σ) , the as-signment D (cid:55)→ M D ( x ) defined for D ∈ Amp( X Σ ) R is a negative valued continuousfunction to the real line R .Proof. The assignment D (cid:55)→ M D ( x ) is the continuous map ξ x post composedwith the continuous map ( − · || − || : ΓΓΓ( G ) R → R . (cid:3) Wall and semi-chamber.
Let X Σ be a projective toric variety. Equa-tion (5.1) implies that the χ D -semistable locus stays constant throughout D ∈ Amp( X Σ ) R . Hence the variation of stratifications of C Σ(1) induced by R -ampledivisors only occurs in the χ D -unstable locus Z (Σ). In addition, we have seenin Section 5.2 that each χ D -stratum is a grouping of { L ( S ) } S ∈ L by the list ofvectors { λ DS } S ∈ L . We therefore deduce that:(1) There is a type one variation (Definition 3.11) between the stratificationsof C Σ(1) induced by
D, D (cid:48) ∈ Amp( X Σ ) R if and only if there is a pair S , S ∈ L such that λ DS (cid:54) = λ DS while λ D (cid:48) S = λ D (cid:48) S .(2) If there is a type two variation (Definition 3.11) between the stratificationsof C Σ(1) induced by
D, D (cid:48) ∈ Amp( X Σ ) R , then there is a pair S , S ∈ L such that || λ DS || < || λ DS || while || λ D (cid:48) S || ≥ || λ D (cid:48) S || . Therefore, the R -ample divisors D ∈ Amp( X Σ ) R such that the stratificationsundergo type one (resp. type two) variations should be captured by the collections { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z χ ∗ D } (resp. { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} )for various Z , Z ∈ L .We now formulate two types of walls that intuitively capture two types ofvariations. Definition 5.9.
Suppose X Σ is a projective toric variety. Let ∅ (cid:54) = H (cid:40) Amp( X Σ ) R be a proper subset. We say H is a type one wall if there are Z and a singleton { ρ } in L such that(1) W Z ∪{ ρ } is a codimension one subspace of W Z , and(2) H = { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z ∪{ ρ } χ ∗ D } . In this case, wesay H is a type one wall with respect to Z and { ρ } .We say H is a type two wall if there are Z , Z ∈ L such that(1) there is no containment between W Z and W Z , and(2) H = { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} . In this case,we say H is a type two wall with respect to Z and Z . Remark 5.10.
Here are some remarks of the definition of walls. First, let Z, { ρ } ∈ L . For the collection { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z ∪{ ρ } χ ∗ D } to be proper it only makes sense for W Z ∪{ ρ } to be a proper subspace of W Z . Since W Z ∪{ ρ } is the vanishing locus of the single linear functional (cid:104) χ D ρ , −(cid:105) : ΓΓΓ( G ) R → R on W Z , it has to be of codimension one in W Z .As for type two walls, the non-containment assumption required for a typetwo wall is due to the following observation: Suppose W Z ⊂ W Z . Then for any D ∈ Amp( X Σ ) R , || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D || is equivalent to Proj W Z χ ∗ D =Proj W Z χ ∗ D . We will see in Proposition 5.11 below that a non-empty collection { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z χ ∗ D } is either the whole ample cone or an intersection of type one walls. Hence whenthere is a containment between W Z and W Z , a non-empty proper subset ofAmp( X ) R of the form { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} belongs to type one walls already. Proposition 5.11.
Let X Σ be a projective toric variety. Let Z and Z be twosubsets of L . Then a non-empty collection H = { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z χ ∗ D } is either the whole ample cone or an intersection of type one walls. On the otherhand, a non-empty collection H (cid:48) = { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} is either the whole ample cone, an intersection of type one walls, or a type twowall.Proof. For the collection H , note that Proj W Z χ ∗ D = Proj W Z χ ∗ D is equivalent toProj W Z χ ∗ D = Proj W Z ∪ Z χ ∗ D and Proj W Z ∪ Z χ ∗ D = Proj W Z χ ∗ D . ARIATION OF STRATIFICATIONS FROM TORIC GIT 29
We may choose a sequence l , . . . , l k in Z such that W Z ∪ l ∪ ... ∪ l k = W Z ∪ Z and similarly r , . . . r m in Z such that W Z ∪ r ∪ ... ∪ r m = W Z ∪ Z . We can now express H as the intersection H = (cid:0) ∩ i { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z ∪ li χ ∗ D } (cid:1) (cid:92)(cid:0) ∩ j { D ∈ Amp( X Σ ) | Proj W Z χ ∗ D = Proj W Z ∪ rj χ ∗ D } (cid:1) . If H is a proper subset of Amp( X Σ ) R , then some subsets that occurred in theabove intersection are proper and are type one walls. The statement for H isproved.For H (cid:48) with H (cid:48) (cid:54) = Amp( X Σ ) R , if there is a containment W Z ⊂ W Z , then H (cid:48) is the same as H . This is an intersection of type one walls as we have seen. Ifthere is no containment between W Z and W Z , then H (cid:48) is a type two wall bydefinition. (cid:3) Therefore, walls should completely capture the divisors D such that the strati-fication undergoes variations. We will rigorously prove that type one walls (resp.type two walls) capture type one (resp. type two) variations at Theorem 5.21 andTheorem 5.22. We now describe the structure of walls. Proposition 5.12.
Let X Σ be a projective toric variety.(1) A type one wall is of codimension 1 in Amp( X Σ ) R . It is the intersectionof a codimension 1 subspace of Pic( X Σ ) R with Amp( X Σ ) R .(2) A type two wall is of codimension at least 1 in Amp( X Σ ) R . It is eitherthe intersection of a linear subspace of Pic( X Σ ) R with Amp( X Σ ) R , orthe intersection of a regular codimension 1 submanifold of Pic( X Σ ) R with Amp( X Σ ) R away from a subspace of codimension at least 2.(3) If X Σ is simplicial, then a type two wall is of codimension 1 in Amp( X Σ ) R .It is the intersection of a regular codimension 1 submanifold of Pic( X Σ ) R with Amp( X Σ ) R away from a subspace of codimension at least 2.Moreover, walls are defined by equations that satisfy the following properties: • the condition that a point (cid:80) ρ a ρ D ρ ∈ Amp( X Σ ) R is in a type one wallcorresponds to a linear equation of a ρ ’s with rational coefficient, • the condition that a point (cid:80) ρ a ρ D ρ ∈ Amp( X Σ ) R is in a type two wallcorresponds to a homogeneous quadratic equation of a ρ ’s with rationalcoefficients.Proof. For (1), let H be the type one wall with respect to Z and { ρ } . Namely, H = { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z ∪{ ρ } χ ∗ D } . Define the linear map ν ρ : Pic( X Σ ) R → R by the compositionPic( X Σ ) R ΓΓΓ( G ) R W Z R ∗ Proj WZ ( − ) (cid:104) χ Dρ , −(cid:105) . Namely, ν ρ ( D ) = (cid:104) χ D ρ , Proj W Z χ ∗ D (cid:105) . We then haveProj W Z χ ∗ D = Proj W Z ∪{ ρ } χ ∗ D ⇔ Proj W Z χ ∗ D ∈ W Z ∪{ ρ } ⇔ (cid:104) χ D ρ , Proj W Z χ ∗ D (cid:105) = ν ρ ( D ) = 0 . Hence H = ker ν ρ ∩ Amp( X Σ ) R . Since H is a proper subset, ker ν ρ is a codimension one subspace of Pic( X Σ ) R . For (2), let H (cid:48) be the type two wall { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} with respect to Z and Z . Notice that H (cid:48) is defined by the vanishing locus V ( Q )of the quadratic form Q associated to the bilinear form B on Pic( X Σ ) R definedby(5.3) B ( χ D , χ D (cid:48) ) = (Proj W Z χ ∗ D , Proj W Z χ ∗ D (cid:48) ) − (Proj W Z χ ∗ D , Proj W Z χ ∗ D (cid:48) ) . By Sylvester’s law of inertia, Q looks like x + · · · + x m − y − · · · y n in a suit-able coordinate of Pic( X Σ ) R . If one of m or n is zero, then V ( Q ) is a linearsubspace. If both m and n are nonzero, then by Theorem 2.5, V ( Q ) is a regularsubmanifold of codimension 1 away from the codimension at least 2 linear subspace V ( x , . . . , x m , y , . . . , y n ).For (3), let H (cid:48) be the type two wall with respect to Z , Z as in (2). When X Σ is simplicial, Pic( X Σ ) Q (cid:39) Cl( X Σ ) Q so the map Pic( X Σ ) R → ΓΓΓ( G ) R defined by D (cid:55)→ χ ∗ D is an isomorphism. Hence the collection { D ∈ Pic( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} is diffeomorphic to the collection { v ∈ ΓΓΓ( G ) R | || Proj W Z v || = || Proj W Z v ||} , which is the same as equi-distant collection { v ∈ ΓΓΓ( G ) R | dist( v, W Z ) = dist( v, W Z ) } . By Proposition 2.6, H (cid:48) is the intersection of a regular submanifold away from asubspace of codimension at least 2 with Amp( X Σ ) R .We now deal with rationality of the equations of walls. For this, fix an integral R -basis { λ , . . . , λ q } ⊂ ΓΓΓ( G ) of W Z for each Z ∈ L . Let D = (cid:80) ρ a ρ D ρ be an elementin the ample cone. Then there exists b , . . . , b q such that (cid:80) i b i λ j = Proj W Z χ ∗ D .We will prove that each b i is a linear sum of a ρ ’s with rational coefficients. If this isproved, then if D is in a type one wall, there are a subset Z and a ρ / ∈ Z such that (cid:104) χ D ρ Proj W Z χ ∗ D (cid:105) = 0 as we have seen. Since (cid:104) χ D ρ , λ i (cid:105) ∈ Z , (cid:104) χ D ρ Proj W Z χ ∗ D (cid:105) = 0is translated to a linear equation of a (cid:48) ρ s with rational coefficients. Similarly, thecondition that each b i is a linear sum of a ρ ’s with rational coefficients implyrationality of the equations of type two walls.For each i = 1 , . . . , q , we have (cid:104) χ D , λ i (cid:105) = ( χ ∗ D , λ i ) = (Proj W Z χ ∗ D , λ i ) = q (cid:88) j =1 b j ( λ i , λ j ) . Letting A be the q × q invertible matrix with integral entries ( λ i , λ j ), we see that(5.4) A · b ... b q = (cid:104) χ D , λ (cid:105) ... (cid:104) χ D , λ q (cid:105) ARIATION OF STRATIFICATIONS FROM TORIC GIT 31
Since each (cid:104) χ D , λ i (cid:105) is an integral combination of a ρ ’s, each b i is a Q -combinationof a ρ ’s. The proposition is proved. (cid:3) Since L is finite, we have finitely many walls. We now define semi-chambersthat are open subsets in the complement of the union of walls. Definition 5.13.
Let X Σ be a projective toric variety. Let { F i } be the defin-ing equations for walls in Amp( X Σ ) R . A semi-chamber is a non-empty opensemialgebraic set of the form { D ∈ Amp( X Σ ) R | ± F i ( D ) > i } . Proposition 5.14.
Each wall and semi-chamber is a cone. The ample cone is afinite union of walls and semi-chambers. Moreover, semi-chambers are mutuallyexclusive.Proof.
That each wall and semi-chamber is a cone follows from the fact that wallsare defined by the vanishing loci of homogeneous polynomials. The rest followsdirectly from the fact that there are finitely many walls and from the definition ofsemi-chambers. (cid:3)
The main results.
Let X Σ be a projective toric variety. In this section weprove that if two R -ample divisors are in the same semi-chamber, then they areSIT-equivalent at Theorem 5.18. Namely, they induce equivalent stratifications of C Σ(1) . We also prove that type one (resp. type two) walls capture type one (resp.type two) variations at Theorem 5.21 (resp. Theorem 5.22).
Lemma 5.15.
