aa r X i v : . [ m a t h . AG ] J u l TORUS INVARIANT CURVES
GEOFFREY SCOTT
Abstract.
Using the language of T-varieties, we study torus invariant curves on a completenormal variety X with an effective codimension-one torus action. In the same way that the T -invariant Weil divisors on X are sums of “vertical” divisors and “horizontal” divisors, so too iseach T -invariant curve a sum of “vertical” curves and “horizontal” curves. We give combinatorialformulas that calculate the intersection between T -invariant divisors and T -invariant curves, andgeneralize the celebrated toric cone theorem to the case of complete complexity-one T -varieties. Introduction A T -variety is a normal complex algebraic variety with an effective action of an algebraic torus.This definition matches the definition of a toric variety, except that the dimension of the torus may beless than the dimension of the variety on which it acts. In particular, any normal algebraic variety isa T -variety when endowed with the trivial action of ( C ∗ ) . We therefore can’t expect to prove muchabout general T -varieties; we usually restrict our attention to complexity-one T -varieties, where thedimension of the torus is exactly one less than the dimension of the variety. In this paper, we studythe T -invariant curves of a complete complexity-one T -variety, find formulas for their intersectionwith T -invariant divisors (using the theory of T -invariant divisors developed by Petersen and S¨ußin [PS]), and prove that the numerical equivalence classes of these curves generate the Mori cone ofthe T -variety.We review the basics of T -varieties in Section 2. Informally speaking, a complexity-one T -varietyis encoded by a family (parametrized by a projective curve Y ) of polyhedral subdivisions of a vectorspace, all with the same tailfan. In Section 3, we describe two kinds of T -invariant curves in a T -variety, vertical curves and horizontal curves. The vertical curves correspond to walls (codimension-one strata) of one of these polyhedral subdivisions, while the horizontal curves correspond to certainmaximal-dimensional cones of the tailfan. We give formulas that calculate the intersection of thesecurves with a T -invariant divisor using the language of Cartier support functions from [PS].In Section 4, we generalize the toric cone theorem, which states that the Mori cone of a toricvariety is generated as a cone by the classes of T -invariant curves corresponding to the walls of itsfan. In our generalization, we show that the Mori cone of a complete complexity-one T -variety isgenerated as a cone by the classes of a finite collection of vertical curves and horizontal curves. Weend the paper with examples in Section 5.2. Primer on T -varieties In this section, we review the basic notation and construction of T -varieties. The presentationfavors brevity over pedogogy; we encourage any reader unfamiliar with T -varieties to read theexcellent survey article [A] for a friendlier exposition to this beautiful topic. The author was partially supported by NSF RTG grant DMS-1045119.The author was partially supported by NSF RTG grant DMS-0943832.
Notation.
Let T ∼ = ( C ∗ ) k be an algebraic torus, and M, N be the character lattice of T andthe lattice of 1-parameter subgroups of T respectively. These lattices embed in the vector spaces N Q := Q ⊗ N M Q := Q ⊗ M and are dual to one another . In classic toric geometry, one studies the correspondence betweencones (and fans) in N Q and the toric varieties encoded by these combinatorial data. Analogously, westudy T -varieties through the correspondence between combinatorial gadgets called p -divisors (anddivisoral fans) and the T -varieties they encode. Informally speaking, a p -divisor is a Cartier divisoron a semiprojective variety Y with polyhedral coefficients; a divisorial fan is a collection of p -divisorswhose polyhedral coefficients “fit together” in a suitable way. To make formal these definitions, webegin by discussing monoids of polyhedra.Let σ be a pointed cone in N Q , and σ ∨ ⊆ M Q its dual. The set Pol + Q ( N, σ ) of all polyhedra in N Q having σ as its tailcone (with the convention that ∅ ∈ Pol + Q ( N, σ )) is a monoid under Minkowskiaddition with identity element σ . Any nonempty ∆ ∈ Pol + Q ( N, σ ) defines a map h ∆ : σ ∨ → Q (1) u min v ∈ ∆ h v, u i called the support function of ∆. A nonempty ∆ ∈ Pol + Q ( N, σ ) also defines a normal quasifan N (∆)in M Q consisting of a cone λ F for each face F of ∆ defined by λ F = { u ∈ σ ∨ | h u, v i = h ∆ ( u ) ∀ v ∈ F } . The figure below shows an example of a polyhedron and its normal quasifan.∆ N (∆) Figure 1.
