Bernt Ivar Utstøl Nødland
Chow groups and pseudoffective cones of complexity- one T-varieties
Received: 2 April 2020 / Accepted: 4 August 2021
Abstract. We show that the pseudoeffective cone ofk-cycles on a complete complexity-one T-variety is rational polyhedral for anyk, generated by classes ofT-invariant subvarieties.
When X is also rational, we give a presentation of the Chow groups of X in terms of generators and relations, coming from the combinatorial data definingXas aT-variety.
1. Introduction
AT-variety is a normal algebraic variety X with an effective action of an alge- braic torus T, defined over an algebraically closed field of characteristic 0. The complexity of aT-variety is defined as dimX −dimT, thusT-varieties of com- plexity zero correspond to the toric varieties. For toric varieties there is a well- known correspondence between the geometry of a variety and combinatorial data coming from the T-action. There has in recent years been developed a similar quasi-combinatorial language for describingT-varieties of higher complexity, start- ing with Altmann and Hausen’s paper [1]. Following this, there have been many papers studying the geometrical and combinatorial properties ofT-varieties (see for instance [3,5,12,13,17,19] and the references in [4]).
Here we will study algebraic cycles on T-varieties of complexity one. Our first main result is a description of the cones of effective cycles:
Theorem 1.1.The pseudoeffective conesEffk(X)of a complete T -variety X of complexity one are rational polyhedral, generated by classes of invariant subvari- eties.
This generalizes Scott’s results on the pseudoeffective cone of curves on aT- variety [20]. In general there are not many examples where all pseudoeffective cones of cycles are known. Our result gives a large class of examples where these are rational polyhedral. WhenXis rational, it is known thatXis a Mori dream space [12], so the statement was previously known for the cone of curves and effective divisors. However, the result applies also in the non-rational cases, showing that while theT-varietiesXare not Mori dream spaces, their cones of curves and divisors B. I. U. Nødland (
B
): Universitetet i Oslo, Oslo, Norwaye-mail: [email protected]
Mathematics Subject Classification (2000)14C25·14L30·14M25
https://doi.org/10.1007/s00229-021-01326-2
are still rational polyhedral. Moreover, even in the caseXis rational, and thus Mori dream, it is not a priori clear that the effective cones of cycles of intermediate dimensions should be rational polyhedral, as was shown by [7, Example 6.10].
Our second main result gives a presentation for the Chow groups ofX, in the caseXis a rational complete complexity-oneT-variety. The result is inspired by two different well-known results. First of all, for toric varieties, Fulton and Sturmfels showed that invariant subvarieties generate the Chow groups, and moreover they described the relations between these generators [9, Proposition 2]. On the other hand, for a rational completeT-variety of complexity one, Altmann and Petersen give an analogous short exact sequence describing its Picard group [5, Corollary 2.3]. These two results give some hints to how the Chow groups of a -oneT-variety might look. Our second main result gives a complete description of these.
To explain the result, we first recall some basic facts aboutT-varieties. If X is a complete rationalT-variety of one it comes equipped with a rational quotient map, that is, a rational map f : X P1, and another complete rationalT-variety of one,X, with aT-equivariant blow-up mapr : X → X which resolves f; thus there is a mapg: X →P1such that f ◦r =gwhenever defined. Following [13], we have thatXcorresponds to a fangiving the general toric fiber ofg, as well as finitely many special fibers which correspond to polyhedral complexes with tailfan . We denote byPthe following set: If the set of points inP1such that the fiber is not general has size at least 2 thenPequals this set. If this set is smaller, thenPis the union of this set with (1 or 2) additional arbitrary points ofP1. In addition we need to keep track of the data of which cones correspond to varieties contracted by r(see Section2for more details).
Letting the dimension ofX ben+1 and be the fan describing the general fiber ofg(which is the toric variety X)we define, for a non-negative integerk, the following sets:
Rk =Cones of dimensionn+1−kcorresponding to subvarieties not contracted byr.
Vk =Faces of dimensionn−kof polyhedral subdivisions corresponding to fibers of points inP, such that the tailcone corresponds to a subvariety not contracted byr.
Tk=Cones of dimensionn−kcorresponding to subvarieties contracted byr.
Theorem 1.2.For a complete rational T-variety X of one there is for any0≤k≤ dimX an exact sequence
F∈Vk+1
M(F)
τ∈Rk+1
(M(τ)⊕ZP/Z)
τ∈Tk+1
M(τ)→ZVk⊕ZRk ⊕ZTk→Ak(X)→0,
where for a rational polyhedral coneτ in NQ,M(τ)=τ⊥∩M. Also M(F)is the character lattice of the toric variety corresponding to F .
The maps will be described below. Fork = n this coincides with the exact sequence of Altmann and Petersen. The above results also generalizes the results of Laface, Liendo and Moraga [17], where they give a presentation of the rational Chow ring of a complete complexity-oneT-variety which is contraction-free, that
is, when X = X. Being contraction-free is however quite restrictive, for instance their rational Chow ring is generated by divisors [17, Lemma 4.1], which is not true in general. We do not have an explicit description of the ring structure of a general complexity-oneT-variety, although we believe that the results in this paper will be useful in describing it. We illustrate our results by studying examples such as toric downgrades (meaning we only remember the action of a codimension one torus on a toric variety), projectivizations of rank-two toric vector bundles and the Grassmannian Gr(2,4), none of which (in general) are contraction-free.