Let X Σ be a projective toric variety. Let D a and D b be in thesame semi-chamber in Amp( X Σ ) R . Then the following three statements hold:(1) For any Z, Z (cid:48) ∈ L , Proj W Z χ ∗ D a = Proj W Z (cid:48) χ ∗ D a implies Proj W Z χ ∗ D =Proj W Z (cid:48) χ ∗ D for all D ∈ Amp( X Σ ) R . In particular, Proj W Z χ ∗ D a = Proj W Z (cid:48) χ ∗ D a ⇔ Proj W Z (cid:48) χ ∗ D b = Proj W Z (cid:48) χ ∗ D b . (2) For any Z, Z (cid:48) ∈ L , || Proj W Z D ∗ a || < || Proj W Z (cid:48) D ∗ a || ⇔ || Proj W Z D ∗ b || < || Proj W Z (cid:48) D ∗ b || .(3) For any Z ⊂ S ∈ L , − Proj W Z χ ∗ D a ∈ Λ D a S ⇔ − Proj W Z χ ∗ D b ∈ Λ D b S .Proof. For (1), suppose Proj W Z χ ∗ D a = Proj W Z (cid:48) χ ∗ D a . Then the collection H = { D ∈ Amp( X Σ ) R | Proj W Z χ ∗ D = Proj W Z (cid:48) χ ∗ D } contains D a so is non-empty.Proposition 5.11 dictates that H is either Amp( X Σ ) R or an intersection of typeone walls. Since D a is in a semi-chamber, H = Amp( X Σ ) R . Hence Proj W Z χ ∗ D =Proj W Z (cid:48) χ ∗ D for all D ∈ Amp( X Σ ) R . Statement (1) is proved.For (2), suppose || Proj W Z χ D a || < || Proj W Z (cid:48) χ D a || . We consider two caseswhere in one case there is a containment between W Z and W Z (cid:48) , and no con-tainment in the other. If there is a containment between W Z and W Z (cid:48) , since || Proj W Z χ ∗ D a || < || Proj W Z (cid:48) χ ∗ D a || , we have W Z (cid:40) W Z (cid:48) . Hence || Proj W Z χ ∗ D b || ≤ || Proj W Z (cid:48) χ ∗ D b || . If || Proj W Z χ ∗ D b || = || Proj W Z (cid:48) χ ∗ D b || , then we would haveProj W Z χ ∗ D b = Proj W Z (cid:48) χ ∗ D b . However, statement (1) implies Proj W Z χ ∗ D a = Proj W Z (cid:48) χ ∗ D a , a contradiction.Hence || Proj W Z χ ∗ D b || < || Proj W Z (cid:48) χ D b || as desired. Suppose there is no con-tainment between W Z and W Z (cid:48) . The collection H (cid:48) = { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z (cid:48) χ ∗ D ||} is either empty or a type two wall. If H (cid:48) = ∅ ,then by continuity of the quadratic Q form (induced by a bilinear form definedsimilarly as in Equation (5.3)) and convexity of Amp( X Σ ) R , || Proj W Z χ ∗ D || < || Proj W Z (cid:48) χ ∗ D || for all D ∈ Amp( X Σ ) R . If H (cid:48) (cid:54) = ∅ , it is a type two wall. As D a , D b are in the same semi-chamber, Q ( D a )and Q ( D b ) have the same sign. Hence || Proj W Z χ ∗ D b || < || Proj W Z (cid:48) χ ∗ D b || as well.The argument can be reversed so statement (2) holds.For (3), let ρ ∈ S and ν ρ : Pic( X Σ ) R → R be the linear map defined by D (cid:55)→ (cid:104) χ D ρ , Proj W Z χ ∗ D (cid:105) . We will prove ν ρ ( D a ) < ⇔ ν ( D b ) < . Suppose ν ρ ( D a ) <
0. The set H (cid:48)(cid:48) := ker ν ρ ∩ Amp( X Σ ) R is either empty, or atype one wall with respect to the subsets Z and Z ∪ { ρ } . If H (cid:48)(cid:48) = ∅ , then bycontinuity of ν ρ and convexity of Amp( X Σ ) R , ν ρ ( D ) < D ∈ Amp( X Σ ) R .If H (cid:48)(cid:48) is a type one wall, then as D a , D b are in the same semi-chamber, ν ρ ( D b ) < ρ ∈ S , we get − Proj W Z χ ∗ D a ∈ σ S if and only if − Proj W Z χ ∗ D b ∈ σ S . The validity of statement(3) is established and the lemma is proved. (cid:3) Lemma 5.16.
Let X Σ be a projective toric variety and let G = Hom Z (Cl( X Σ ) , C × ) .Order the vectors in ΓΓΓ( G ) R by their norms. If D a and D b are in the same semi-chamber in Amp( X Σ ) R , then there is an order preserving bijection Ξ : (cid:91) S ∈ L Λ D a S → (cid:91) S ∈ L Λ D b S defined by − Proj W Z χ ∗ D a (cid:55)→ − Proj W Z χ ∗ D b for all subsets Z ⊂ S such that − Proj W Z χ ∗ D a ∈ Λ D a S and for all S ∈ L . Moreover,for any subcollection { S , . . . S l } ⊂ L , Ξ restricts to an order preserving bijectionbetween ∪ li =1 Λ D a S i and ∪ li =1 Λ D b S i .Proof. Statement (1) in Lemma 5.15 implies the map Ξ is well defined and injective.Statement (3) in Lemma 5.15 implies that the image of Ξ lands inside ∪ S ∈ L Λ D b S and that Ξ is surjective. Statement (2) in Lemma 5.15 implies that Ξ is orderpreserving. The same logic works for any subcollection { S , . . . , S l } ⊂ L . (cid:3) Proposition 5.17.
Let X Σ be a projective toric variety and suppose D a , D b arein the same semi-chamber in Amp( X Σ ) R . Let S ∈ L and Z be a subset of S .Then λ D a S = − Proj W Z χ ∗ D a ⇔ λ D b S = − Proj W Z χ ∗ D b . Proof.
This is a direct result of Lemma 5.16, applied to the single subset S . (cid:3) We are now ready to prove
Theorem 5.18.
Let X Σ be a projective toric variety. If two R -ample divisors on X Σ are in the same semi-chamber, then they are SIT-equivalent.Proof. Suppose D a and D b are two R -ample divisors on X Σ and they are inthe same semi-chamber. Let S , S be two elements in L . We first prove that L ( S ) and L ( S ) are in the same χ D a -stratum if and only if they are in thesame χ D b -stratum. This would give rise to a bijection between the set of χ D a -strata and the set of χ D b -strata that preserves each stratum as sets. For this, ARIATION OF STRATIFICATIONS FROM TORIC GIT 33 pick subsets Z , Z of S and S respectively so that λ D a S = − Proj W Z χ ∗ D a and λ D a S = − Proj W Z χ ∗ D a . By Proposition 5.17, we have λ D b S = − Proj W Z χ ∗ D b and λ D b S = − Proj W Z χ ∗ D b . Suppose x, y are in the same χ D a -stratum. We then have λ D a S = λ D a S . By Lemma 5.16, λ D b S = λ D b S . Hence x, y are in the same χ D b -stratum.The argument can be reversed. Hence we obtain a bijection between the set of χ D a -strata and the set of χ D b -strata that preserves strata as sets.Finally, the bijection preserves the strict partial order by Lemma 5.16, appliedto the entire collection L . (cid:3) The rest of the section is to prove that each type of walls capture a type ofvariation. We consider line segments connecting pairs of R -ample divisors thatinduce different stratifications. Since we do not know if a semi-chamber is convexor not, Lemma 5.19 below cannot be applied to establish the bijection part ofLemma 5.16. Lemma 5.19.
Let X Σ be a projective toric variety and D, D (cid:48) be two elements in
Amp( X Σ ) R such that the line segment DD (cid:48) does not intersect any type one wallsproperly. Then for any D (cid:48)(cid:48) ∈ DD (cid:48) and any S ∈ L , the assignment Ψ S : Λ DS → Λ D (cid:48)(cid:48) S defined by − Proj W Z χ ∗ D (cid:55)→ − Proj W Z χ ∗ D (cid:48)(cid:48) is a bijection.Proof. It comes down to show the following two statements:(1) For any Z ⊂ S , − Proj W Z χ ∗ D ∈ Λ DS if and only if − Proj W Z χ ∗ D (cid:48)(cid:48) ∈ Λ D (cid:48)(cid:48) S .(2) For any Z , Z ⊂ S , − Proj W Z χ ∗ D = − Proj W Z χ ∗ D if and only if − Proj W Z χ ∗ D (cid:48)(cid:48) = − Proj W Z χ ∗ D (cid:48)(cid:48) .Note that statement (1) implies the image of Ψ S lands in Λ D (cid:48)(cid:48) S and that Ψ S issurjective. Statement (2) implies Ψ S is well-defined and injective. The proofs areentirely analogous to Lemma 5.15. To be careful we supply the proofs here.For statement (1), let Z ⊂ S . Define ν ρ : Pic( X Σ ) R → R for every ρ ∈ S by ν ρ ( ˜ D ) = (cid:104) χ D ρ , Proj W Z χ ∗ ˜ D (cid:105) where D ρ ∈ Cl( X Σ ) is the torus invariant divisor corresponding to ρ . We need toshow that ν ρ ( D ) < ⇔ ν ρ ( D (cid:48)(cid:48) ) < ρ ∈ S. For this, suppose ν ρ ( D ) <
0. The set H = ker ν ρ ∩ Amp( X Σ ) R is then eitherempty or a type one wall. Hence If H is empty then by continuity of ν ρ andconvexity of Amp( X Σ ) R , ν ρ ( D (cid:48)(cid:48)(cid:48) ) < D (cid:48)(cid:48)(cid:48) ∈ Amp( X Σ ) R . In particular, ν ρ ( D (cid:48)(cid:48) ) <
0. If H is a type one wall but ν ρ ( D (cid:48)(cid:48) ) ≥
0, then by continuity of ν ρ on the line segment DD (cid:48)(cid:48) , the type one wall H must intersect DD (cid:48)(cid:48) properly,contradicting the assumption that DD (cid:48) does not intersect any type one wallproperly. The argument can be reversed to show the converse. Since we can dothis for any Z ⊂ S , statement (1) is proved.For statement (2), assume Proj W Z χ ∗ D = Proj W Z χ ∗ D for some Z , Z ⊂ S .Then the set H (cid:48) = { ˜ D ∈ Amp( X Σ ) R | Proj W Z χ ∗ ˜ D = Proj W Z χ ∗ ˜ D } is either the entire Amp( X Σ ) R or an intersection of type one walls by Proposi-tion 5.11. Hence H (cid:48) either contains the entire line segment DD (cid:48) or intersects itat the point D . Since DD (cid:48) does not intersect any type one wall properly, it can only be the case that DD (cid:48) ⊂ H (cid:48) , in which case Proj W Z χ ∗ D (cid:48) = Proj W Z χ ∗ D (cid:48) isautomatic. The argument can be reversed to show the converse. Statement (2)and therefore the lemma is proved. (cid:3) Lemma 5.20.
Let X Σ be a projective toric variety, D, D (cid:48) be two elements in
Amp( X Σ ) R and S ∈ L . Suppose there are two subsets Z , Z ⊂ S such that thefollowing conditions hold:(1) At D , λ DS = − Proj W Z χ ∗ D (cid:54) = − Proj W Z χ ∗ D .(2) At D (cid:48) , λ D (cid:48) S = − Proj W Z χ ∗ D (cid:48) (cid:54) = − Proj W Z χ ∗ D (cid:48) .Then the line segment DD (cid:48) intersects a type one wall properly.Proof. Suppose on the contrary that DD (cid:48) does not intersect any type one wallproperly. Then by Lemma 5.19, both − Proj W Z χ ∗ D (cid:48)(cid:48) and − Proj W Z χ ∗ D (cid:48)(cid:48) areelements of Λ D (cid:48)(cid:48) S for all D (cid:48)(cid:48) ∈ DD (cid:48) . This together with the assumptions of thelemma imply that there is no containment between W Z and W Z . To see this,if say W Z ⊂ W Z , then || Proj W Z χ ∗ D || ≤ || Proj W Z χ ∗ D || , contradicting theassumption that λ DS = − Proj W Z χ ∗ D (cid:54) = − Proj W Z χ ∗ D . Similarly we cannot have W Z ⊂ W Z for the assumptions at D (cid:48) .Let − Proj W Z χ ∗ D , . . . , − Proj W ZN χ ∗ D be N distinct elements in Λ DS wherethere is no containment between W Z i and W Z j for any i (cid:54) = j . We can considerthe N − || Proj W Z χ ∗ ˜ D || = || Proj W Zi χ ∗ ˜ D || for ˜ D ∈ Amp( X Σ ) R and for i = 2 . . . . , N. Since λ DS = − Proj W Z χ ∗ D , we have(5.6) || Proj W Z χ ∗ D || > || Proj W Zi χ ∗ D || for all i = 2 , . . . , N. Hence each of the N − D . In particular, none of these conditions includes the wholeline segment DD (cid:48) .By continuity of the quadratic form associated to the bilinear form defined at(5.3), the condition for i = 2 in (5.5) defines a type two wall that intersects theline segment DD (cid:48) properly. Hence the set A of points where the type two wallsfrom (5.5) intersect DD (cid:48) properly is not empty. Note that D / ∈ A .Since a type two wall intersects a line properly at at most two points, the set A isfinite. We can therefore pick the point D a ∈ A that is closest to D . By Lemma 5.19and the construction of a , it must be the case that λ D (cid:48)(cid:48) S = − Proj W Z χ ∗ D (cid:48)(cid:48) for all D (cid:48)(cid:48) in the half open interval DD a \ D a . We claim that λ D a S = − Proj W Z χ ∗ D a . If this is not the case then there is a j (cid:54) = 1 such that λ D a S = − Proj W Zj χ ∗ D a (cid:54) = − Proj W Z χ ∗ D a . Hence || Proj W Zj χ ∗ D a || > || Proj W Z χ ∗ D a || . By continuity, there will be a point D b ∈ DD a \ D a such that || Proj W Zj χ ∗ D b || = || Proj W Z χ ∗ D b || , contradicting theconstruction of a . The claim is proved.However, if the point a is the intersection of DD (cid:48) with the type two wall definedfor k in (5.5), then the uniqueness imposed by Theorem 2.1 forcesProj W Z χ ∗ D a = Proj W Zk χ ∗ D a . ARIATION OF STRATIFICATIONS FROM TORIC GIT 35
This implies D a is in the intersection of type one walls defined by the conditionProj W Z χ ∗ ˜ D = Proj W Zk χ ∗ ˜ D for ˜ D ∈ Amp( X Σ ) R , contradicting our assumption that DD (cid:48) intersects no type one walls properly. Thelemma is proved. (cid:3) Theorem 5.21.