A polyhedron and its normal quasifan
Proposition 2.1. ( [AH03] , Lemma 1.4 and Proposition 1.5) The support function h ∆ is a well-defined map whose regions of linearity are the maximal cones of N (∆) . Moreover, any function inHom ( σ ∨ , Q ) whose regions of linearity define a quasifan can be realized as h ∆ for some ∆ . Let Pol Q ( N, σ ) be the Grothendieck group of Pol + Q ( N, σ ). Let Y be a semiprojective variety, withCaDiv( Y ) its group of Cartier divisors. An element D ∈
Pol Q ( N, σ ) ⊗ Z CaDiv( Y )is a polyhedral divisor with tailcone σ if it has a representative of the form D = P D P ⊗ P for some D P ∈ Pol + Q ( N, σ ) and P prime . We will describe a procedure for constructing an affine T -variety In this paper, when a picture of N Q is juxtaposed with a picture of M Q , the reader may assume that the basesfor these vector spaces have been chosen so that the pairing between them is the standard dot product. Because σ (not ∅ ) is the identity element of Pol Q ( N, σ ), the summation notation in this sentence implies thatonly finitely many of the polyhedral coefficients D P differ from σ ORUS INVARIANT CURVES 3 from a certain kind of polyhedral divisor (called a p-divisor ); this construction will involve takingthe spectrum of the global sections of a sheaf of rings defined over a subset of Y . This subset, calledthe locus of D , is Loc( D ) := Y \ ∪ D P = ∅ P. The evaluation of D at u ∈ M ∩ σ ∨ is the Q -Cartier divisor D ( u ) := X D P = ∅ h D P ( u ) P (cid:12)(cid:12) Loc( D ) . We say that D is a p -divisor if D ( u ) is semiample for all u ∈ σ ∨ and big for all u in the interior of σ ∨ . The direct sum of the sheaves defined by the evaluations D ( u ) is an M -graded sheaf of rings O ( D ) := M u ∈ σ ∨ ∩ M O Loc( D ) ( D ( u )) χ u over Loc( D ). There are two different T -varieties encoded by the p -divisor D g T V ( D ) := Spec Loc( D ) O ( D ) and T V ( D ) := Spec Γ(Loc( D ) , O ( D ))where the torus action is given by the M -grading on O ( D ). All affine T -varieties can be constructedthis way. Theorem 2.2. ( [AH03] , Corollary 8.14) Every normal affine variety with an effective torus actioncan be realized as T V ( D ) for some p -divisor D Similar to the way that a fan of a non-affine toric variety can be obtained by “gluing together”the cones constituting an affine cover, so too can a non-affine T -variety be encoded by “gluingtogether” the p -divisors constituting an affine cover. To make formal these concepts, we first definethe intersection of two p -divisors D , D ′ on Y as the p -divisor D ∩ D ′ := X ( D P ∩ D ′ P ) ⊗ P .We say that D ′ is a face of D if D ′ P ⊆ D P for each P and the induced map T V ( D ′ ) → T V ( D ) is anopen embedding. A finite set S of p -divisors on Y is a divisoral fan if for any D , D ′ ∈ S , D ∩ D ′ isan element of S and is a face of both D and D ′ . We define T V ( S ) and g T V ( S ) to be the T -varietiesobtained by gluing together the T -varieties { T V ( D ) } D∈S and { g T V ( D ) } D∈S according to these facerelations. This process is detailed in [AHS08].2.2.