2. Preliminaries onT-varieties
The papers [1,3] give a general framework for describingT-varieties of any com- plexity, we briefly recall the set-up. Denote byT (K∗)na torus of dimensionn and letM andNdenote the lattices of characters and one-parameter subgroups of T, respectively.Kis an algebraically closed field of characteristic 0.
Recall that any polyhedroncan be decomposed as a Minkowski sumσ+P, whereσis a unique polyhedral cone, called the tailcone, andPis a polytope. Fixing a polyhedral coneσ ⊂NQ, we consider the semigroup under Minkowski addition
Pol+Q(N, σ)= {⊂NQ|is a polyhedron with tailconeσ}
We also allow∅as an element of Pol+Q(N, σ). LetYbe a normal and semiprojective variety (meaning it is projective over some affine variety) and let CDiv(Y)denote the group of Cartier-divisors onY. We consider “divisors” of the form
D=
Z
Z ⊗Z,
whereZ is an element of Pol+Q(N, σ)and theZareQ-Cartier divisor onY, such that only finitely manyZ differ from the tailcone. Foru ∈ σ∨∩M, we may consider the evaluation
D(u)=
Z|Z=∅
minZ,uZ ∈CDivQ(Y),
which is a finite sum, since the minimum for anyZsuch thatZequals the tailcone is 0. We callDa p-divisor on(Y,N)ifD(u)is semiample for allu ∈σ∨∩M, as well as big foruin the interior ofσ∨∩M. To a p-divisorDwe can associate the sheaf of ringsOY(D)=
u∈σ∨∩MOY(D(u)). ThenX =Spec(Y,OY(D))is an affine T-variety of complexity dimY. Also X = SpecYOY(D)is T-variety of complexity dimY and there is an equivariant mapr : X → X. We say thatX is contraction-freeifX =X.
Altmann and Hausen [1] shows that any affine T-variety arises from a p-divisor in this way.
IfDandDboth are p-divisors on(Y,N)we define their intersectionD∩D as having coefficientZ∩Z onZ. We say thatD⊂DifZ ⊂Z for allZ. In that case we say thatDis a face ofDif the induced mapX(D)→ X(D)is an open embedding; there is a technical condition [3, Proposition 3.4] for checking this, which we do not recall here.
Definition 2.1.A finite setSof p-divisors on(Y,N)is called a divisorial fan if the intersection of any two p-divisors is a common face of both andSis closed under intersections.
The condition on the intersections comes from the fact that one can glue the varietiesX(D),X(D)along the open setX(D∩D). Then Altmann, Hausen and Süss [3] show that anyT-variety arises from a divisorial fan in this way.
While the arguments in this paper are mostly geometric, the perspective of a divisorial fan will be useful. Also, in complexity one, there is a different perspective, due to Ilten and Süss [13], which has the advantage of avoiding the technical condition about the open embeddingsX(D)→ X(D). We will alternate between these perspectives, depending on what is most convenient.
We now specialize to the case whereY is a curve, denoted byC, which we can assume is a smooth curve [1, Corollary 8.12]. Also we generally denote the divisor Z by p, since divisors on curves correspond to sums of points. For a p-divisorD onCwe define its degree degDas the Minkowski sum
pp. Definition 2.2.Amarked fansy divisor onC is a formal sum =
Sp⊗ [p]
together with a fan ⊂NQand a subsetK ⊂such that
(1) EachSpis a complete polyhedral subdivison ofNQsuch that tail(Sp)=for allp.
(2) Ifσ ∈ Khas full dimension thenDσ =
Dσp⊗ [p]is a p-divisor, whereDσp
is the polyhedron inSpwith tail(Dσp)=σ.
(3) for a full dimensional coneσ ∈ K and a faceτ ofσ,τ ∈ K if and only if degDσ ∩τ = ∅.
(4) Ifτ is a face ofσ, thenτ ∈ K implies thatσ ∈ K. The subsets degDσ glue to a subset deg⊂NQ.
Then [13] shows that any complete T-variety of complexity one corresponds to a marked fansy divisor. In other words, in this case we only need to remember the polyhedral subdivisions of the fibers as well as the setKwhich records which subvarieties are contracted byr.
Given a complete rational complexity oneT-variety defined by a divisorial fan S, one can associate a marked fansy divisor by taking the polyhedral subdivisions given bySand lettingK consist of all conesσ such that there existsD∈Swith tailconeσ and such that no coefficients ofDequals∅. The varietyX is given as a marked fansy divisor by the same subdivisions as forX but withK = ∅.
The intuition here is the following: For any pointp ∈Ythe fiber ofgis the toric variety (possibly non-reduced, non-irreducible) corresponding to the polyhedral subdivison at the point p. The fan defines a toric variety which is the general fiber, but at some points there might be other fibers. Nevertheless these fibers are all unions of toric varieties. The T-action on X restricted to a fiber ofgis just the T-action on the fiber as a union of toric varieties.