Let X Σ be a projective toric variety. If D, D (cid:48) are two elementsin
Amp( X Σ ) R such that the stratifications induced by D and D (cid:48) undergo a typeone variation, then the line segment DD (cid:48) intersects a type one wall properly.Proof. Without loss of generality, we may pick a pair S , S of elements in L suchthat L ( S ) and L ( S ) are in different χ D -strata but in the same χ D (cid:48) -stratum.Let Z , Z (resp. Z (cid:48) , Z (cid:48) ) be subsets of S and S so that λ DS = − Proj W Z χ ∗ D , λ DS = − Proj W Z χ ∗ D (resp. λ D (cid:48) S = − Proj W Z (cid:48) χ ∗ D (cid:48) , λ D (cid:48) S = − Proj W Z (cid:48) χ ∗ D (cid:48) . )Suppose on the contrary that the line segment DD (cid:48) intersects no type one wallsproperly. By Lemma 5.19, we then have(1) Proj W Z χ ∗ D (cid:48)(cid:48) (cid:54) = Proj W Z χ ∗ D (cid:48)(cid:48) for all D (cid:48)(cid:48) ∈ DD (cid:48) , and(2) Proj W Z (cid:48) χ ∗ D (cid:48)(cid:48) = Proj W (cid:48) Z χ ∗ D (cid:48)(cid:48) for all D (cid:48)(cid:48) ∈ DD (cid:48) .We claim that Proj W Zi χ ∗ D (cid:54) = Proj W Z (cid:48) i χ ∗ D for at least one of i = 1 ,
2. Suppose not,then by Item 2, we haveProj W Z χ ∗ D = Proj W Z (cid:48) χ ∗ D = Proj W Z (cid:48) χ ∗ D = Proj W Z χ ∗ D , contradicting the assumption that L ( S ) and L ( S ) are in different χ D -strata.The claim is proved.Suppose Proj W Z χ ∗ D (cid:54) = Proj W Z (cid:48) χ ∗ D . By Lemma 5.19,Proj W Z χ ∗ D (cid:48) (cid:54) = Proj W Z (cid:48) χ ∗ D (cid:48) as well. By Lemma 5.20 (applied to S ), the line segment DD (cid:48) intersects a typeone wall properly, a contradiction. The same arguments works for the case thatProj W Z χ ∗ D (cid:54) = Proj W Z (cid:48) χ ∗ D . The theorem is proved. (cid:3) Theorem 5.22.
Let X Σ be a projective toric variety and D, D (cid:48) be two elementsin
Amp( X Σ ) R . If the stratifications induced by D and D (cid:48) undergo a type twovariation and the line segment DD (cid:48) intersects no type one walls properly, then DD (cid:48) intersects a type two wall properly.Proof. Without loss of generality, we may pick a pair S , S of elements in L sothat M D ( x ) > M D ( y ) but M D (cid:48) ( x ) ≤ M D (cid:48) ( y ) for all x ∈ L ( S ) , y ∈ L ( S ) . By Corollary 5.8, there exists a D (cid:48)(cid:48) ∈ DD (cid:48) such that M D (cid:48)(cid:48) ( x ) = M D (cid:48)(cid:48) ( y ) for all x ∈ L ( S ) and y ∈ L ( S ). Suppose M D (cid:48)(cid:48) ( x ) = || Proj Z (cid:48)(cid:48) χ ∗ D (cid:48)(cid:48) || and M D (cid:48)(cid:48) ( y ) = || Proj W Z (cid:48)(cid:48) χ ∗ D || where Z (cid:48)(cid:48) , Z (cid:48)(cid:48) are subsets of S and S respectively. Let us considerthe collection H = { ˜ D ∈ Amp( X Σ ) R (cid:12)(cid:12) || Proj W Z (cid:48)(cid:48) χ ∗ ˜ D || = || Proj W Z (cid:48)(cid:48) χ ∗ ˜ D ||} . Obviously D (cid:48)(cid:48) ∈ H ∩ DD (cid:48) . We will show that H is a type two wall thatintersects DD (cid:48) properly. For this, we prove that(1) there is no containment between W Z (cid:48)(cid:48) and W Z (cid:48)(cid:48) , and (2) H ∩ DD (cid:48) (cid:40) DD (cid:48) . Note that statement (1) and the fact that D (cid:48)(cid:48) ∈ H implies that H is either theentire ample cone or a type two wall. Statement (2) implies that H is a type twowall and H intersects the line segment DD (cid:48) properly.To prove (1), suppose there is a containment, say W Z (cid:48)(cid:48) ⊂ W Z (cid:48)(cid:48) . Then the con-dition that || Proj W Z (cid:48)(cid:48) χ ∗ D (cid:48)(cid:48) || = || Proj W Z (cid:48)(cid:48) χ ∗ D (cid:48)(cid:48) || actually implies Proj W Z (cid:48)(cid:48) χ ∗ D (cid:48)(cid:48) =Proj W Z (cid:48)(cid:48) χ ∗ D (cid:48)(cid:48) , causing a type one variation at D (cid:48)(cid:48) on the line segment DD (cid:48) .By Theorem 5.21, the line segment DD (cid:48)(cid:48) intersects a type one wall properly, acontradiction to the assumption about the line segment DD (cid:48) . Statement (1) isproved.For statement (2), let Z , Z be subsets of S and S respectively such that λ DS = − Proj W Z χ ∗ D and λ DS = − Proj W Z χ ∗ D . We claim that Proj W Zi χ ∗ D =Proj W Z (cid:48)(cid:48) i χ ∗ D for both i = 1 , i = 1, then by Lemma 5.19, Proj W Z χ ∗ D (cid:48)(cid:48) (cid:54) =Proj W Z (cid:48)(cid:48) χ ∗ D (cid:48)(cid:48) as well. Lemma 5.20 (applied to S ) then implies that the linesegment DD (cid:48)(cid:48) intersects a type one wall properly, a contradiction to the assumptionabout DD (cid:48) . The same argument works for the case i = 2. Hence the claim is true.Now suppose on the contrary, that H ∩ DD (cid:48) = DD (cid:48) . The claim together withthe assumption that H ∩ DD (cid:48) = DD (cid:48) then imply M D ( x ) = || Proj W Z χ ∗ D || = || Proj W Z (cid:48)(cid:48) χ ∗ D || = || Proj W Z (cid:48)(cid:48) χ ∗ D || = || Proj W Z χ ∗ D || = M D ( y )for all x ∈ L ( S ) and y ∈ L ( S ), a contradiction. Statement (2) and therefore thetheorem is proved. (cid:3) With Proposition 5.12, Proposition 5.14, Theorem 5.18, Theorem 5.21, andTheorem 5.22, we obtain the summary:
Theorem 5.23.
Let X Σ be a projective toric variety. There are two types of wallsin the cone of ample divisors, called type one walls and type two walls respectively.The following properties hold for walls:(1) There are finitely many walls.(2) A type one (resp. type two) wall is a rational hyperplane (resp. homoge-neous quadratic hypersurface).(3) Type one walls capture type one variations in the following sense: Let D, D (cid:48) be two R -ample divisors on X Σ . Then the stratifications inducedby D and D (cid:48) undergo a type one variation only if the line segment DD (cid:48) intersects a type one wall properly in the ample cone. Namely, DD (cid:48) crossesa type one wall.(4) Type two walls capture type two variations in the following sense: Let D, D (cid:48) be two R -ample divisors on X Σ . If the stratifications induced by D and D (cid:48) undergo a type two variation and if the line segment DD (cid:48) intersectsno type one walls properly, then DD (cid:48) intersects a type two wall properlyin the ample cone. Namely, DD (cid:48) crosses a type two wall.Away from the walls, the ample cone is decomposed into semi-chambers such thatthe following properties hold:(1) There are finitely many semi-chambers.(2) A semi-chamber is a cone, possibly not convex. ARIATION OF STRATIFICATIONS FROM TORIC GIT 37 (3) The stratification stays constant in a semi-chamber in the following sense:If two R -ample divisors are in the same semi-chamber, then they induceequivalent stratifications of C Σ(1) . Variation of stratification - an elementary example.
The followingexample illustrates the two types of variations. This is from the smooth toricsurface obtained by blowing up P C at a point.5.5.1. Set up.
Consider the complete fan u u u u τ τ τ τ where(1) u = ( − , u = (0 , u = (1 , u = (0 , − τ = Cone ( u , u ),(2) τ = Cone ( u , u ),(3) τ = Cone ( u , u ),(4) τ = Cone ( u , u ).We will compute the stratification of C Σ(1) = C induced by all ample divisors on X Σ . Let us first compute the ample cone of X Σ .5.5.2. The ample cone.
To begin with, sequence (4.2) in this case is 0 → Z → Z → Cl( X Σ ) → Z → Z is defined by the matrix − − . There-fore, we have D = D and D + D = D in Cl( X Σ ). One also checks that { D , D } is linearly independent in Cl( X Σ ) so that it forms a Z -basis of Cl( X Σ ).With respect to this basis sequence (4.2) is represented by(5.7) 0 → Z → Z → Z → Z → Z is given by the matrix (cid:18) (cid:19) . Let D = a D + a D be a divisor. Then we have(1) m τ = ( a − a , − a ),(2) m τ = (0 , − a ),(3) m τ = (0 , (4) m τ = ( a , D is ample if and only if (cid:104) m τ , u (cid:105) = a − a > , (cid:104) m τ , u (cid:105) = a > , (cid:104) m τ , u (cid:105) = − a > − a , (cid:104) m τ , u (cid:105) = a > , (cid:104) m τ , u (cid:105) = 0 > − a , (cid:104) m τ , u (cid:105) = 0 > − a , (cid:104) m τ , u (cid:105) = 0 > − a , (cid:104) m τ , u (cid:105) = a > . This system of inequalities is essentially a > a >
0. We now compute severalthings about G .5.5.3. The group G . Applying Hom Z ( − , C × ) to sequence (5.7), we see that G (cid:39) ( C × ) injects into ( C × ) Σ(1) = ( C × ) by the formula( t , t ) (cid:55)→ ( t , t , t , t · t ) . Applying Hom Z ( − , Z ) to sequence (5.7), we see that ΓΓΓ( G ) (cid:39) Z injects into Z bythe matrix with respect to the dual basis { D ∗ , D ∗ } of ΓΓΓ( G ). It followsthat the inner product ( − , − ) on ΓΓΓ( G ) R satisfies( aD ∗ + bD ∗ , cD ∗ + dD ∗ ) = ac + bd + ac + ( a + b ) · ( c + d ) . From now on we will write a divisor on X Σ as a tuple ( a , a ) instead of a D + a D . Similarly we write an element in ΓΓΓ( G ) R as a tuple ( a, b ) instead of aD ∗ + bD ∗ .5.5.4. Stratifications induced by R -ample divisors. We are now ready to analyzethe stratification induced by all R -ample divisors. Here the primitive collectionsare { ρ , ρ } and { ρ , ρ } so L = {∅} , { } , { } , { } , { } , { , } , { , } . The idea is to write the one parameter subgroup λ DS that is χ D -adapted to L ( S )as a piecewise function of D for each S ∈ L . For brevity, when we write λ DS , Λ DS σ S and W S , we do not separate elements in S by commas. We will write L S instead of L ( S ) and without commas as well.It is computed that(5.8) − χ ∗ D = ( − a + a , a − a . and − Proj W χ ∗ D = (0 ,
0) as W = 0For S = ∅ , note that L ∅ is the origin in C and σ ∅ = ΓΓΓ( G ) R . Hence it isobvious that λ D ∅ = − χ ∗ D .For S = { } , { } , or { , } , we have σ S = { ( a, b ) ∈ ΓΓΓ( G ) R | a ≥ } . Looking atEquation (5.8), − χ ∗ D ∈ Λ DS if and only if − a + a ≥
0. This is not possible inthe ample cone. It is computed that − Proj W S χ ∗ D = (0 , − a ) . We conclude that λ D = λ D = λ D = (0 , − a . ARIATION OF STRATIFICATIONS FROM TORIC GIT 39
For S = { } , we have σ = { ( a, b ) ∈ ΓΓΓ( G ) R | b ≥ } . Looking at Equation (5.8),we have − χ ∗ D ∈ Λ D if and only if a − a ≥
0, which is possible in the amplecone. It is computed that − Proj W χ ∗ D = ( − a , . We conclude that λ D = (cid:40) − χ ∗ D if a − a ≥ , or − Proj W χ ∗ D = ( − a ,
0) otherwise . For S = { } , we have σ = { ( a, b ) ∈ ΓΓΓ( G ) R | a + b ≥ } . Looking at Equa-tion (5.8), we have − χ ∗ D ∈ Λ D if and only if − ( a + 2 a ) ≥
0, which is impossiblein the ample cone. It is computed that − Proj W χ ∗ D = ( − a + a , a − a ). Weconclude that λ D = − Proj W χ ∗ D = ( − a + a , a − a . For S = { , } , we have σ = { ( a, b ) ∈ ΓΓΓ( G ) R | a + b, b ≥ } . By what we had, − χ ∗ D , − Proj W χ ∗ D are not in Λ D , but − Proj W χ ∗ D ∈ Λ D . On the other hand W ⊂ W so that || Proj W χ ∗ D || ≤ || Proj W χ ∗ D || . We conclude that λ D = − Proj W χ ∗ D = ( − a + a , a − a . We obtain the following description of the set of χ D -strata for each ampledivisor D = ( a , a ): If a − a ≥
0, then • L ∅ ∪ L = V ( x , x , x ) is the χ D -stratum indexed by − χ ∗ D , • L ∪ L ∪ L = V ( x , x ) − V ( x , x ) is the χ D -stratum indexed by(0 , − a ), and • L ∪ L = V ( x , x ) − V ( x ) is the χ D -stratum indexed ( − a + a , a − a )If a − a <
0, then • L ∅ = V ( x , x , x , x ) is the χ D -stratum indexed by − χ ∗ D , • L = V ( x , x , x ) − V ( x ) is the χ D -stratum indexed by ( − a , • L ∪ L ∪ L = V ( x , x ) − V ( x , x ) is the χ D -stratum indexed by(0 , − a ), and • L ∪ L = V ( x , x ) − V ( x ) is the χ D -stratum indexed by ( − a + a , a − a ).Crossing the linear wall a − a = 0, we see that the stratification undergoes atype one variation. There are type two variations among the order of the stratawhich we now describe.When a − a ≥
0, we only need to compare the order between the χ D -stratumindexed by (0 , − a ) and ( − a + a , a − a ). We have || ( − a + a , a − a || = a − a √ ≥ a √ > a √ || (0 , − a || . Hence the χ D -strata are ordered as V ( x , x , x ) > V ( x , x ) − V ( x ) > V ( x , x ) − V ( x , x ) . When a − a <
0, we have to compare the order between the χ D -strataindexed by ( − a , , (0 , − a ) and ( − a + a , a − a ). Note that we always have || ( − a , || = a √ > a − a √ || ( − a + a , a − a || . It remains to compare (0 , − a ) , ( − a + a , a − a ) and ( − a , , (0 , − a ).It is computed that || (0 , − a || > || ( − a + a , a − a || ⇔ a < (1 + (cid:114)
32 ) · a
10 CHI-YU CHENG and || (0 , − a || > || ( − a , || ⇔ a < (cid:114) · a . Summary.
We may divide the ample cone into six regions where each regioncorresponds to a distinct class of stratification of C (see Table 1). a a a = 3 a a = (1 + (cid:113) ) a a = (cid:113) a a = a Here the line defined by a = a and the a -axis form the boundary of the amplecone. The two blue lines a = (cid:113) a and a = (1 + (cid:113) ) a were labeled as 5 and3. They are the type two walls with respect to { } , { } and { } , { } respectively.The red line a = 3 a is the type one wall with respect to ∅ , { } . Table 1.