Geometry of
T V ( S ) and g T V ( S ) . Because g T V ( D ) is defined as the relative spectrum of asheaf of rings on Loc( D ), there is a natural projection map π : g T V ( D ) → Loc( D ) ⊆ Y . Because T V ( D ) is defined as the spectrum of the global sections of the structure sheaf on g T V ( D ), we alsohave a natural map p : g T V ( D ) → Γ( g T V ( D ) , O g T V ( D ) ) ∼ = T V ( D ). Given a divisoral fan S , the maps π, p corresponding to the different D ∈ S glue into maps g T V ( S ) T V ( S ) Y π p
In this subsection, we describe the fibers of p and π . In particular, we will notice that for y ∈ Y , thereduced fiber π − ( y ) is a union of irreducible toric varieties, and that the contraction map p identifies Some authors define a “ Q -Cartier” divisor to be a Weil divisor with a Cartier multiple. Our Q -Cartier divisorsare elements of Q ⊗ Div ( Y ) having a Cartier multiple (so may have rational coefficients). The pedantic reader isinvited to replace all instances of “ Q -Cartier divisor” in this paper with “ Q -Cartier Q -divisor”. GEOFFREY SCOTT certain disjoint torus orbits of g T V ( S ). Many of these results simplify when T V ( S ) is a complexity-one T -variety; because this is the only case we will need for later sections, we will henceforth assumethat Y is a projective curve. The reader interested in higher-complexity T -varieties should read [A]for the more general results.In [P], the author describes the reduced fibers of π using the language of dappled toric bouquets .We begin by reviewing this language. Definition 2.3.
The fan ring of a quasifan Λ in M Q is C [Λ] := M u ∈| Λ |∩ M C χ u with multiplication defined by χ u χ v = (cid:26) χ u + v if u, v ∈ λ for some cone λ ∈ Λ0 otherwiseFor a nonempty ∆ ∈ Pol + Q ( N, σ ) and a cone λ F of its inner normal quasifan N (∆), let M λ F := { u ∈ λ F ∩ M | h ∆ ( u ) ∈ Z } . Remark 2.4.
In other papers, M λ F is defined differently: when ∆ ⊗ [ P ] appears as a summand in a p -divisor, the elements u ∈ M λ F are required to satisfy the condition that h ∆ ( u )[ P ] is locally principalat P . In the complexity-one case, this condition coincides with our condition that h ∆ ( u ) ∈ Z .Finally, let S ∆ ⊆ | Λ(∆) | ∩ M consist of those u such that S ∆ ∩ λ F = M λ F for every cone λ F ∈ N (∆). S ∆ can be thought of as a conewise-varying sublattice of M . The figure below showsan example of S ∆ for a given ∆; the elements of S ∆ ⊆ M are in bold. ∆ ⊆ N Q S ∆ ⊆ M Figure 2. S ∆ is a conewise-varying sublattice of M Definition 2.5.
The dappled fan ring of ∆ is the following subring of C [ N (∆)] C [ N (∆) , S ∆ ] := M u ∈ S ∆ C χ u Definition 2.6.
The dappled toric bouquet encoded by ∆ is the variety
T B (∆) := Spec( C [ N (∆) , S ∆ ]).Given a polyhedral complex Σ = { ∆ } in N Q , the dappled toric bouquet encoded by Σ is the variety T B (Σ) obtained by gluing the { T B (∆) } ∆ ∈ Σ according to the face relations among the polyhedra. ORUS INVARIANT CURVES 5
Observe that
T B (∆) and
T B (Σ) have a natural torus action induced by the M -grading of thedappled fan rings. For a T -variety T V ( S ) over Y and a point y ∈ Y , the polyhedra {D y } D∈S fittogether into a polyhedral complex S y of N Q . Proposition 2.7. [ [P] , Prop 1.29] Let S be a p-divisor on the smooth projective curve Y . Thereduced fiber π − ( y ) of π : g T V ( S ) → Y is equivariantly isomorphic to T B ( S y ) . Motivated by Proposition 2.7 to study the geometry of non-affine toric bouquets, we constructa fan for each vertex of a polyhedral subdivision Σ of N Q ; the toric varieties they encode will beprecisely the irreducible components of T B (Σ). For a vertex v ∈ Σ, define the lattice M v = { u ∈ M | h u, v i ∈ Z } Because M v is a sublattice of M , N is a sublattice of N v := M ∨ v ⊆ N Q . Let i v : N Q → ( N v ) Q bethe map induced by this inclusion. As ∆ ranges over all polyhedra in Σ containing v , the cones i v ( Q ≥ · (∆ − v )) form a fan F v in ( N v ) Q . For any cone σ = i v ( Q ≥ · (∆ − v )) of F v , the semigroup σ ∨ ∩ N ∨ v is isomorphic to the semigroup λ ∆ ∩ S ∆ . Because this isomorphism commutes with the gluingdata induced by the face relations, we have the following description of the irreducible componentsof T B (Σ).