Example 2.3.Figure 1 shows polyhedral subdivisions for a complexity-one T- variety X of dimension three, with three special fibers. Its tailfan is the fan of the blow up ofP1×P1in two points contained inP1×0. The subdivisions alone
Fig.1.PolyhedralsubdivisonsdefiningaT-varietyXofdimensionthree
Fig. 2.Everything except the gray area is deg
do not define X, we also need to specify the subsetK. If we for instance choose thatKcontains all maximal cones of, we get from Fig.2that all rays except the one generated by(0,−1)are also inK.
This choice ofK corresponds to the projectivization of a rank-two non-split toric vector bundle overP1×P1, see Section6.
3. Subvarieties ofT-varieties of complexity-one
Given a fan, we denote by X the associated toric variety, by B the torus- invariant boundary ofXand byTthe torus. Forσ ∈, letV(σ)denote the T- invariant subvariety ofXof codimension dimσassociated toσ. It is a toric variety with torus T(σ), whose lattice of one-parameter subgroups is N(σ) = N/Nσ, whereNσ is the lattice generated byσ∩N.
Fix a complete T-variety of complexity-oneXwith tailfan. On the varietyX there are various subvarieties arising from the combinatorial structure defining it.
For a point p ∈ C and a faceF of the polyhedral complex Spdefining the fiber g−1(p)there is the T-orbit orb(p,F)of dimension codimF, the closure is denoted byZp,F =orb(p,F)⊂X.
We denote the generic point ofC by η. For any cone σ ∈ there is a T- invariant subvarietyBσ which is given by the closure of orb(η, σ), in other words BσdominatesCand in the general fiber is given by the subvarietyV(σ)⊂X. We define B = ∪ρ∈(1)Bρ. ThenBis a finite union of divisors on X, each of which themselves is a T-variety of complexity-one [12, Proof of Proposition 4.12]. This fact will be important in the proof of Theorem3.1.
Fix any rayρ ∈ . Locally,X is given by a p-divisor D =
Z DZZ with tailconeσ. For a rayρ, letπ : NQ → NQ/Qρ denote the projection sendingρ to 0. Then Dρ =
Zπ(DZ)Z is a new p-divisor onY with tailconeπ(σ ); this definesBρ as a T-variety [12, Proof of Proposition 4.12].
OnX some of the subvarietiesBσ are contracted, namely those whereσ is in K. In this caser(Bσ)is contracted to a variety of dimension one less. We denote the corresponding orbit closurer(Bσ)in X byWσ. We also denoter(Zp,F)by Wp,F, for any p,F. Whenσ /∈ K then Bσ is not contracted and we also denote the corresponding subvarieties ofX by the same symbol.
We are interested in studying the pseudoeffective cones Effk(X)inside the group Nk(X)of cycles modulo numerical equivalence, which by definition is the closure of the cone of effectivek-cycles.
Theorem 3.1.The pseudoeffective cones Effk(X) of a complete T-variety X of complexity one are rational polyhedral, generated by classes of invariant subvari- eties.
Proof. We first show the statement for the contraction-free case, in other words whenX =X. Fix a subvarietyVof dimensionkofXand choose a basisv1, ..., vn
of the lattice N. Letλi be the one-parameter subgroup ofT corresponding tovi. We have a corresponding actionK∗×X → Xfor eachλi. LetV1be the flat limit as t goes to zero ofλ1·V. Similarly letVi be the flat limit as t goes to zero of λi ·Vi−1. ThenVnis an effective cycle numerically equivalent toV and invariant under the entire torus action. Now for any irreducible componentW ofVnwe have two possibilities: EitherW dominatesCor it is contained in a fiberXy.
If it is contained in a fiberXy, then since it is irreducible it is contained in an irreducible component of the fiber. There are only finitely many components which are different from X. In each of these there are only finitely many irreducible invariant subvarieties. Thus ifW is contained in one of these special fibers it has to equal one of these finitely many subvarieties. If it is contained in a general fiber then, since the mapX →C is flat, it is algebraically equivalent to a cycle in one fixed general fiber. In particular it is also numerically equivalent. ThusW can be written as an effective sum of finitely many generators.
If it is not contained in a fiber, then we claim it has to be contained in the boundary divisorB. Indeed, if not, then there is a pointy∈Csuch thatXy X and such that there isxin(Xy\By)∩Wy =T∩Wy. ThenTacting onxwill be a set of dimensionnin the fiberXy. But thenWyhas to have dimension at least n, since it is invariant under the entire torus action. However by assumption it is a proper closed subset of X which has dimensionn, so its dimension must be smaller thann.
Thus whenW is not contained in a fiber, it is contained in Bwhich is a finite union ofT-varieties of complexity one of smaller dimension thanX. Since we are onXwe know that no varieties are contracted byr, in particular any p-divisor must have∅as a coefficient. Thus any p-divisor for a componentBρofB, which by the above description is given by projecting the polyhedral coefficients of a p-divisor forX, will also have an empty coefficient. In particularBρ=Bρ, in other words it is also contraction-free. A T-variety of dimension one and complexity one is simply a curve. The proposition is obviously true for any curve, thus by induction it is true for any contraction-freeT-variety of complexity one.