Stratifications induced by R -ample divisors in eacharea. Here represents the origin in C .Area Stratification1 V ( x , x , x ) > V ( x , x ) − V ( x ) > V ( x , x ) − V ( x , x )2 > V ( x , x , x ) − V ( x ) > V ( x , x ) − V ( x ) > V ( x , x ) − V ( x , x )3 > V ( x , x , x ) − V ( x ) > V ( x , x ) − V ( x ) > V ( x , x , x ) − V ( x ) > V ( x , x ) − V ( x , x )4 > V ( x , x , x ) − V ( x ) > V ( x , x ) − V ( x , x ) > V ( x , x ) − V ( x )5 > V ( x , x , x ) − V ( x ) > V ( x , x ) − V ( x ) > V ( x , x ) − V ( x , x ) > V ( x , x ) − V ( x )6 > V ( x , x ) − V ( x , x ) > V ( x , x , x ) − V ( x ) > V ( x , x ) − V ( x )We see that:(1) There are six areas corresponding to six SIT-equivalence classes of R -ampledivisors.(2) The stratification does not change inside a semi-chamber.(3) Type two walls capture type two variations. More specifically crossing atype two wall swaps the order of a pair of strata and on the wall the strictpartial order breaks up into two linearly ordered chains.(4) The type one wall captures a type one variation. More specifically crossingthe type one wall downwards combines L with into a single strata. ARIATION OF STRATIFICATIONS FROM TORIC GIT 41
Relations to the structure of the fan.
A VSIT adjunction.
In this section we prove that toric VSIT is intrinsicto the two combinatorial structures of a fan: its primitive collections and therelations among its ray generators. For this we formulate an equivalence betweencomplete fans that capture the two combinatorial structures.
Definition 5.24.
Let N and N be two free abelian groups of the same rank.Let Σ ⊂ ( N ) R , Σ ⊂ ( N ) R be two complete fans. We say Σ and Σ are amplyequivalent if there is a bijection Ψ : Σ (1) → Σ (1) between the rays such that thefollowing two properties hold:(1) Ψ preserves primitive collections. That is, for any subset C ⊂ Σ (1), Ψ( C )is a primitive collection of Σ if and only if C is a primitive collection ofΣ .(2) Ψ preserves relations among the ray generators. That is, for any tupleof integers { a ρ } ρ ∈ Σ (1) , (cid:80) ρ a ρ u ρ = 0 if and only if (cid:80) ρ a ρ u Ψ( ρ ) = 0 where u ρ (resp. u Ψ( ρ ) ) represents the ray generator of ρ (resp. Ψ( ρ )).We will show that for any pair of amply equivalent fans, the toric varieties associ-ated to them possess isomorphic Q -ample cones. To construct such isomorphisms,we first need Proposition 5.25.
Let N and N be two free abelian groups of rank n . Let Σ ⊂ ( N ) R , Σ ⊂ ( N ) R be two amply equivalent fans with the bijection Ψ :Σ (1) → Σ (1) that preserves their primitive collections and relations among theirray generators. Then there is a Q -linear isomorphism Φ : ( N ) Q → ( N ) Q suchthat(1) Φ( u ρ ) = u Ψ( ρ ) for all ρ ∈ Σ (1) , and(2) if Φ R is the extension of Φ over R , then a cone σ ⊂ ( N ) R is a cone in Σ if and only if Φ R ( σ ) is a cone in Σ . Moreover, if G i = Hom Z (Cl( X Σ i ) , C × ) , then(3) Φ induces an isomorphism ΓΓΓ( G ) Q (cid:39) ΓΓΓ( G ) Q .Proof. To construct the isomorphism Φ, pick n Q -linearly independent ray gener-ators { u ρ , . . . , u ρ n } from Σ . We may do so as Σ is complete. We then have a Q -linear map Φ : ( N ) Q → ( N ) Q defined by sending u ρ i to u Ψ( ρ i ) . By the assump-tion on the relation among ray generators, { u Ψ( ρ ) , . . . , u Ψ( ρ n ) } is independentover Q as well. The map Φ is therefore an isomorphism.We now show that Φ( u ρ ) = u Ψ( ρ ) for any other u ρ . Since { u ρ , . . . , u ρ n } isa Q -basis of ( N ) Q , there exist a nonzero K ∈ Z and integers a i such that K · u ρ = (cid:80) i a i u ρ i . By the assumption on the relation among ray generators, wealso have K · u Ψ( ρ ) = (cid:80) i a i u Ψ( ρ i ) . Therefore, we get K · Φ( u ρ ) = Φ( K · u ρ ) = (cid:88) i a i Φ( u ρ i ) = (cid:88) i a i u Ψ( ρ i ) = K · u Ψ( ρ ) . Hence we have Φ( u ρ ) = u Ψ( ρ ) for all ρ . This proves statement(1).For statement (2) on the cones, it is sufficient to show that Φ R ( σ ) ∈ Σ ( n )for any σ ∈ Σ ( n ) because we can apply the result to Φ − R on the cones in Σ ( n )and because a fan is completely determined by its maximal cones. We can furtherreduce to show that for any σ ∈ Σ ( n ), there exists a cone σ ∈ Σ ( n ) such that Φ R ( σ ) ⊂ σ . For then we can apply the result to Φ − R on σ and get anotherinclusion σ ⊂ Φ − R ( σ ) ⊂ ˜ σ for some ˜ σ ∈ Σ ( n ) . The maximality of σ as a cone in Σ would imply σ = Φ − R ( σ ) , which in turnimplies Φ R ( σ ) = σ .Let σ be a cone in Σ ( n ) and L = σ (1) be the set of rays in σ . Thenany subset of L (including L ) does not form a primitive collection of Σ . Byassumption any subset of Ψ( L ) does not form a primitive collection of Σ .Therefore, either(1) { u Ψ( ρ ) } ρ ∈ L is contained the list of ray generators of some cone in Σ , or(2) there is a proper subset L (cid:40) L such that { u Ψ( ρ ) } ρ ∈ L is not containedin the list of ray generators of any cone in Σ .Suppose condition (1) fails and let L (cid:40) L be the subset from condition (2). AsΨ( L ) does not form a primitive collection in Σ , the same two conditions canbe said about Ψ( L ). By construction of L , condition (1) fails for Ψ( L ) also.Hence we obtain a proper subset L (cid:40) L such that (1) fails for Ψ( L ). Thisprocess cannot continue indefinitely as L is a finite set. Therefore, condition (1)holds for Ψ( L ) in the first place. Namely, { u Ψ( ρ ) } ρ ∈ L is contained the list of raygenerators of some cone in Σ Let σ be a cone in Σ whose list of ray generators contains { u Ψ( ρ ) } ρ ∈ L . Thenclearly Φ R ( σ ) ⊂ σ . The validity of statement (2) is established.For statement (3), let Σ(1) be an indexing set of Σ and let { e i } i ∈ Σ(1) be thestandard basis of Q Σ(1) . Define the Q -linear maps π : Q Σ(1) → ( N ) Q (cid:0) resp. π : Q Σ(1) → ( N ) Q (cid:1) by e i (cid:55)→ u ρ i (cid:0) resp. e i (cid:55)→ u Ψ( ρ i ) (cid:1) . Then Φ is compatible with these two maps by statement (1). Each π i is surjectivewith kernel ΓΓΓ( G i ) Q by statement (4) from Proposition 4.1. Hence the isomorphismΦ fits into the following commutative diagram of short exact sequences(5.9) 0 ΓΓΓ( G ) Q Q Σ(1) ( N ) Q
00 ΓΓΓ( G ) Q Q Σ(1) ( N ) Q ϕ Φ . Since the right and middle vertical map are isomorphisms, the induced map ϕ : ΓΓΓ( G ) Q → ΓΓΓ( G ) Q is an isomorphism. (cid:3) Ample equivalence has the following implication when a complete fan is amplyequivalent to a smooth complete fan:
Corollary 5.26.
Let Σ ⊂ ( N ) R , Σ ⊂ ( N ) R be two amply equivalent completefans with Σ smooth. If Φ : ( N ) Q → ( N ) Q is the Q -linear isomorphismconstructed in Proposition 5.25, its restriction to N factors through N as a Z -linear monomorphism N (cid:44) → N . Moreover, Σ is simplicial and the followingstatements are equivalent:(1) One of the maximal cones in Σ is smooth.(2) The restriction of Φ to N factors through N as a Z -isomorphism N (cid:39) N .(3) Σ is a smooth fan. ARIATION OF STRATIFICATIONS FROM TORIC GIT 43
In this case, ϕ induces a toric isomorphism X Σ (cid:39) X Σ .Proof. Since Σ is smooth and complete, there is a maximal cone σ ∈ Σ whoseray generators form a Z -basis for N . Hence every element in N is a Z -combinationof ray generators of σ . By the construction of Φ, we have Φ( N ) ⊂ N . Namely,the restriction of Φ to N factors through N .Let ϕ : N → N be the factorization. Since Φ : ( N ) Q → ( N ) Q is anisomorphism, ϕ is injective.That Σ is simplicial follows directly from statement (2) in Proposition 5.25that every cone in Σ is the image of a smooth cone in Σ under the isomorphismΦ R .Finally, ϕ is an isomorphism if and only if ϕ sends a Z -basis of N to a Z -basisof N . The equivalence of the three statements now follows from statement (2) inProposition 5.25. (cid:3) Here is an example of a pair of amply equivalent fans where one of them issmooth but the other is only simplicial.
Example 5.27.
Let Σ ⊂ R be the complete fan with ray generators u = (1 , u = (0 , u = ( − , −
1) and let Σ ⊂ R be the complete fan with raygenerators v = (2 , v = (1 , −
1) and v = ( − , − (cid:80) i u i = (cid:80) i v i = (0 , u i ) (cid:55)→ Cone ( v i )is a bijection between rays that preserves relations among the ray generators.Moreover, both Σ (1) and Σ (1) are the only primitive collections for Σ and Σ respectively. Hence Σ and Σ are amply equivalent but Σ is not smooth.Equation (5.10) below is an equality that we will use extensively. Let k bea field. Suppose V i , W i for i = 1 , k with a perfect pairing (cid:104)− , −(cid:105) i : V i × W i → k that identifies V i and W i as dualsrespectively. Then for any k -linear map f : W → W and v ∈ V , w ∈ W , wehave(5.10) (cid:104) v , f ( w ) (cid:105) = (cid:104) f ∗ ( v ) , w (cid:105) where f ∗ : V → V is the dual of f .We may now make sense of the term ”ample equivalence” between fans. Weshow that two complete toric varieties where their fans are amply equivalent haveisomorphic ample cones over the field of rational numbers (and therefore over thefield of real numbers as well). Proposition 5.28.