Proposition 2.8.
The irreducible components of
T B (Σ) are equivariantly isomorphic to the toricvarieties { T V ( F v ) } where the set ranges over the vertices v of Σ . For example, the polyhedral complex in Figure 3 encodes a toric bouquet with three irreducibletoric components. We have drawn the lattices N v not as a square grid, but in a way that thesublattice N ⊆ N v (in bold) is a square grid so that the angles between the polyhedra are preserved.In the example, one fan encodes P and the other fans encode weighted projective spaces. F (1 / , / F (1 / , / F (2 , / Figure 3.
Components of a toric bouquetGiven a divisorial fan S , its tailfan tail( S ) is the fan consisting of the tailcones of the p -divisorscomprising S . Because the coefficients D y of a p -divisor D differ from its tailcone for only finitely GEOFFREY SCOTT many y , the polyhedral subdivisions S y differ from tail( S ) for only finitely many y . By Proposition2.8, the fiber of π over y ∈ Y is equal to T V (tail( S )) for all but finitely many y and specializes to a(possibly non-reduced) union of toric varieties at these finitely many points.By the discussion above, the familiar orbit-cone correspondence for toric varieties translates into acorrespondence between T -orbits in g T V ( S ) and pairs ( y, F ) where y ∈ Y and F ∈ S y . To understand T V ( S ), we will describe how the map p identifies certain of these orbits in different fibers. We firstconsider the case of an affine T -variety. For a p -divisor D with tailcone σ and a u ∈ σ ∨ ∩ M , thesemiample divisor D ( u ) defines a map ξ u : Loc( D ) → Proj M k ≥ Γ(Loc( D ) , D ( ku )) . Theorem 2.9. ( [AH03] , Theorem 10.1) The map p : g T V ( D ) → T V ( D ) induces a surjection { ( y, F ) : y ∈ Y, F is a face of D y } → { T − orbits in TV( D ) } that identifies the orbits corresponding to ( y, F ) and ( y ′ , F ′ ) iff λ F = λ F ′ ⊆ M Q and ξ u ( y ) = ξ u ( y ′ ) for some (equivalently, for any) u ∈ relint( λ F ) . In the non-affine case, the gluing maps among { T V ( D ) } D∈S are prescribed by the face relationsbetween the p -divisors, which identifies precisely those T -orbits in T V ( D ) and T V ( D ′ ) correspondingto the faces { ( y, D y ∩ D ′ y ) } y ∈ Y .3. T -invariant Curves and Intersection Theory In this section, we study the intersection theory of complete complexity-one T -varieties over aprojective curve Y . For the rest of the paper, all T -varieties are complete, complexity-one T -varieties over a projective curve Y . The “completeness” condition translates into thecombinatorial requirement that |S y | = N Q for all y . Motivated by the correspondence between T -invariant Cartier divisors and Cartier support functions introduced in [PS], we define the notion of a Q -Cartier support function to encode Q -Cartier torus invariant divisors. We will describe two kindsof T -invariant curves – vertical curves and horizontal curves – then give formulas that compute theintersection of these curves with a T -invariant Q -Cartier divisor. Definition 3.1.
Given a nontrivial ∆ ∈ Pol + Q ( N, σ ) and an affine ϕ : ∆ → Q , the linear part of ϕ is the function lin ϕ : σ → Q n ϕ ( p + n ) − ϕ ( p )where p is any point in ∆. If h σ i ⊆ N Q is the subspace spanned by σ , the function lin ϕ extendsuniquely to a linear function h σ i → Q , which will also be written lin ϕ without risk of confusion. Definition 3.2.