Now we consider the case of the generalX. By the same argument as above, any subvariety V of X of dimensionkcan be written as an effective sum of T- invariant irreducible subvarieties. By [1, Theorem 10.1] any irreducibleT-invariant subvariety ofX is the image of aT-invariant subvariety of X underr. From this we get thatVcan be written as an effective sum of the images of the generators of Effk(X). This proves the theorem also forX.
In fact the above proof gives some information about the generators of the pseudoeffective cones ofX:
Corollary 3.2.Let X be a complete contraction-free T-variety of complexity one, in other words such thatX=X . Then the pseudoeffective coneEffk(X)is generated by Bτ, wheredimτ =n+1−k and Zy,F, where y∈Y anddimF =n−k.
Proof. The proof of Theorem3.1implicitly describes the generators: Any effective irreduciblek-cycleW of dimension less than dimX has to either be contained in someBρ,ρ∈(1)or in some fiberXy.
If it is contained in a fiberXythen, since the fiber is a union of irreducible toric varieties, it has to be contained in one of them, say the one corresponding to the vertexv of Sy. After possibly replacing any compact face containingv with the corresponding cone emanating fromvwe have a complete fan with vertexv defining the toric variety. Now we know that as a cycle inX,W can be written as an effective sumW ≡
Fi∈(n−k)aiV(Fi). Letting ji :V(Fi)→ X denote the inclusion, we have that(ji)∗V(Fi)=Zv,Fi. Thus in this caseW can be written as a positive sum of the finitely manyZv,F.
Next we note, as in the proof of Theorem3.1, that any boundary divisor in X is also contraction-free. If an effective cycle is contained in some boundary divisor Bρwe argue by induction on codimW. If codimW =1 then sinceWis contained inBρand they are both irreducible of the same dimension, we must haveW =Bρ, hence we are done. If codimW >1 then by inductionWcan be written as a positive sum ofBp(τ)andZv,p(F), wherep:NQ→ NQ/Qρis the projection. Denoting by j :Bρ → Xthe inclusion we have that j∗(Bp(τ))= Bτ and j∗(Zv,p(F))= Zv,F.
Thus we are done by induction.
To study the generators of the pseudoeffective cones for generalXwe need to also take into account the varieties that are contracted byr. First of all, it follows from [19, Proof of Proposition 3.13] that for any invariant subvarietyZof a T-variety X we have that
dimg(Z)+dimr(Z)−dim(Z)≥0.
This fact, combined with the fact that we always have dim(Z)≥dim(r(Z)), implies that if dim(g(Z)) =0 thenZ is not contracted. Moreover if dim(g(Z))=1 and Z is contracted then dim(r(Z)) = dim(Z)−1. If Bσ has codimensionk in X (equivalentlyσ is of dimensionk), then if Bσ is contracted byrtoWσ (meaning σ ∈ K) it has to be contracted to a subvariety of codimensionk+1 inX.
We will describe the images of the contracted varieties. We denote byμ(F)the smallest positive integer inN/Nσ such thatvFis a lattice point.
Lemma 3.3.Letσ ∈and consider Bσ ⊂X as a complexity-one T -variety with respect to the torus T(σ). Then for a face Zq,F with tailconeσ we have that the generic stabilizer group of the corresponding divisor in Bσis cyclic of orderμ(F). Proof. By the proof of [12, Proposition 4.12] Bσ is described as aT-variety by projecting the coefficients from N toN/Qσ. By [12, Proposition 4.11] the order of the generic stabilizer group is cyclic of the stated order.
We can determine the difference of generic stabilizers of a general fiberZp,σ
and its imager(Zp,σ). Since the mapr is equivariant anyt ∈T in the stabilizer of Zp,F will be in the stabilizer of the image r(Zq,F), for any F with tailcone σ. Each Zp,F is a toric variety with an effective action of a corresponding torus.
For a general fiber this torus has lattice of one-parameter subgroupsN/Nσ, while in general it will be a lattice containing this: The latticeLF generated by the unit vectors together with the vertexvFinN/Nσ. By [1, Proposition 5.2 ii] the stabilizer of a point inXis given by a torus associated to a certain lattice. For a point inZp,F
this lattice isLFby [1, Proposition 7.10]. Since allZp,F are mapped to the same image by the proof of Lemma3.4, the stabilizer ofr(Zq,σ)will thus simply be the group generated by the stabilizers of all Zp,F. Letting L be the lattice generated by all integer linear combinations of the unit vectors together with set ofvF, for all Zp,Fwith the tailcone ofFequal toσ, we obtain a latticeLcontaining the lattice N/Nσ. The difference of stabilizers[Stab(Zq,σ):Stab(r(Zq,σ))]will thus equal the lattice index[L :N/Nσ]. We denote this number bysσ.
Lemma 3.4.Assumeσ ∈and thatσ ∈ K . Then all faces Zp,FwithtailF=σ are mapped to Wσ under the contraction map r . Moreover we have the equality μ(F)r∗([Zp,F])=sσ[Wσ]of numerical classes, for F with tailconeσ.