Let Σ and Σ be two amply equivalent fans and G i = Hom Z (Cl( X Σ i ) , C × ) for i = 1 , . Let ϕ : ΓΓΓ( G ) Q → ΓΓΓ( G ) Q be the isomorphism from statement (3) of Proposi-tion 5.25. Then the dual of ϕ is an isomorphism ϕ ∗ : Cl( X Σ ) Q → Cl( X Σ ) Q thatrestricts to Pic( X Σ ) Q (cid:39) Pic( X Σ ) Q with ϕ ∗ (Amp( X Σ ) Q ) = Amp( X Σ ) Q .Proof. Let Ψ : Σ (1) (cid:39) Σ (1) be the bijection between rays that preserves theprimitive collections and relations among the ray generators. By statment (4)from Proposition 4.1, the short exact sequences dual to those in diagram (5.9) fit into the following commutative diagram:0 ( M ) Q Q Σ(1)
Cl( X Σ ) Q
00 ( M ) Q Q Σ(1)
Cl( X Σ ) Q Φ ∗ ϕ ∗ where ϕ ∗ is an isomorphism and sends a class of Q -Weil divisor (cid:80) i q i D Ψ( ρ i ) inCl( X Σ ) Q to the class of (cid:80) i q i D ρ i in Cl( X Σ ) Q . We now show that ϕ ∗ restrictto an isomorphism Pic( X Σ ) Q (cid:39) Pic( X Σ ) Q .For this it is enough to show that ϕ ∗ (Pic( X Σ ) Q ) ⊂ Pic( X Σ ) Q becausewe may apply the result to the inverse of ϕ ∗ and obtain another inclusion( ϕ ∗ ) − (Pic( X Σ ) Q ) ⊂ Pic( X Σ ) Q . We then havePic( X Σ ) Q ⊂ ( ϕ ∗ ) − (Pic( X Σ ) Q ) ⊂ Pic( X Σ ) Q so that Pic( X Σ ) Q ) = ( ϕ ∗ ) − (Pic( X Σ ) Q ) , which in turn implies ϕ ∗ (Pic( X Σ ) Q ) =Pic( X Σ ) Q .Let D = (cid:80) i q i D Ψ( ρ i ) be a Q -Cartier divisor on X Σ . We need to construct aninteger K > K · ϕ ∗ ( D ) is Cartier on X Σ . For this, first let us choosean integer K (cid:48) > K (cid:48) · q ρ ∈ Z for all ρ , and(2) K (cid:48) · D is Cartier on X Σ .Write K (cid:48) · D as (cid:80) i a i D Ψ( ρ i ) . Then by Theorem 4.2, for every maximal cone σ inΣ , there exists an m σ ∈ M such that (cid:104) m σ , u Ψ( ρ i ) (cid:105) = − a i for all Ψ( ρ i ) ∈ σ (1) . Now choose an integer K (cid:48)(cid:48) large enough so that K (cid:48)(cid:48) · Φ ∗ ( m σ ) ∈ M for all σ ∈ Σ ( n ). We claim that K (cid:48) · K (cid:48)(cid:48) · ϕ ∗ ( D ) is Cartier on X Σ .For this, let σ be a maximal cone in Σ . Then by Proposition 5.25, Φ( σ ) is amaximal cone in Σ . Let (cid:104)− , −(cid:105) i : ( M i ) Q × ( N i ) Q → Q be the Q -extension of thenatural perfect pairing. Then by Equation (5.10), we have (cid:104) K (cid:48)(cid:48) · Φ ∗ ( m Φ( σ ) ) , u ρ i (cid:105) = (cid:104) K (cid:48)(cid:48) · m Φ( σ ) , u Ψ( ρ i ) (cid:105) = − K (cid:48)(cid:48) · a i for all ρ i ∈ σ (1) . By Theorem 4.2, K (cid:48)(cid:48) · K (cid:48) · ϕ ∗ ( D ) is Cartier on X Σ , proving the equality ϕ ∗ (Pic( X Σ ) Q ) =Pic( X Σ ) Q . The statement on ample cones can be proved similarly using Theo-rem 4.2 and Equation (5.10). The proposition is proved. (cid:3)
We are now ready to prove that the variation of stratifications from toric GITis intrinsic to the primitive collections and relations among the ray generators ofa fan. If two complete fans Σ , Σ are amply equivalent, then by Proposition 4.8,the unstable loci in their GIT quotient constructions can be identified. We willwrite Z (Σ) as the common unstable locus. Theorem 5.29 (VSIT Adjunction) . Suppose Σ and Σ are two amply equivalentfans. Let G i = Hom Z (Cl( X Σ i ) , C × ) for i = 1 , and ϕ : ΓΓΓ( G ) Q (cid:39) ΓΓΓ( G ) Q be the isomorphism from Proposition 5.25. Then theassignment of the strata (5.11) S ϕ ( χ D ) λ S χ D ϕ ∗ ( λ ) defined for all D ∈ Amp( X Σ ) Q and λ ∈ ΓΓΓ( G ) Q is an equivalence of stratificationof Z (Σ) . The theorem also holds for D ∈ Amp( X Σ ) R and λ ∈ ΓΓΓ( G ) R . ARIATION OF STRATIFICATIONS FROM TORIC GIT 45
Proof.
Recall that we had a diagram on short exact sequences0 ΓΓΓ( G ) Q Q Σ(1) ( N ) Q
00 ΓΓΓ( G ) Q Q Σ(1) ( N ) Q ϕ Φ . Let || − || i be the norm on ΓΓΓ( G i ) R . Recall that the norms on ΓΓΓ( G i ) comes fromrestricting the standard norm on ΓΓΓ(( C × ) Σ(1) ) . Therefore, for every λ ∈ ΓΓΓ( G ) Q ,we have || λ || = || ϕ ( λ ) || . Because of this and Equation (5.10), we get(5.12) (cid:104) ϕ ∗ ( χ D ) , λ (cid:105) G || λ || = (cid:104) χ D , ϕ ( λ ) (cid:105) G || ϕ ( λ ) || for all χ D ∈ Cl( X Σ ) Q and λ ∈ ΓΓΓ( G ) Q where (cid:104)− , −(cid:105) G i : χχχ ( G i ) Q × ΓΓΓ( G i ) Q → Q denotes the Q -extension of the naturalpairing.Recall that for any x ∈ Z (Σ) and D ∈ Amp( X Σ ) R , λ Dx is the vector in ΓΓΓ( G ) R that is χ D -adapted to x (Definition 5.4). Also note that when D is in Amp( X Σ ) Q ,the vector λ Dx is in ΓΓΓ( G ) Q .Equation (5.12) and the fact that ϕ preserves norms imply that for any x ∈ Z (Σ), λ = λ ϕ ∗ ( D ) x ⇔ ϕ ( λ ) = λ Dx for all D ∈ Amp( X Σ ) Q , λ ∈ ΓΓΓ( G ) Q . This in turn implies that the assignment (5.11) is a bijection of strata that preserveseach stratum as sets. That assignment (5.11) preserves strict partial order followsfrom the fact that ϕ preserves norms. The proof easily extends over R . Thetheorem is proved. (cid:3) Topological properties of the strata.
In this section we supply some topolog-ical properties of the strata. We first note a property satisfied by the primitivecollections of a fan.
Proposition 5.30.
Let Σ ⊂ N R be a complete fan. Then the union of primitivecollections is Σ(1) . Proof.
Let ρ ∈ Σ(1) be a ray. We will construct a primitive collection that contains ρ . Since Σ is complete, there exists a σ ∈ Σ max such that − u ρ is in σ. Then since σ is strongly convex, Cone ( u ρ ) ∩ σ = { } . Hence ˜ σ := Cone ( u ρ , σ ) is a coneproperly containing σ . This implies ˜ σ is not contained in σ (cid:48) for any σ (cid:48) ∈ Σ max .This implies { ρ } ∪ σ (1) is not contained in any σ (cid:48) (1). We make take S ⊂ σ (1) tobe a subset minimal with respect to this property: { ρ } ∪ S is not contained inany σ (cid:48) (1) for any σ (cid:48) ∈ Σ max . It follows from the definition of primitive collections(Definition 4.5) that { ρ } ∪ S is a primitive collection. (cid:3) Note that the analysis we did in Section 3.5 can be extended over the amplecone. In particular, Theorem 3.17 can be applied to stratifications induced by R -ample divisors to derive the Theorem 5.31.
Let D be an R -ample divisor of the projective toric variety X Σ .Then for each λ ∈ Λ D , the stratum S χ D λ is smooth, irreducible and open inside thezariski closure S χ D λ . More precisely, S χ D λ = V ( { x ρ | ρ ∈ S } ) − ∪ R V ( { x ρ | ρ ∈ R } ) where S is a subset of Σ(1) containing some primitive collection and R runsthrough some collection of subsets of Σ(1) − S . Proof.
By Theorem 3.17, each χ D -stratum is of the form S χ D λ = V ( { x ρ | ρ ∈ S } ) ∩ (cid:0) ∪ j ( D ( A j ) (cid:1) where each A j is contained in the complement Σ(1) − S . Since S χ D λ = V ( { x ρ | ρ ∈ S } ) ⊂ Z (Σ) is irreducible, by Proposition 4.8, there is a primitive collection C such that V ( { x ρ | ρ ∈ S } ) ⊂ V { ( x ρ | ρ ∈ C } ). Equivalently this means C ⊂ S . Thetheorem is proved. (cid:3) Corollary 5.32.
Let D be an R -ample divisor of the projective toric variety X Σ .Then for every primitive collection C ⊂ Σ(1) , there is a unique λ C ∈ Λ D suchthat S χ D λ C = V ( { x ρ | ρ ∈ C } ) . Proof.
By Proposition 4.6, each primitive collection C corresponds to the irre-ducible component V ( { x ρ | ρ ∈ C } ) of Z (Σ). We also have Z (Σ) = (cid:91) λ ∈ Λ D S χ D λ = (cid:91) λ ∈ Λ D S χ D λ . Since each S χ D λ is irreducible, the maximal S χ D λ ’s correspond to the irreduciblecomponents of Z (Σ). Namely, for each primitive collection C , there is a λ C ∈ Λ D such that S χ D λ C = V ( { x ρ | ρ ∈ C } ) . The uniqueness of λ C follows from the fact that the boundary of a stratum onlycontains higher strata. (cid:3) We can also derive the
Corollary 5.33.
Let D be an R -ample divisor of the projective toric variety X Σ .For every primitive collection C ⊂ Σ(1) , let λ C be as in Corollary 5.32. We have = (cid:92) λ ∈ Λ D S χ D λ = (cid:92) primitive collection C S χ D λ C where stands for the origin in C Σ(1) .Proof.
By Proposition 5.30, the intersection (cid:84) C V ( { x ρ | ρ ∈ C } ) taken over allprimitive collections C of Σ(1) is the origin. By Corollary 5.32, we have (cid:84) C S χ D λ C = (cid:84) C V ( { x ρ | ρ ∈ C } ). Moreover, each χ D -stratum’s closure S χ D λ is a linear subspaceso ∈ (cid:84) λ ∈ Λ D S χ D λ . In sum, we then have ∈ (cid:92) λ ∈ Λ D S χ D λ ⊂ (cid:92) C S χ D λ C = (cid:92) C V ( { x ρ | ρ ∈ C } ) = . The corollary is proved. (cid:3) The computer program
Given a projective toric variety, the computer program works out the(1) ample cone (Section 6.1, Section 6.2),(2) potential real one parameter subgroups that index the unstable strata inits GIT quotient construction (Section 6.3),(3) walls in the ample cone (Section 6.4),(4) stratification induced by a particular ample divisor (Section 6.5), and(5) visualization of the wall and semi-chamber decomposition of the amplecone if the ample cone is less than three dimensional (Section 6.6).
ARIATION OF STRATIFICATIONS FROM TORIC GIT 47
The computer program is solely sageMath where a number of packages for toricvarieties are already available, such as constructing fans and enumerating theirprimitive collections. Although there are computer programs in SageMath andMacaulay2 that compute ample cones, the output is difficult to utilize internallyin our program. Hence it became more desirable that we design the ample conecomputation by ourselves. The first challenge is to figure out the basis of thePicard group for a toric variety. Again the output from SageMath and Maculay2is not easy to dispose of within our program. We therefore formulate in Section 6.1a particular class of projective toric varieties whose basis of the Picard group isfairly easy to describe and with which the computer program performs correctlyand self-containedly.However, the ampleness condition and the equations of walls can be computedin terms of divisors of the form D = (cid:80) ρ a ρ D ρ without referencing to any basis ofthe Picard group (Theorem 4.2, Proposition 5.12). Because of this, our programoutputs ampleness conditions and walls as inequalities and equations respectivelyin terms of { a ρ } ρ ∈ Σ(1) . An advantage to this is that users have the flexibility tospecialize the outputs to any basis they have at their hands.We will walk through the computer program with the easiest non-trivial examplegiven by the projective toric variety P C × P C . It comes from the fan Σ ex in R consisting of the four rays(1) ρ = Cone ( e ),(2) ρ = Cone ( − e ),(3) ρ = Cone ( e ),(4) ρ = Cone ( − e ),and the four maximal cones(1) Cone ( ρ , ρ ),(2) Cone ( ρ , ρ ),(3) Cone ( ρ , ρ ),(4) Cone ( ρ , ρ ) . At the start of the program, the user is asked to enter a fan as a list of thecoordinates of rays, followed by list of ambient indicies for each maximal cone inthe fan. In the case of Σ ex , the list of rays is (1 , , ( − , , (0 , , (0 , −
1) and thelist of ambient indicies is (0 , , (1 , , (1 , , (0 , . Computing the Picard group.
We consider projective toric varieties X Σ whose Picard groups satisfy(6.1) Pic( X Σ ) R (cid:39) (cid:77) ρ/ ∈ σ (1) R · D ρ for some maximal cone σ ∈ Σ . We will see in Corollary 6.4 that condition (6.1) includes all simplicial projectivetoric varieties.However, the computer program does not detect if condition (6.1) holds whena fan is given. The output of ample cone may not be correct if condition (6.1)fails. Users may use a characterization given as Corollary 6.2 and Theorem 4.2 todetermine if their projective toric variety satisfies condition (6.1). First, we need
Proposition 6.1.
Let Σ be a fan with a full dimensional cone σ . Then thenatural map CDiv T N ( X Σ ) → Pic( X Σ ) induces an isomorphism ϕ : { D = (cid:88) ρ a ρ D ρ ∈ CDiv T N ( X Σ ) | a ρ = 0 for all ρ ∈ σ (1) } (cid:39) Pic( X Σ ) . Proof.