Let S be the divisorial fan of a complexity-one T -variety over Y . A Q -Cartiersupport function is a collection of affine functions { h D ,y : |D y | → Q } D∈S y ∈ Y with rational slope and rational translation such that(1) For a fixed y ∈ Y , the functions { h D ,y } D∈S define a continuous function h y : |S y | → Q .That is, h D ,y and h D ′ ,y agree on D y ∩ D ′ y for D , D ′ ∈ S .(2) For each D ∈ S with complete locus, there exists u ∈ M, f ∈ K ( Y ) and N ∈ Z > such that N h D ,y ( v ) = − ord y ( f ) − h u, v i for all y ∈ Y and all v ∈ N Q . ORUS INVARIANT CURVES 7 (3) If D y , D ′ y ′ have the same tailcone, then lin h D ,y = lin h D ′ ,y ′ .(4) For a fixed D , h D ,y differs from lin h D ,y for only finitely many y .A Q -Cartier support function is called a Cartier support function if each h D ,y has integral slope andintegral translation and N = 1 in condition (2). We write CaSF ( S ) and Q CaSF ( S ) to denote theabelian group (under standard addition of functions) of Cartier support functions and Q -Cartiersupport functions respectively.For any T -invariant Cartier divisor D on T V ( S ) and any p -divisor D ∈ S , we can always find anopen cover { U i } of Y for which there exists Cartier data for D (cid:12)(cid:12) T V ( D ) of the form ( T V ( D (cid:12)(cid:12) U i ) , f i χ u i )(see proof of [PS], Prop 3.10 for details). These Cartier data define functions { h D ,y ( v ) = − ord y ( f i ) − h u i , v i} y ∈ U i which agree on the overlaps of the U i to define h D ,y for all y . In this way, we can define a Cartiersupport function for any Cartier divisor on T V ( S ). Proposition 3.3. ( [PS] , Prop 3.10) Let T − CaDiv ( S ) denote the group of T -invariant Cartierdivisors on T V ( S ) . This association of a Cartier support function to a T -invariant Cartier divisordefines an isomorphism of groups T − CaDiv ( S ) ∼ = CaSF ( S )If { h D ,y } is the Cartier support function for N D , where
N > D is a T -invariant Q -Cartierdivisor, then { N − h D ,y } is a Q -Cartier support function. In this way, we can associate a Q -Cartiersupport function to any T -invariant Q -Cartier divisor on T V ( S ). The following is an immediatecorollary of Proposition 3.3. Corollary 3.4.
Let T − Q CaDiv ( S ) denote the group of T -invariant Q -Cartier divisors on T V ( S ) .Then the association described above is an isomorphism of groups T − Q CaDiv ( S ) ∼ = Q CaSF ( S )3.1. Vertical Curves.
Toward the goal of describing the intersection theory of a T -variety, we studyits T -invariant curves. We start with vertical curves , which are images (under p ) of a T -invariantcurve contained in a single fiber of π .Recall from Proposition 2.8 that for y ∈ Y , the reduced fiber π − ( y ) has as its irreducible compo-nents a toric variety for each vertex v of S y . A toric variety has a T -invariant curve correspondingto each wall of its fan (by taking the closure of the corresponding torus orbit). Translating thisfact into the context of toric bouquets, we call a codimension-one element of a polyhedral complex a wall if it can be realized as the intersection of two top-dimensional polyhedra; there is a T -invariantcurve in a toric bouquet for each wall of the corresponding polyhedral complex. In this section, westudy the curves in g T V ( S ) and T V ( S ) corresponding to these T -invariant curves.Fix a T -variety T V ( S ) and a point y ∈ Y . Let τ ∈ S y be a wall of the polyhedral complex S y , let D , D ′ ∈ S be two p -divisors for which τ = D y ∩ D ′ y , let λ τ, D ⊆ M Q be the cone in N ( D y ) dual to τ ,and let u τ, D be the semigroup generator of M λ τ, D . As usual, unweildy notation obfuscates a simplepicture: if D and D ′ have polyhedral coefficients over y as shown in Figure 4 ( τ is the horizontalplane in a single orthant at a height of 2 / Z · u τ, D = M λ τ, D ∪ M λ τ, D′ consistsof the bold elements of the vertical axis of M ∼ = Z shown on the right. A wall of a fan is a codimension-one cone that can be realized as the intersection of two top-dimensional cones. GEOFFREY SCOTT D y D ′ y u τ, D u τ, D ′ Figure 4.