Proof. This follows from [1, Theorem 10.1]: Two facesZp,FandZy,Gof the same polyhedral divisorDare identified underr if they have the same normal coneλ and ifD(u)=0 for someu in the relative interior ofλ(ifC =P1this is in fact equivalent toZp,F being identified withZy,G).
Letσ ∈Kbe a maximal cone. There is a polyhedral divisorDσwith tailconeσ and no empty coefficients. For any coefficientσpofDσthe associated normal cone is the point 0, moreoverD(0)=0, thusWp,σp is identified withWq,σq for any pointsp,q ∈C. In particular, the entireBσinXis mapped to any fixed subvariety of the formWp,σp.
Ifτ ∈ Kis not maximal we have that∩τσdegDσ∩τ = ∅by Definition2.2(3).
This implies that in any fiberSpthe intersection∩τσσp= ∅, which implies that in any fiber there is only one faceFpwith tailconeτ. LetDτbe the p-divisor having the corresponding facesFpas coefficients. We haveFp=τ+Qp, whereQpis some polytope inNQ. Lettingvpbe a vertex ofQp, we have thatFp=τ+vp+(Qp−vp). Since there is only one face ofSpwith tailconeτ, the polytopeQp−vpmust be contained in the linear span ofτ. This implies that all Fphave the same normal coneλ. We wish to show that
Dτ(u)=
minFp,u =0
foru in the relative interior ofλ. We have that minFp,u = vp,u, sinceu by definition is normal to any point in the linear span ofτ. Since degDτ =degDσ∩ τ ⊂τ we must also have that
vp ∈τ, thus
pvp,u =0, which is what we wanted to show. Thus Zp,Fp is identified with Zq,Fq for any p,q ∈ C, thus the horizontal subvarietyBτ maps to anyWp,Fp.
For the last claim, we have that for a general fiberqthe maprrestricted toZq,σ
is finite of ordersσ, thusr∗(Zq,σ)=sσWσ. For a subvarietyZp,Fof a special fiber
the stabilizer group will have orderμ(sσF), thusr∗(Zp,F)= μ(sσF)Wσ, which proves
the claim.
Proposition 3.5.Let X be a complete T-variety of complexity one. Then the pseu- doeffective coneEffk(X)is generated by the following classes:
• Bτ, wheredimτ =n+1−k andτ /∈ K
•Wy,F, where y∈Y and F ⊂Syhas dimension n−k andtail(F) /∈K
•Wσ whereσ has dimension n−k andσ ∈ K .
Proof. As noted earlier Effk(X)is the image of Effk(X)viar∗, thus we know that r(Bτ)andr(Zp,F)generate Effk(X)as above. However we can omitr(Bτ)for τ ∈(n+1−k)∩K sincer∗(Bτ)=0 in this case. Fixingσ ∈ Kof dimension n−kwe have by the lemma above that all classesr∗(Zp,F), for any pointpand F with tailconeσ, are proportional, thus it is more convenient to only remember the single representativeWσ instead of all the differentWp,F.
4. Chow groups
We now assume X is a rational and complete T-variety of complexity one with tailfan. Recall thatr denotes the contraction mapX → X. We say that a cone σ ∈ is contracted byr ifσ ∈ K. Also recall that P is the following set: If the set of points inP1such that the fiber ofris not general, has size at least 2 thenP equals this set. If this set is smaller, then P is the union of this set with (1 or 2) additional arbitrary points ofP1. Fixkand, inspired by the above result, define the following sets
Rk =Cones in of dimensionn+1−knot contracted byr corresponding to subvarietiesBσ.
Vk =Faces of dimensionn−kof fibers of points in P, such that the tailcone is not contracted corresponding to subvarietiesWp,F, forp∈ P, tailF ∈/ K. Tk =Cones inof dimensionn−kcontracted byrcorresponding to subvarieties Wσ.
Note that we will show in the proof below that invariant subvarieties corre- sponding to faces of points that have general fibers are in the group generated by the other subvarieties (inside the Chow-group), thus we do not require these as generators.
Then there is a surjection
ZRk∪Vk∪Tk → Ak(X)→0 (1) Fork = nthis is the surjection for the Picard group of X given in [5] (note that Tnis always empty). Altmann and Petersen also describe the relations between the generators:
0→ZP/Z⊕M →ZVn∪Rn →Pic(X)→0.
Recall that P is the set of points pinP1where the polyhedral subdivisonSpdo not equal. If P has size less than 2 then we simply add points so that P has
size 2. This corresponds to fixing aT-invariant structure onP1. Without loss of generality we assume∞ ∈ P. The first map is given by the following: generators forZP/Zcorrespond to principal divisors[p] − [∞]onP1, one such generator is mapped to
p,vμ(v)Wp,v−
∞,vμ(v)W∞,v. A characterm∈ Mis mapped to
vμ(v)m, vWp,v+
Rnm, ρBρ. Hereμ(v)is the smallest integer such that μ(v)vis a lattice point.
Theorem 4.1.For a rational complete complexity-one T-variety X there is for any 0≤k≤dimX an exact sequence
F∈Vk+1
M(F)
τ∈Rk+1
(M(τ)⊕ZP/Z)
τ∈Tk+1
M(τ)→ZVk⊕ZRk⊕ZTk→Ak(X)→0.