Let D = (cid:80) ρ a ρ D ρ ∈ Pic( X Σ ). Then there exists m σ ∈ M as in Theo-rem 4.2. Then D ∼ D − div( χ m σ ) in Pic( X Σ ) and the later is in the image of ϕ . This shows ϕ is surjective. To prove that ϕ is injective, suppose ϕ ( D ) = 0where D = (cid:80) ρ/ ∈ σ (1) a ρ D ρ . Then by the short exact sequence (4.3), there existsan m ∈ M such that D = div( χ m ). In particular, (cid:104) m, u ρ (cid:105) = 0 for all ρ ∈ σ (1).Since σ is full dimensional, m = 0, This proves injectivity. (cid:3) Here is a characterization about when (6.1) holds:
Corollary 6.2.
Let X Σ be a toric variety and let σ ∈ Σ be a full dimensionalcone. Then the following statements are equivalent:(1) The inclusion Pic( X Σ ) ⊂ Z Σ(1) induces
Pic( X Σ ) R (cid:39) (cid:76) ρ/ ∈ σ (1) R · D ρ ,(2) the inclusion Pic( X Σ ) ⊂ Z Σ(1) induces
Pic( X Σ ) Q (cid:39) (cid:76) ρ/ ∈ σ (1) Q · D ρ , and(3) D ρ is Q -Cartier for all ρ / ∈ σ (1) .Proof. There is an inclusion Pic( X Σ ) ⊂ Z Σ(1) due to Proposition 6.1. Statement(1) and (2) are equivalent because R is faithfully flat over Q . The equivalence of(2) and (3) follows from Proposition 6.1. (cid:3) To see why the class of toric varieties that satisfies (6.1) includes projectivesimplicial toric varieties, we need the
Proposition 6.3.
Let X Σ be a toric variety. The following statements areequivalent:(1) X Σ is simplicial.(2) Every D ∈ Cl( X Σ ) is Q -Cartier. Namely, for every D ∈ Cl( X Σ ) , thereexists an integer K such that K · D ∈ Pic( X Σ ) .(3) Pic( X Σ ) is of finite index in Cl( X Σ ) . Namely, there exists an integer K such that K · D ∈ Pic( X Σ ) for all D ∈ Cl( X Σ ) .(4) The inclusion Pic( X Σ ) ⊂ Cl( X Σ ) induces Pic( X Σ ) Q (cid:39) Cl( X Σ ) Q .Proof. Statements (2),(3),(4) are equivalent because Cl( X Σ ) is finitely generatedand tensoring Q is flat and kills torsion. The only non-trivial part is to prove (2)is equivalent to (1). For this, let σ be a cone in Σ. Viewing σ itself as a fan, wehave the following short exact sequence M Z σ (1) Cl( X σ ) 0whose dual is 0 Cl( X σ ) ∨ Z σ (1) N ψ σ by (4.5). We see that X Σ is simplicial if and only if ψ σ is injective. This isequivalent to requiring that Cl( X σ ) ∨ = 0, which in turn is equivalent to Cl( X σ )is torsion. Because there are only finitely many σ and { X σ } σ ∈ Σ covers X Σ , (1)implies (2). For the converse, it is enough to show that Cl( X Σ ) is torsion forall σ ∈ Σ . Since the restriction Cl( X Σ ) → Cl( X σ ) is surjective, we may write adivisor in Cl( X σ ) as D | X σ for some divisor D on X Σ . By assumption, there existsa K > K · D is Cartier. Then K · D | X σ is also Cartier. The resultfollows from the fact that affine normal toric varieties does not have non-trivialPicard group. See proposition 4.2.2 in [CLS11]. (cid:3) Corollary 6.4.
Let Σ be simplicial and σ ∈ Σ be a full dimensional cone.We have that Pic( X Σ ) Q (cid:39) Cl( X Σ ) Q has Q -basis { D ρ } ρ/ ∈ σ (1) . If Σ is smooth, Pic( X Σ ) (cid:39) Cl( X Σ ) is free with Z -basis { D ρ } ρ/ ∈ σ (1) . ARIATION OF STRATIFICATIONS FROM TORIC GIT 49
Proof.
By Proposition 6.3, every D ρ is Q -Cartier. Using statement (3) fromCorollary 6.2, we get the result. (cid:3) Hence we get that the class of toric varieties that satisfies (6.1) includes simplicialprojective toric varieties. For the fan Σ ex , condition (6.1) is satisfied say for thecone Cone( ρ , ρ ). For brevity, for a ray ρ i , instead of writing D ρ i (resp. a ρ i ), wewrite D i (resp. a i ). With these notations, Pic( X Σ ex ) R has basis D , D . In fact, D = O (1 ,
0) and D = O (0 ,
1) on P C × P C so that a D + a D = O ( a , a ) . Computing the ample cone.
Recall that Σ max means the collection ofmaximal cones in Σ. The logic behind the computation of ample cones is mainlythe following:
Proposition 6.5.
Let Σ be a complete fan and D = (cid:80) ρ a ρ D ρ be a Cartier divisor.For every σ ∈ Σ max and every ρ (cid:48) / ∈ σ (1) , fix a relation (cid:80) ρ ∈ σ (1) b ρ u ρ = u ρ (cid:48) with b ρ ∈ Q for all ρ ∈ σ (1) . Then D is ample if and only if (cid:80) ρ ∈ σ (1) b ρ a ρ < a ρ (cid:48) forall σ ∈ Σ max and ρ (cid:48) / ∈ σ (1) .Proof. Let Σ be a fan in N R and M be the dual lattice of M . Since Σ is complete,all maximal cones in Σ are full dimensional. Hence we can solve for rationalrelations for every ray outside of a maximal cone. Fix a relation (not necessarilyunique) (cid:80) ρ ∈ σ (1) b ρ u ρ = u ρ (cid:48) for each maximal cone σ and each ρ (cid:48) / ∈ σ (1). Since D is Cartier, for every σ ∈ Σ max , there exists an unique m σ ∈ M such that (cid:104) m σ , u ρ (cid:105) = − a ρ for all ρ ∈ σ (1) by Theorem 4.2. In order for D to be ample, wealso need (cid:104) m σ , u ρ (cid:48) (cid:105) > − a ρ (cid:48) for all ρ (cid:48) / ∈ σ (1). But (cid:104) m σ , u ρ (cid:48) (cid:105) = (cid:88) ρ ∈ σ (1) (cid:104) m σ , b ρ u ρ (cid:105) = − (cid:88) ρ ∈ σ (1) a ρ b ρ . The result follows. (cid:3)
Therefore, computing the ample cone breaks down into the following steps:(1) Loop through each maximal cone σ in the fan and solve relations u ρ (cid:48) = (cid:80) ρ ∈ σ (1) b ρ u ρ for each ρ (cid:48) / ∈ σ (1).(2) Produce the list of normal vectors of the inequalities { a ρ (cid:48) − (cid:88) ρ ∈ σ (1) b ρ a ρ > (cid:12)(cid:12) σ ∈ Σ max , ρ (cid:48) / ∈ σ (1) } . Call this list L gen .Now we know when a Cartier divisor (cid:80) ρ ∈ Σ(1) a ρ D ρ is ample. Therefore the laststep is to,(3) Specialize L gen to Pic( X Σ ) R . Call this list L red .The gen in the list produced in step (2) stands for generic as we look at divisors (cid:80) ρ a ρ D ρ without restricting to any basis of Pic( X Σ ) R . To go from step (2) tostep(3), if condition (6.1) is satisfied for a maximal cone σ , then it is simply settingthe ρ -th component of vectors in L gen to be 0 for every ρ ∈ σ (1). In the programwe simply remove the ρ -th component of the vectors in L gen for ρ ∈ σ (1). We nowhave a reduced list of normal vectors. Call it L red . The vectors in L red can beeither thought of as coefficients of strict inequalities for a divisor (cid:80) ρ/ ∈ σ (1) a ρ D ρ tobe ample, or as the normal vectors of the supporting hyperplanes for the nef coneNef( X Σ ). Take Σ ex for example, the fan is smooth so all divisors are Cartier. The onlyrelations we have are u ρ + u ρ = 0 and u ρ + u ρ = 0 . Hence the divisor (cid:80) i =0 a i D i is ample if and only if a + a > a + a > . The output of L gen is the list (1 , , , , (0 , , ,
1) of normal vectors. We alsoknow that D = O (1 ,
0) and D = O (0 ,
1) form a basis for Pic( X Σ ) where(6.1) is satisfied for the cone σ = Cone ( ρ , ρ ). In this case L red = (1 , , (0 , a D + a D is ample if and only if a , a > Potential real one parameter subgroups.
This is the core part of theprogram as its output would be used internally to compute the stratificationinduced by an ample divisor, to enumerate walls, and to plot walls in the amplecone. The reader may want to review the notations introduced in Section 5.1.Specifically, this part of the program computes − Proj W Z χ ∗ D ∈ ΓΓΓ( G ) R for anysubset Z ∈ L and for any D = (cid:80) ρ a ρ D ρ . The adjective potential comes from thefact we deduced in Section 5.2 that the real one parameter subgroup in ΓΓΓ( G ) R that is χ D -adapted to L ( S ) for an S ∈ L is of the form − Proj W Z χ ∗ D for somesubset Z of S . We note that Remark 6.6.
The description of the subspace W Z ⊂ ΓΓΓ( G ) R is rather simple.Viewing ΓΓΓ( G ) R as a subspace of R Σ(1) via the inclusion from (4.5), we have thata point v ∈ R Σ(1) is in W Z if and only if v ∈ ΓΓΓ( G ) R and v ρ = 0 for all ρ ∈ Z .Computing potential real one parameter subgroups breaks down roughly intothe following steps:(1) Enumerate primitive collections of the fan Σ, and compute the set L .(2) Compute a Q -basis for W Z for each Z ∈ L .(3) For each Z ∈ L , solve Equation (5.4) to obtain − Proj W Z χ ∗ D .Each output − Proj W Z χ ∗ D is a vector in R Σ(1) whose components are Q -combination of the a ρ ’s. The user may specialize the results to the ample cone.We now explain how each step is carried out.For step (1), the primitive collections can be obtained by the .primitive collections()method applied to the fan. The set L being the union of the power set of Σ(1) − C over all primitive collections C , can be obtained by elementary set operations inSageMath.For step (2), note that the Q -extension of sequence (4.2) is(6.2) 0 Q dim Σ Q Σ(1)
Cl( X Σ ) Q B B ⊥ where B is a matrix whose ρ -th row is the coordinate of the ray generator u ρ in Z Σ(1) . The dual of the above sequence is(6.3) 0 ΓΓΓ( G ) Q Q Σ(1) Q dim Σ ( B ⊥ ) t B t . Hence ΓΓΓ( G ) R is the space of vectors u ∈ R Σ(1) such that B t · u = 0 . Let us aug-ment the matrix B by inserting column vectors e ρ for ρ ∈ Z . Call the augmentedmatrix B Z . Then by Remark 6.6, W Z is the space of vectors v ∈ R Σ(1) such that( B Z ) t · v = 0. The .right kernel() method when applied to ( B Z ) t , yields a matrix C whose columns consist of a Q -basis of W Z , finishing step (2). The matrix C is ARIATION OF STRATIFICATIONS FROM TORIC GIT 51 called the right kernel of ( B Z ) t simply because C consists of vectors v such that( B Z ) t · v = 0.For step (3), suppose W Z has the Q -basis { λ , . . . , λ q } ⊂ Q Σ(1) . WriteProj W Z χ ∗ D = (cid:80) qj =1 b j λ j . It then comes down to solve Equation (5.4) for each b j .This is no difficult task for SageMath, finishing step (3). We now explain the datastructure that is used to store the list of potential real one parameter subgroupsas the list is used extensively in later parts of the program.6.3.1. Data structure of potential real one parmaeter subgroups.
The list of poten-tial real one parameter subgroups has the following data structure. Each entry inthe list is a list [ v, l, || v || ]where v is a column vector in Q Σ(1) and l is a list of sets in L such that v = − Proj W Z χ ∗ D for all Z ∈ l . The reason why we included the squares || v || in thedata is that they will be used to compute stratifications and to determine thestrict partial order between strata (See Section 6.5).Now take Σ ex to demonstrate the ideas. For Σ ex , the primitive collections are[0 ,
1] and [2 , L = [[] , [2] , [3] , [2 , , [0] , [1] , [0 , ∅ . For brevity, for a subset Z ∈ L , say Z = [0 , W [0 , (resp. B [0 , ), we write W (resp. B ).In this case the matrix B = − − in sequence (6.2). Taking the subset Z = [0 , ∈ L for example, we have B = − − . Note that the sum of the first and the fourth column of M yields the thirdcolumn. This implies ( B ) t , ( B ) t , and ( B ) t have the same right kernel, whichis computed as . Solving Equation (5.4), we getProj W χ ∗ = Proj W χ ∗ D = Proj W χ ∗ D = − / ∗ a − / ∗ a − / ∗ a − / ∗ a . Here is the full output of potential real one parameter subgroups for X Σ ex . − / ∗ a − / ∗ a − / ∗ a − / ∗ a , [[0] , [1] , [0 , , / ∗ a + a ∗ a / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a , [[2] , [3] , [2 , , / ∗ a + a ∗ a / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a , [[]] , / ∗ a + a ∗ a / ∗ a +1 / ∗ a + a ∗ a / ∗ a The first entry of the list says that Proj W χ ∗ D = Proj W χ ∗ D = Proj W χ ∗ D for all D ∈ R Σ(1) , as was discussed.Let us specialize the results to the ample cone Amp( X Σ ex ) R . Since condition(6.1) is satisfied for the cone Cone( ρ , ρ ), we can first specialize the results toPic( X Σ ex ) R by setting a = a = 0. Here is the resulting list:(6.4) − / ∗ a − / ∗ a , [[0] , [1] , [0 , , / ∗ a − / ∗ a − / ∗ a , [[2] , [3] , [2 , , / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a , [[]] , / ∗ a + 1 / ∗ a . Finally, as was discussed earlier, D = a D + a D is ample if and only if a , a >
0. This implies − Proj W χ ∗ D (and therefore − Proj W χ ∗ D , − Proj W χ ∗ D )is on the ray R > · (0 , , − −
1) whenever D is ample. Similarly, − Proj W χ ∗ D (andtherefore − Proj W χ ∗ D , − Proj W χ ∗ D ) is on the ray R > · ( − , − , ,
0) whenever D is ample. Hence we know concretely what indivisible one parameter subgroupis on R > · ( − Proj W χ ∗ D ) or R > · ( − Proj W χ ∗ D ) for all ample D . In general, itis not possible to write the indivisible one parameter subgroup on the ray of theprojection − Proj W Z χ ∗ D for any Z ∈ L abstractly in terms of D .6.4. Enumerating walls.