The sublattice Z · u τ, D corresponding to the wall τ = D y ∩ D ′ y If g ∈ K ( Y ) is Cartier data for D ( u τ, D ) in some neighborhood of y , the mapsΓ(Loc( D ) , O ( D )) → C [ z ] f χ u (cid:26) u / ∈ M λ τ , D ( g k f )( y ) z k u = ku τ, D Γ(Loc( D ′ ) , O ( D ′ )) → C [ z − ] f χ u (cid:26) u / ∈ M λ τ , D ′ ( g − k f )( y ) z − k u = − ku τ, D glue together to induce a map(2) P → T V ( S )the image of which we will call the vertical curve C τ,y . Proposition 3.5.
The vertical curve C τ,y is the image under p of the closure of the torus orbit in T B ( S y ) ⊆ g T V ( S ) corresponding to the wall τ .Proof. For any affine open U ⊆ Y containing y , Map 2 factors(3) P → π − ( U ) ⊆ g T V ( S ) p → T V ( S )where P → π − ( U ) is given byΓ( U, O ( D )) → C [ z ](4) f χ u (cid:26) u / ∈ M λ τ , D ( g k f )( y ) z k u = ku τ, D Γ( U, O ( D ′ )) → C [ z − ] f χ u (cid:26) u / ∈ M λ τ , D ′ ( g − k f )( y ) z − k u = − ku τ, D Therefore, it suffices to show that the image of P → π − ( U ) is the closure of the torus orbit in T B ( S y ) ⊆ g T V ( S ) corresponding to the wall τ . To do so, we recall some relevant details about theisomorphism between the reduced fibers of π and a dappled toric bouquet (see [AH03], Proposition7.10 for details). This isomorphism is constructed by first choosing a collection of functions { g D ( u ) ∈ K ( Y ) } u ∈ S ∆ such that, after possibly shrinking U ,div( g D ( u ) ) (cid:12)(cid:12) U = D ( u ) (cid:12)(cid:12) U and g D ( u + u ′ ) = g D ( u ) g D ( u ′ )ORUS INVARIANT CURVES 9 Then the isomorphism between the fiber and the dappled toric bouquet is induced byΓ( U, O ( D )) → C [ N ( D y ) , S D y ](5) f χ u (cid:26) ( g D ( u ) f )( y ) χ u if u ∈ S D y D ′ ). On the other hand, the closure of the torus orbit corresponding to τ in thetoric bouquet is parametrized by gluing the maps C [ N ( D y ) , S D y ] → C [ z ](6) χ u (cid:26) z k if u = ku τ, D , k ∈ Z C [ N ( D ′ y ) , S D ′ y ] → C [ z − ] χ u (cid:26) z − k if u = − ku τ, D , k ∈ Z (cid:3) To find a formula that calculates the intersection between C τ,y and a T -invariant Cartier divisor D , we pick Cartier data for D that includes two sets of the form { ( T V ( D (cid:12)(cid:12) U ) , f χ u ) , ( T V ( D ′ (cid:12)(cid:12) U ′ ) , f ′ χ u ′ ) } where U, U ′ ⊆ Y are open sets containing y . The Cartier support function for D includes h D ,y = − ord y ( f ) − h u, v i and h D ′ ,y = − ord y ( f ′ ) − h u ′ , v i . Because h D ,y and h D ′ ,y agree on τ , it must be the case thatord y ( f ) − ord y ( f ′ ) + h u − u ′ , v i = 0for all v ∈ τ . In particular, u − u ′ ∈ Q · u τ, D . Moreover, since h u − u ′ , v i = ord y ( f ′ ) − ord y ( f ) ∈ Z ,it must be the case that h u − u ′ , v i ∈ Z for v ∈ τ . Therefore, u − u ′ = ku τ, D for some k ∈ Z ,and the quotient of the two Cartier data is f χ u /f ′ χ u ′ = ( f /f ′ ) χ ku τ, D . Under the parametrizationof C τ,y in Equation 2, this rational function pulls back to ( g k f /f ′ )( y ) z k on C τ,y ∼ = P , where g isCartier data for D ( u τ, D ). Therefore, the degree of the pullback of D onto C τ,y is k . This is precisely µ − τ h u − u ′ , n τ, D i , where µ τ is the index of Z · u τ, D in Q · u τ, D ∩ M and n τ, D ∈ N is any representativeof the generator of N/ ( u τ, D ) ⊥ that pairs positively with u τ, D (equivalently, n τ, D is any element of N such that h n τ, D , u τ, D i = µ τ ). h D, C τ,y i = µ − τ h u − u ′ , n τ, D i or, using the language of Cartier support functions,(7) h D, C τ,y i = µ − τ (lin h D ′ ,y − lin h D ,y )( n τ, D )By linearity, the same formula applies when D is a T -invariant Q -Cartier divisor. Example 3.6.