The maps are given as follows:
Any F ∈ Vk+1corresponds to an invariant subvariety of an irreducible com- ponent of a fiber p (possibly several components, if so pick one). We denote the character lattice of the corresponding toric variety by M(F). Thenm ∈ M(F) maps to
dimG=n−k,F⊂G tailG∈/K
m, vF,GZp,G+
dimG=n−k,F⊂G tailG∈K
m, vF,GstailG
μ(G)WtailG,
wherevF,GgeneratesN(F)/N(G). This is the usual notion of rational equivalence on a toric variety.
Ifτ ∈ Rk+1thenBτ itself corresponds to a T-variety of complexity one. The map comes from this structure, as in the exact sequence of Altmann and Petersen.
Explicitly a generator ofZP/Zcorresponds to a divisor[p]−[∞]and it is mapped to
F,tailF=τ
μ(vF)Wp,F−
F,tailF=τ
μ(vF)W∞,F,
wherevF is the vertex corresponding to the image of F in the divisorial fan cor- responding toBτ as a T-variety (see Section3). A characterm∈M(τ)is mapped to
F∈Vk,tailF=τ
μ(vF)m, vFWp,F+
σ∈Rk,τ⊂σ
m,σ¯Bσ,
whereσ¯ is the ray which is the image ofσ inN/Nτ.
Ifτ ∈Tk+1then every coneσ ∈containingτ will also be contracted byr.
In particular any suchσof dimensionn−klies inTk. Thusm∈M(τ)maps to
dimσ=n−k,τ⊂σ
m, vτ,σWσ.
Again this is just the usual notion of rational equivalence on a toric variety.
Proof. As already noted Proposition3.5implies that the final map is surjective.
An invariant subvarietyW of X corresponding to some face of a polyhedral subdivision could be non-normal. By [1, Theorem 10.1] the corresponding sub- variety of X is the normalizationW. Since the mapr is the normalization when restricted to any invariant subvariety, we get thatrrestricted to any invariant sub- variety is generically 1 : 1. This implies by [11, Ch. 5, Proposition 3.3] that the normalization is in this case bijective. By [10, Example 1.2.3] we thus have that the orders of vanishing of any rational function onW is the same as the orders of vanishing onW. In particular they can be computed using the usual techniques on normal toric varieties.
By construction of the maps giving the relations, we see that they are given by choosing an invariant subvarietyZof dimensionk+1 and choosing a rational function onZ and taking its divisor. Thus by definition these will give relations in
Ak(X), thus the composition of the two maps are zero.
By [8, Theorem 1] the canonical homomorphismATk(X)→ Ak(X)is an iso- morphism, whereATk(X)is the T-stable Chow group ofX. By definition this is the quotientZkT(X)/RkT(X)of the groupZkT(X)generated by T-invariant subvarieties of X modulo the subgroup RkT(X)generated by divisors of eigenfunctions on T- invariant(k+1)-dimensional subvarieties ofX. In particular this implies that all relations inAk(X)come from divisors on T-invariant subvarieties.
In the exact sequence above we have by construction all such subvarieties and relations, except that we omit T-invariant subvarieties of general fibers such that the tailcone is not contracted. Lettingqbe a point with general fiber we setP=P∪{q}, thus we now considerqas a special fiber. We show that every new subvariety and every new relation in the corresponding exact sequence is already generated by those in the exact sequence fromP. Thus we may omit any general fiber, hence the sequence in the theorem is exact. This is similar to Altmann-Petersen’s proof for the case of divisors [5, Corollary 2.3].
For any coneσ ∈ (n−k)where σ /∈ K we get the subvariety Wq,σ as a summand in Vk. There will also be a new relation coming from considering σ ∈ Rk+1 and observing thatZP/Zhas rank one more thanZP/Z. The extra relation expressesWp,F in terms of the other generators, thus we may omitWq,σ
as a generator inVk, as well as the corresponding relation.
There is one more class of relations coming from addingq: Each coneτ of dimensionn−k−1 withτ ∈ K gives an element inVk+1. Anym∈ M(τ)gives the relation
τσ,dimσ=n−k
m, vτ,σWq,σ
Now we already have relations saying that
Wq,σ =
HS∞,tailH=σ
μ(H)W∞,H. Thus we need to show that the relation
τσ,dimσ=n−k
m, vτ,σ
HS∞,tailH=σ
μ(H)W∞,H
is in the group of relations generated by sequence in the theorem.
Pick a faceGofS∞of dimensionn−k−1 with tailconeτ. ThenG=τ+Q, whereQis a polytope. Lettingvbe a vertex ofQwe have thatG=τ+v+(Q−v).
Since dimτ =dimG we must have that(Q−v)is contained in the linear span ofτ. After taking the quotientN/Qτ the image ofGwill thus be the vertexv, the¯ image ofv. Lete1, ...,ekbe aZ-basis forv¯⊥ande0,e1, ...,ekaZ-basis forM(τ).
We have thatM(G)is generated byμ(G)e0,e1, ...,ek. This implies that the lattice index[N(τ): N(G)]equalsμ(G). We then have the relation
GH,dimH=n−k
μ(G)ei, vG,HW∞,H.