Type one walls.
Recall a type one wall is a non-empty proper collectionof D ∈ Amp( X Σ ) R such that Proj W Z χ ∗ D = Proj W Z ∪{ ρ } χ ∗ D where Z, { ρ } ∈ L and W Z ∪{ ρ } is a codimension one subspace of W Z . The main reasoning behind typeone wall computation is the following elementary observation:
ARIATION OF STRATIFICATIONS FROM TORIC GIT 53
Proposition 6.7.
Let V be a finite dimensional real vector space and V ∨ be itsdual. Let σ ⊂ V be a full dimensional polyhedral cone and f ∈ V ∨ . Then thefollowing statements are equivalent:(1) The hyperplane H f = { v ∈ V | f ( v ) = 0 } intersects the interior of σ ,(2) Neither f nor − f is in the dual cone σ ∨ . Enumerating type one wall breaks down into the following steps:(1) Find Z, { ρ } of sets in L where W Z ∪{ ρ } is a codimension one subspace of W Z .(2) Let ν ρ : Pic( X Σ ) R → R be defined by D (cid:55)→ (cid:104) χ D ρ , Proj W Z χ ∗ D (cid:105) . We then apply Proposition 6.7 to the vector space Pic( X Σ ) R , the coneNef( X Σ ), and the function ν ρ to test if ker ν ρ ∩ Amp( X Σ ) R = ∅ . If not,then ker ν ρ ∩ Amp( X Σ ) R is a type one wall.For step (1), in the program each W Z is represented as the column space ofthe right kernel of B tZ (See Section 6.3). SageMath can count the dimensions of W Z and W Z ∪{ ρ } to finish step (1).For step (2), if Z, { ρ } is a pair obtained in step (1), the function ν ρ is describedby the ρ -th component of Proj W Z χ ∗ D stored in the list of potential real oneparameter subgroups. Next, the nef cone was already known in the first step ofthe program outlined in Section 6.2. Taking V and σ in Proposition 6.7 to bePic( X Σ ) R and Nef( X Σ ) respectively, we apply the .dual() method in SageMathto return the dual of Nef( X Σ ). The .contains() method can then be used todetermine if ± ν ρ is in Nef( X Σ ) ∨ , finishing step (3).For the fan Σ ex , consider the pair of subsets ([] , [0]). It can be checked that W is a codimension one subspace of ΓΓΓ( G ) R . The condition that χ ∗ D = Proj W χ ∗ D amounts to setting the first component of χ ∗ D to 0. Referring back to list (6.4),this amounts to setting − a = 0, which is impossible in the ample cone. In fact,there are no type one walls for the fan Σ ex .6.4.2. Type two walls.
Recall a type two wall with respect to Z , Z ∈ L is of theform { D ∈ Amp( X Σ ) R | || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D ||} where W Z (cid:54)⊂ W Z and W Z (cid:54)⊂ W Z . To detect the containment, we would usethe .is subspace() method. The equations of type two walls are immediate as wealready recorded the norm of each potential real one parameter subgroups. Thenon-trivial task is to determine if the equation has solutions inside the amplecone. Due to computational difficulties, we currently do not test if a quadratichomogeneous polynomial has a solution in the ample cone or not. Instead, welist equations || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D || for any pair W Z , W Z withoutcontainment. Doing this does not affect visualization of the decomposition ofample cone. SageMath can plot graphs in a certain region so anything outside theample cone will not be plotted.6.4.3. Data structure of walls.
We store walls as a list where each entry is a list[ f, l ] . We explain what f and l are for each type of wall.For type one walls, the l in the entry [ f, l ] is a list of pairs ( Z, { ρ } ) defining atype one wall with respect to Z and { ρ } with equation f = 0. For type two walls, the l in the entry [ f, l ] is a list of pairs ( l , l ) of lists of sets in L and f is thehomogeneous polynomial such that f = 0 corresponds to the type two walls withrespect to Z and Z for all Z ∈ l and for all Z ∈ l .Take Σ ex for example, we have already seen that there is no type one wall. Asfor type two walls, note that the only pair of subspaces without containment is W and W . The output is − / ∗ a − a ∗ a − / ∗ a + 1 / ∗ a + a ∗ a / ∗ a , ([[0] , [1] , [0 , , [[2] , [3] , [2 , . Setting a = a = 0, we specialize the result to Pic( X Σ ex ) R . The equation becomes − / ∗ a + 1 / ∗ a = 0. Since a , a > − a + a = 0. Moreover, the pair of lists in the end tells us that the equation − a + a = 0 corresponds to the condition || Proj W Z χ ∗ D || = || Proj W Z χ ∗ D || for any Z ∈ [[0] , [1] , [0 , Z ∈ [[2] , [3] , [2 , Computing stratifications with respect to ample divisors.
Given anample divisor D on X Σ , the program(1) computes the stratification of Z (Σ) induced by χ D , and(2) plots the Hasse diagram of the stratification as a poset.Moreover, given two ample divisors D and D (cid:48) , the program determines if χ D and χ D (cid:48) induce equivalent stratifications (Definition 3.10). We explain briefly howeach of the three functions works.6.5.1. Stratification induced by an ample divisor.
Given D ∈ Amp( X Σ ), computingthe stratification induced by χ D boils down to two steps:(1) Compute the real one parameter subgroup λ DS that is χ D -adapted to L ( S )for each S ∈ L .(2) Group { L ( S ) } S ∈ L by { λ DS } S ∈ L .As was discussed in Section 5.2, step (1) is equivalent to finding the longestvector in Λ DS = {− Proj W Z χ ∗ D (cid:12)(cid:12) − Proj W Z χ ∗ D ∈ σ S , Z ⊂ S } . This is done by a combination of basic set operations and sorting on the list ofpotential real one parameter subgroups obtained earlier. Step (2) is also just asequence of set operations.6.5.2.
Stratification as a poset.
We store the stratification as a poset in SageMathwhere each node is a tuple of sets in L , corresponding to a stratum. For example,if a node is the tuple ( S , . . . , S k ) of subsets in L , then it corresponds to thestratum ∪ ki =1 L ( S i ) . The strict partial order is defined by the norms of strata’sindexing real one parameter subgroups.We also supply the feature to visualize posets. This is simply done by applyingthe .plot() method for posets in SageMath. We demonstrate the visualization forthe fan Σ ex . ARIATION OF STRATIFICATIONS FROM TORIC GIT 55
The stratification induced by the ample divisors ( a , a ) = (2 , , , . Each node in the diagram is a tuple. The empty tuple (() , ) corresponds to thestratum L ( ∅ ), which is the origin. The tuple ((0 , ) , (1 , ) , (0 , L ( { } ) ∪ L ( { } ) ∪ L ( { , } ) and similarly for others.For some reason sageMath is not very consistent in drawing arrows or justan edge. In any case, higher order strata are placed higher in the diagram. InSection 6.4, we calculated that a = a defines a type two wall. Note that thedivisor ( a , a ) = (1 ,
1) is on the type two wall and (2 ,
1) (1 ,
2) are on differentsides of the wall. According to the outputs, crossing this type two wall swaps theorderings of a pair of strata and on the wall the order between these two stratacome together, breaking up the poset into two chains.6.5.3.
Comparing stratifications induced by two ample divisors.
To determineif the stratifications induced by two ample divisors
D, D (cid:48) are equivalent, weconstruct the poset for each ample divisor given. Then there is the .is isomorphic()method in SageMath available to compares if the two posets are isomorphic.By Definition 3.10, χ D and χ D (cid:48) induce equivalent stratifications if the posetsconstructed here for D and D (cid:48) are isomorphic.6.6. Visualizing walls and semi-chambers in the ample cone.
The programcan plot the ample cone and its walls when the dimension of the ample cone isless than three. For two dimensional ample cones, the .implicit plot() method canbe applied directly to plot walls in the ample cone. For three dimensional amplecones the task boils down to the two steps:(1) Produce a two dimensional affine hyperplane H ⊂ Pic( X Σ ) R such thatCone ( H ∩ Amp( X Σ ) R ) = Amp( X Σ ) R . (2) Use the implcit plot() function in SageMath to plot walls in the region H ∩ Amp( X Σ ) R . The result is a two dimensional slice of the original picture.For step(1), Theorem 4.4 tells us that the ample cone is the interior of the nefcone for projective toric varieties. Hence it is sufficient to find an affine hyperplane H that meets all the rays of the nef cone Nef( X Σ ). This can be achieved with thehelp of Proposition 6.8.
Let σ be a full dimensional, strongly convex polyhedral coneand σ ∨ be its dual. Then (1) The sum of ray generators of σ is in the interior of σ .(2) Let m ∈ σ ∨ . Then H m ∩ σ = { } if and only if m is in the interior of σ ∨ . We now explain how Proposition 6.8 can be applied. We already had the nefcone Nef( X Σ ) in earlier stages of the program described in Section 6.2. SinceSageMath can compute the dual and the list of ray generators of a polyhedralcone, we have the list of ray generators for the dual cone Nef( X Σ ) ∨ . Let f be thesum of ray generators of Nef( X Σ ) ∨ .Since the nef cone Nef( X Σ ) is full dimensional and strongly convex in Pic( X Σ ) R by Theorem 4.4, so is Nef( X Σ ) ∨ . We see from Proposition 6.8 that f is in theinterior of Nef( X Σ ) ∨ and H f ∩ Nef( X Σ ) = { } . In particular, f does not vanishon the ray generators of Nef( X Σ ).Let u ρ be any ray generator of Nef( X Σ ). If u ρ (cid:48) is any other ray generator, wethen have f ( f ( u ρ ) f ( u ρ (cid:48) ) u ρ (cid:48) ) − f ( u ρ ) = 0 . This means the affine plane H ⊂ Pic( X Σ ) R defined by the vanishing locus of thefunction v (cid:55)→ f ( v ) − f ( u ρ )contains all rays of Nef( X Σ ), finishing step (1).For step (2), the linear equation for H allows us to solve one of the variablein terms of the other two. Hence we can rewrite the equations of walls and theinequalities that define the ample cone in two variables. With these we thenuse the .implicit plot() function to plot walls within the region H ∩ Amp( X Σ ) R ,finishing step (2).The result is a two dimensional slice of the ample cone and its decompositionby walls and semi-chambers. Interested readers may look at Section 7.2 for atwo dimensional slice of the wall and semi-chamber decomposition of a threedimensional ample cone.7. Examples and counter examples
The highlights of this section are examples where(1) there are walls that are both type one and type two (Section 7.1.5),(2) two semi-chambers can be contained in a single SIT-equivalence class(Section 7.1.6),(3) for each primitive collection C , the one parameter subgroup λ C in Corol-lary 5.32 does not relate to the primitive relation we cooked up in Equa-tion (4.6) (Section 7.1.7), and(4) a semi-chamber does not have to be convex (Section 7.2).For brevity, in the discussion that follows, whenever S ∈ L is a concrete setlike { , } , instead of writing L ( S ) as L ( { , } ), we will simply write L withoutcommas separating the elements in S . Likewise for σ S and W S λ DS and Λ DS , wewill not include commas. However, we will keep the notation L ( S ) when we talkabout an abstract S .7.1. Blow-up of the Hirzebruch surface at a point.
This is a rich examplewhere phenomena (1),(2),(3) listed in the beginning of Section 7 can be found.
ARIATION OF STRATIFICATIONS FROM TORIC GIT 57
The set up.
Let Σ ⊂ R be the smooth complete fan whose rays are givenby ρ =Cone ( e ) ρ =Cone ( e ) ρ =Cone ( − e ) ρ =Cone ( − e + e ) ρ =Cone ( − e + 2 e ) . The primitive collection consists of(1) { ρ , ρ } (2) { ρ , ρ } (3) { ρ , ρ } (4) { ρ , ρ } (5) { ρ , ρ } .Moreover, L = [[] , [1] , [2] , [1 , , [4] , [1 , , [2 , , [1 , , , [3] , [1 , , [2 , , [1 , , , [0] , [0 , , [0 , , [3 , , [0 , , , [0 , , [0 , , , [0 , , [0 , , ρ , ρ ), Pic( X Σ ) is free withbasis D , D , D . The computer program computes that a divisor (cid:80) i =2 a i D i isample if and only if a + a > a > a > . The potential real one parameter subgroups.