For any H with tailconeσ we have thatμ(G)vG,H =μ(H)vτ,σ, by Lemma4.2 (see below), thus the relation equals
GH,tailH=σ
μ(H)ei, vτ,σW∞,H+
GH,tailH=τ
μ(G)ei, vG,HW∞,H, where allHin the sums are of dimensionn−k. Now, ifHcontainsG, has dimension n−kand tailconeτthen the imageHinN/Qτ is a compact edgeehavingGhas one of its vertices. The other vertex also corresponds to someGwith tailconeτ and of dimensionn−k−1. There is a similar relation to the above, corresponding toG. Nowμ(G)vG,Hequals a primitive generator inN/τ∩Nfor the linear space spanned bye. Similarlyμ(G)vG,H equals a primitive generator the same linear space, but with different sign. Thus if we sum the relations fromGandGthe term W∞,H will cancel. Thus if we sum all the relations corresponding to all possible suchG’s once, we see that the resulting relation is
G
GH,dimH=n−k,tailH=σ
μ(H)ei, vτ,σW∞,H
By grouping together terms corresponding to the same coneσwe see we can write this relation as
τσ,dimσ=n−k
ei, vτ,σ
HS∞,tailH=σ
μ(H)W∞,H
which is the relation we wanted to show form=ei. Since this is true for anyi it
will also follow for anymby linearity.
Lemma 4.2.Assumeτ σ are cones satisfying dimτ +1 = dimσ. Assume G H are faces of some Sp, withtailG =τ,tailH =σ,dimG =dimτ and dimH=dimσ. Thenμ(G)vG,H =μ(H)vτ,σ.
Proof. We may assumee1, ...,ekis a basis forN(H)ande1, ...,ek+1is a basis for N(G).N(τ)is a sublattice ofN(H)and there is an upper triangular integer matrix Bsuch that{bi =Bei}is a basis forN(τ). In particular the indexμ(G)= [N(G): N(τ)]equals the product of the diagonal entriesβi ofB. We may also assume that b1, ...,bk is a basis forN(σ). By definitionvτ,σ is a generator of N(τ)/N(σ), in
our chosen basis we can choose it as the image ofbk+1in the quotient. Similarly the image ofek+1is a generator ofN(G)/N(H). Thus we see thatβk+1vG,H =vτ,σ. We also have thatμ(H)= [N(H):N(σ)] =β1· · ·βk. In particularβk+1= μ(μ(GH),
which proves the statement.
Notation 4.3.We denote the number of elements of Rk,Vk,Tkby rk, vk,tk, respec- tively.
Example 4.4.LetX =Gr(2,4)which we can identify with the quadric V(p12p34−p13p24+p14p23)⊂P5.
The groupS L4(K)acts on Gr(2,4)and contains the subgroup of diagonal matrices which is a 3-dimensional torus and which acts effectively on Gr(2,4)making it into aT-variety of complexity one. The rational quotientY is naturally identified with the moduli space of marked genus 0 curvesM0,4 P1(see [15]). The iden- tification is defined as follows. We now think of Gr(2,4)as the space of lines in P3=P(V)whereV has basisx1, ...,x4. Ifl ∈Gr(2,4)is a general line then the intersectionsl∩ {xi =0}will give four points p1,p2,p3,p4onl. Then the cross ratio CR(p1,p2,p3,p4)defines the rational map toP1. In this way we obtain all values inP1except{0,1,∞}. The points 0,1,∞are obtained by the non-general lines in the setsV(p14)∪V(p23),V(p12)∪V(p34),V(p13)∪V(p24), respectively.
In coordinates the map is given as follows: On the open affine setD(p12)a point in Plucker coordinates maps to(p13p24 : p23p14)∈P1. The indeterminancy locus is given when both coordinates equals 0, we see that this locus consists of the eight planes
Tk+=V({pi j|k∈ {i,j}}) Tk−=V({pi j|k∈ {i,/ j}}).
We can resolve the quotient map by blowing up the union of the eight planes to get a mapX =Bl Gr(2,4)→P1. We will reinterpret this example in the language ofT-varieties.
The paper [2] exhibits a divisorial fan for Gr(2,4): Let N = Z4/Z and let e1,e2,e3,e0 denote the image of the standard basis vectors of Z4, thus e0 = −e1−e2−e3. The tailfan is the toric threefold with maximal cones Cone(±e1,±e2,±e3 ±e4|there are exactly 2 pluses and 2 minuses). This has 6 maximal cones, 12 cones of dimension two and 8 rays. The special fibers correspond to the boundary divisorsM0,4\M0,4of reducible genus 0 curves, of which there are three. They correspond to partitions(({1,4}, {2,3}), ({1,2},{3,4}),({1,3},{2,4}), we may assume these correspond to the points 0,1,∞, respectively.
The fiber over 0 corresponds to replacing the origin with the compact edge f23
with vertices(0,0,0)and(−1,−1,0). Similarly in the fiber over 1 we insert the edge f12 with vertices (0,0,0)and(−1,0,−1)and over∞ the edge f13 with vertices(1,1,1)and(1,0,0)(the polyhedra which is written in [2, Theorem 4.2]
is a shifted version of the above, with rational coefficients. By [2, p.8 Remark 2]
the true p-divisor correspond to a shift turning all polyhedra into lattice polyhedra,
which is what we have done.) The p-divisor containing the edge fi jhas the empty coefficient over the other two special fibers. For a cone σ of the faces of the special fibers with tailconeσ all belong to the same p-divisor. See Fig.3.
Thus we get the following numbers:
r3=0, v3=6,t3=0 r2=0, v2=3,t2=8 r1=0, v1=0,t1=12 r0=0, v0=0,t0=6.
The six invariant subvarietiesV(pi j)correspond toV3.V2corresponds to the sub- varietiesV(p12,p34), up to permutation, whilet2=8 says exactly that there are eight invariant subvarieties of codimension two which are blown up by r, they correspond to the setsTk±. Thus we get an exact sequence
Z18→Z11→ A2(Gr(2,4))→0
The Z11 has generators W12,W13,W23 corresponding to the edges fi j, and E±i corresponding to the rays±ei. TheZ18corresponds to six copies of Z3, one for each vertex of a special fiber. Fixing for example the vertexv =(0,0,0)of f13, we have that this is the vertex of a toric variety with rays with directions
1 0 0 ,
0 1 0 ,
0 0
−1 ,
1 1 1 ,
−1−1 0
. The relations we get from this will be
div(1,0,0)=E+1 +E0−−W13
div(0,1,0)=E+2 +E0−−W13
div(0,0,1)=E−3 −E0−.
Doing this for all vertices we see that all Ei− are identified, call this class E−, similarly allEi+are identified, call thisE+and allWi j are identified, call thisW. Then this gives a presentation
A2(Gr(2,4)) Z(E+,E−,W)/(E++E−−W)
which we see is isomorphic to a well known presentation of this group, namely Z(s1,1,s2,s12)/(s1,1+s2−s12)
Here thes1,s2,s1,1correspond to Schubert cycles in the Chow ring of Gr(2,4).
Similarly there is an exact sequence
Z22→Z12 → A1(Gr(2,4))→0
The generators ofZ12correspond to the 12 two-dimensional faces of, they are of the form Cone(ei,−ej). Denote the corresponding generator by Zi,−j. The
Fig.3.PolyhedralsubdivisonsdefiningGr(2,4).Fromeachofthenon-zeroverticesofthepolyhedralsubdivisionsthereisemanatingathree-dimensional conewithfourgenerators
relations come from two types: three copies of Z2 coming from the edges fi j. Writing out the relations we see that they giveZi,−j =Zj,−i. Also there are eight copies ofZ2corresponding to the rays of, they give relationsZi,−j = Zx,−k. Combining these we see that all Zi,−j are identified, thus A1(Gr(2,4))is one- dimensional, as expected.
5. Toric downgrades
In this section we study the example of downgrading a toric variety to only consider it as a T-variety of complexity one. We use the methods described in [14]. Not surprisingly the exact sequence of Theorem4.1coincides with the exact sequence of Fulton-Sturmfels.
We now consider a toric variety coming from a fanliving inZn+1⊗Q. We choose a splittingZn+1=Zn⊕Z=N⊕Zand considerXonly with the action ofTN. We then have an exact sequence
0→ N→ N⊕Z→Z→0
TheZcorresponds to the quotient, which for us will beP1. By construction there will only be two special fibers, over 0 and ∞. We denote by s the projection N⊕Z→ Nand byφthe mapN⊕Z→Z. For a coneσ ∈we get a polyhedral divisor with tailconeσ∩Nand coefficients(σ∩φ−1(1))over[0]and coefficient s(σ∩φ−1(−1))over[∞].
We consider the vector spaceV =Qn+1=Zn+1⊗Qwith basisv1, ..., vn+1
and denote the last coordinate hyperplane by H = {v =
tivi|tn+1 = 0}and H≥0= {v =
tivi|tn+1≥ 0},H>0 = {v =
tivi|tn+1>0}and similarly for H≤0,H<0.
Lemma 5.1.A coneσ ∈of dimension n−k+1corresponds to (1)an element of Rkif and only ifσ ⊂H .
(2)an element of Vkif and only ifσ ⊂H≤0orσ ⊂H≥0, butσis not contained in H .
(3)an element of Tkif and only ifσ intersects both H>0and H<0.
Proof. Ifσ ⊂Hthen we see that the tailcone of the associated polyhedral divisor isσ. Moreover we see that this divisor will have∅as coefficient over[0]and[∞], thus it will not be an element of K.
Ifσ ⊂ H≤0then the coefficient of [0] will be empty thus the tailcone will not lie in K. Moreover the coefficient over[∞]will have dimensionn −k as it will equal a compact polyhedron with vertices corresponding to generators having strictly positive last coordinate plus the tailcone which corresponds to rays with zero last coordinate.
Ifσ intersects both H>0and H<0, then first of all we see that there will be no∅coefficients. Moreover the tailcone will be the intersection ofσ with H. In particular it will have dimensionn−k.
Since any coneσ belongs to only one of the three categories the only if state-
ments follow as well.