For a point D = (cid:80) i =2 a i D i ∈ Amp( X Σ ) R , the potential real one parameter subgroups in ΓΓΓ( G ) R ⊂ R Σ(1) are(1) − χ ∗ D = − / ∗ a − / ∗ a − / ∗ a − / ∗ a / ∗ a − / ∗ a − / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a , (2) − Proj W χ ∗ D = − / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a , (3) − Proj W χ ∗ D = − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a , (4) − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a / ∗ a / ∗ a − / ∗ a , (5) − Proj W χ ∗ D = − / ∗ a − / ∗ a − / ∗ a / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a , (6) − Proj W χ ∗ D = − / ∗ a − / ∗ a − / ∗ a / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a , (7) − Proj W χ ∗ D = − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a ,(8) − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a / ∗ a − / ∗ a ,(9) − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a / ∗ a / ∗ a − / ∗ a ,(10) − Proj W χ ∗ D = − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a − / ∗ a , (11) − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a ,(12) − Proj W χ ∗ D = − Proj W χ ∗ D = − Proj W = − Proj W χ ∗ D = − / ∗ a − / ∗ a ,(13) − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a , ARIATION OF STRATIFICATIONS FROM TORIC GIT 59 (14) − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a / ∗ a / ∗ a − / ∗ a .Although there are 21 subsets in L , we listed the seventeen − Proj W Z χ ∗ D where W Z (cid:54) = 0.7.1.3. Stratification induced by ample divisors.
For each S ∈ L , we may describethe vectors λ DS that is χ D -adapted to L ( S ) as a piecewise function of D ∈ Amp( X Σ ) R like we did in Section 5.5. To get the stratification induced by χ D one simply groups { L ( S ) } S ∈ L by { λ DS } S ∈ L .Take S = [0] ∈ L for example. By the list of potential real one parametersubgroups provided in Section 7.1.2, for − χ ∗ D to be inside σ , we must have − / ∗ a − / ∗ a − / ∗ a ≥ λ D = − Proj W χ ∗ D . Let us look at one more examplewhere S = [1 , − χ ∗ D / ∈ σ and − Proj W χ ∗ D / ∈ σ . However, − Proj W χ ∗ D ∈ σ if − / ∗ a / ∗ a ≥
0, which is is possible in the ample cone. We thenconclude that λ D = (cid:40) − Proj W χ ∗ D if − / ∗ a / ∗ a ≥ , or − Proj W χ ∗ D otherwise . Below is a complete list of piecewise descriptions of the real one parameter subgroup λ DS that is χ D -adapted to L ( S ) for all S ∈ L and for all D ∈ Amp( X Σ ) R .(1) S = ∅ , λ D ∅ = − χ ∗ D .(2) S = [0] ,λ D = − Proj W χ ∗ D .(3) S = [1], λ D = (cid:40) − χ ∗ D if − / ∗ a / ∗ a ≥ , or − Proj W χ ∗ D otherwise . (4) S = [1 , , , [2 , , [1 , ,λ DS = − Proj W χ ∗ D (5) S = [3], λ D = − Proj W χ ∗ D .(6) S = [4], λ D = (cid:40) − χ ∗ D if − / ∗ a / ∗ a − / ∗ a ≥ , or − Proj W χ ∗ D otherwise.(7) S = [0 , ,
4] or [3 , λ DS = − Proj W χ ∗ D .(8) S = [1 , ,
3] or [2 , λ DS = − Proj W χ ∗ D .(9) S = [0 , ,
4] or [0 , λ DS = − Proj W χ ∗ D . (10) S = [0 , , λ D = − Proj W χ ∗ D . (11) S = [1 , λ D = (cid:40) − Proj W χ ∗ D if − / ∗ a / ∗ a ≥ , or − Proj χ ∗ D otherwise.(12) S = [0 , λ D = (cid:40) − Proj W χ ∗ D if 1 / ∗ a − / ∗ a / ∗ a ≥ , or − Proj W otherwise.(13) S = [0 , λ D = (cid:40) − Proj W χ ∗ D if − / ∗ a / ∗ a − / ∗ a ≥ , or − Proj W χ ∗ D otherwise.(14) S = [1 , λ D = (cid:40) − Proj W χ ∗ D if − / ∗ a / ∗ a ≥ , or − Proj W χ ∗ D otherwise.(15) S = [0 , λ D = − Proj W χ ∗ D .7.1.4. The list of type one Walls.
The program returns the following list of typeone walls. We refer the reader back to Section 6.4 for relevant data structures.(7.1) 2 / ∗ a − / ∗ a , [ (cid:0) [4] , [1] (cid:1) ] − / ∗ a / ∗ a − / ∗ a , [ (cid:0) [0] , [3] (cid:1) , (cid:0) [0] , [4] (cid:1) ]1 / ∗ a − / ∗ a / ∗ a , [ (cid:0) [] , [4] (cid:1) ]1 / ∗ a − / ∗ a , [ (cid:0) [3] , [1] (cid:1) , (cid:0) [] , [1] (cid:1) ] . We do not include type two walls here as it is rather long and complicated. Instead,we only cite type two walls that are used as we go along.7.1.5.
A wall can be of type one and type two.
We examine the type one wall W , := { D ∈ Amp( X Σ ) R | Proj W χ ∗ D = Proj W χ ∗ D } . According to the type one wall list (7.1), this corresponds to the hyperplane − / ∗ a / ∗ a − / ∗ a . Let W +0 , = { ( a , a , a ∈ Amp( X Σ ) R | − / ∗ a / ∗ a − / ∗ a > } ,W − , = { ( a , a , a ∈ Amp( X Σ ) R | − / ∗ a / ∗ a − / ∗ a < } . With the piecewise information provided in Section 7.1.2, we deduced the following:(1) When D ∈ W − , , L is in the stratum indexed by − Proj W χ ∗ D , and L ∪ L ∪ L is in the different stratum indexed − Proj W χ ∗ D .(2) When D ∈ W , , Proj W χ ∗ D = Proj W χ ∗ D . Hence L ∪ L ∪ L ∪ L all come together in the same stratum.(3) When D ∈ W +0 , , L is in the stratum indexed − Proj W χ ∗ D , separatedfrom L ∪ L ∪ L , which is in the stratum indexed by − Proj W χ ∗ D For concreteness we consider the three ample divisors(1) D − = 45 D + 65 D + 55 D ∈ W − , ,(2) D = 450 D + 775 D + 550 D ∈ W , ,(3) D + = 45 D + 80 D + 55 D ∈ W +0 , . ARIATION OF STRATIFICATIONS FROM TORIC GIT 61
Here are the stratifications induced by D − , D and D + respectively. . We refer the reader back to Section 6.5.2 for how to understand the graphicrepresentation of a stratification. Notice how L and L move in the diagrams.To understand why the chain breaks up into two at D ∈ W , , we investigatethe type two wall Z , := { D ∈ Amp( X Σ ) R | || Proj W χ ∗ D || = || Proj W χ ∗ D ||} . Using the computer program, we see that Z , corresponds to the condition − / ∗ a + 2 / ∗ a ∗ a − / ∗ a ∗ a . Since a > − a + 2 a − a = 0 , which is exactly the defining equation for W , . Notice how L , L move in thediagrams. Crossing this wall results in two types of variations. This is an examplewhere a wall can be of both type.7.1.6. Two semi-chambers can be contained in an SIT-equivalence class.
In thisexample we are also able to see that the stratifications induced by ample divisorsin two different semi-chambers can be the same. To establish such a pair of semi-chambers, it is sufficient to identify a wall H together with two ample divisors D, D (cid:48) such that the following conditions hold:(1) Both D and D (cid:48) are not on any walls. Namely, they are in some semi-chambers.(2) D and D (cid:48) are on different sides of H. Namely, if F = 0 is the definingequation for H , we require that F ( D ) · F ( D (cid:48) ) < D and D (cid:48) induce equivalent stratifications.Indeed, by the definition of a semi-chamber (Definition 5.13), any pair ( D, D (cid:48) ) ofample divisors that satisfy the first two conditions are in different semi-chambers.
We provide two walls here where the above three conditions are met. The firstone is given by the type two wall Z , = { D ∈ Amp( X Σ ) R | || Proj W χ ∗ D || = || Proj W χ ∗ D ||} . The equation for Z , is3 / ∗ a − / ∗ a + 1 / ∗ a ∗ a / ∗ a = 0 . We chose this wall because of the two reasons. First, by the piecewise infor-mation provided earlier, there are no subsets S ∈ L with λ DS = − Proj W χ ∗ D throughout the ample cone. Second, it is checked that there are no other wallswhose equation is the same as the equation for Z , . We therefore expect Z , never swaps the ordering of any pair of strata. Indeed, it can be checked bythe computer that the pair of ample divisors D = 30 D + 92 D + 70 D and D (cid:48) = 30 D + 99 D + 70 D and the wall Z , satisfy the three conditions requiredin the beginning.Here is a visualization of Z , , drawn as a blue curve in a 2D slice of the amplecone. . The slice is the intersection of the ample cone with the hyperplane a a a − a a , a > a > − a W = 2,dim W = 1 and W ⊕ W = ΓΓΓ( G ) R . By Equation (2.5), Z , looks like aquadric cone of the form x = y + y , giving the slice of Z , the curviness wesee in the figure.The second wall is more interesting. It has two connected components whereproducing a pair of divisors that meet the three conditions is only possible aroundone component. The wall is given by Z , := { D ∈ Amp( X Σ ) R | || Proj W χ ∗ D || = || Proj W χ ∗ D ||} . This corresponds to the hypersurface − / ∗ a + 2 / ∗ a ∗ a − / ∗ a − / ∗ a ∗ a / ∗ a ∗ a − / ∗ a = 0 . Note that since dim W = dim W = 2 and W + W = ΓΓΓ( G ) R , Equation (2.5)implies that Z , is a union of two hyperplanes. The line where these twohyperplanes meet is contained in the supporting hyperplane of the nef cone. Thewall splits up into two components in the ample cone (See figure (7.2)).To fully describe wall crossing behavior for Z , , we will need other two typeone walls:(1) − a a χ ∗ D = Proj W χ ∗ D , and ARIATION OF STRATIFICATIONS FROM TORIC GIT 63 (2) − ∗ a ∗ a − ∗ a χ ∗ D =Proj W χ ∗ D .The visualization of these three walls in the ample cone is given by(7.2)The union of blue lines is Z , . The vertical red line is defined by 1 − ∗ a − a a ∗ a ∗ a − − ∗ a ∗ a − ∗ a a , a
3) = (1 / , Claim 7.1.
It is only possible to produce a pair of ample divisors around thecomponent of Z , bounded in region one to meet the three conditions mentionedin the beginning of Section 7.1.6. Let D and D (cid:48) be two ample divisors on different sides of Z , . We have toestablish the impossibility for D, D (cid:48) to satisfy the three conditions in the followingtwo cases:(1) Both D and D (cid:48) are in region two.(2) D is in region one and D (cid:48) is in region two.For reader’s convenience, we clip the piecewise information specifically for S =[1] , [4]: • S = [1], λ D = (cid:40) − χ ∗ D if − / ∗ a / ∗ a ≥ , or − Proj W χ ∗ D otherwise. • S = [4], λ DS = (cid:40) − χ ∗ D if − / ∗ a / ∗ a − / ∗ a ≥ , or − Proj W χ ∗ D otherwise.Hence in region two, L is contained in the strata indexed by − Proj W χ ∗ D while L is contained in the stratum indexed by − Proj W χ ∗ D . Moreover, it canbe checked from the list of type one walls (7.1) that Proj W χ ∗ D (cid:54) = Proj W χ ∗ D forany D ∈ Amp( X Σ ) R . Therefore, the following two facts are immediate: • L and L are in different strata around the component of Z , in regiontwo, and • crossing Z , in region two swaps the ordering between the stratum thatcontains L and the stratum that contains L .Therefore, if D, D (cid:48) are divisors on different sides of Z , in region two, thestratifications induced by D and D (cid:48) are never equivalent. The impossibility forcase (1) is established.For case (2), note that the origin L ∅ is always in the stratum indexed by − χ ∗ D .Hence on the negative side of the vertical red line, L ∅ and L are in differentstratum while on the positive side, they come together in a stratum. Therefore D and D (cid:48) in case (2) never induce equivalent stratifications. Claim 7.1 is proved.We now produce a pair of ample divisors D, D (cid:48) in region one that satisfy thethree conditions mentioned in the beginning of Section 7.1.6. We take the twoample divisors D = 430 D + 960 D + 570 D and D (cid:48) = 430 D + 955 D + 570 D .It can be verified by the computer that • Both D and D (cid:48) are not on any walls. • Both D and D (cid:48) are in region one. Namely, they both satisfy − a a > • D and D (cid:48) are on different sides of Z , , but • D and D (cid:48) induces the same stratification.The reason why it is possible to produce such a pair in region one is that inregion one, L is in the stratum indexed by − Proj W χ ∗ D while L is in the stratumindexed by − χ ∗ D . Therefore, the stratum that contains L is always higher thanthe stratum that contains L . Namely, Z , no longer swaps the ordering betweenthe stratum that contains L and the stratum that contains L .7.1.7. A counter example about primitive relations.
Recall in Corollary 5.32 weproved that for every R -ample divisor D ∈ Amp( X Σ ) R and for every primitivecollection C ⊂ Σ(1), there exists a unique vector λ C ∈ Λ D such that S χ D λ C = V ( { x ρ | ρ ∈ C } ) . We will show that − λ C need not be a positive multiple of the primitive relation(Definition 4.9) for C , even when Σ is smooth. In fact, we will show by thisexample that such λ C depends on D while the primitive relation of C does not.For this, take the primitive collection C = { , } . Since u ρ + u ρ = u ρ , the primitive relation of C is(1 , − , , , ∈ ΓΓΓ( G ) R . According to the piecewise information, the locally closed subvariety L ∪ L ∪ L ∪ L = V ( x , x ) − V ( x )is in the stratum indexed by − Proj W χ ∗ D . We concluded that λ C = − Proj W χ ∗ D where we calculated earlier that − Proj W χ ∗ D = − / ∗ a / ∗ a − / ∗ a / ∗ a / ∗ a − / ∗ a . ARIATION OF STRATIFICATIONS FROM TORIC GIT 65
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