Let D , D ′ ∈ S be p -divisors that have the slices shown in Figure 4. Suppose thatwith respect to the standard basis given by the coordinate axes in the picture, a T -invariant divisor D has the following Cartier support functions h D ,y ( v ) = −
10 + h (9 , , , v i and h D ′ ,y ( v ) = 0 + h (9 , , , v i Then h D, C τ,y i = 3 − h (0 , , − , (0 , , i = − Horizontal Curves.
Let σ be a full-dimensional cone of tail( S ). Because the T -varieties westudy are complete, every S y contains a polyhedron with tailcone σ . Such a polyhedron correspondsto a fixed point in the fiber π − ( y ). Taking the union (as y varies) of these fixed points defines acurve e C σ ⊆ g T V ( S ). Theorem 2.9 shows that p contracts e C σ precisely if there is some D ∈ S withtailcone σ and complete locus. In this case, we say that σ is marked . Definition 3.7.
A cone σ of tail( S ) is marked if σ is the tailcone of a p -divisor D ∈ S with completelocus.When σ is unmarked, Theorem 2.9 shows that no distinct points of e C σ are identified by p . Towardthe goal of finding an intersection formula for these horizontal curves C σ := p ( e C σ ), we parametrizethem. Let T V ( S ) be a T -variety and let σ be an unmarked full-dimensional cone of tail( S ). For D ∈ S with tailcone σ , we have a map of rings ϕ D : Γ( T V ( D ) , O ( D )) → Γ(Loc( D ) , O Y )(8) f χ u (cid:26) f if u = 00 otherwiseBecause each { Loc( D ) | tail( D ) = σ } is affine, these glue into a map s σ : Y ֒ → T V ( S )where we used the fact that S is complete (so |S y | = N Q for all y ) to deduce that Y is covered by { Loc( D ) | tail( D ) = σ } . The map s σ factors through g T V ( S ). By carefully following the isomorphismbetween the fibers of π and the corresponding toric bouquets (as in the proof of Proposition 3.5),we see that the image of s σ indeed equals the horizontal curve p ( e C σ ).We can use this parametrization to find an intersection formula for T -invariant divisors andhorizontal curves. Fix a cone σ of tail( S ) of full dimension and a T -invariant Cartier divisor D with Cartier support function { h D ,y } . Because σ has full dimension, there is a unique u σ ∈ M andcollection of integers { a y ∈ Z } y ∈ Y such that for each D with tailcone σ and each y ∈ Loc( D ), h y, D ( v ) = − a y − h u σ , v i . We can find Cartier data for D whose open sets and rational functions are of the form( T V ( D (cid:12)(cid:12) U ) , f D ,U χ u σ )for open sets U ⊆ Y . Then ord y ( f D ,U ) = a y for all D with tailcone σ and y ∈ U . When σ is unmarked, the open sets U appearing in the Carter data are affine, and the pullback of thetransition function f D ,U f − D ′ ,U ′ χ onto the curve C σ ∼ = Y is the function f D ,U f − D ′ ,U ′ on U ∩ U ′ . Thatis, the functions f D ,U appearing in the Cartier data for D are themselves the Cartier data for thepullback of D onto C σ ∼ = Y . As a Weil divisor, the pullback of D onto C σ is P a y [ y ]; we call thisdivisor D σ . Definition 3.8.
Given a Q -Cartier support function { h D ,y } , a cone σ of full dimension in tail( S ),and a point y , there is a unique a y ∈ Z such that for every D with tailcone σ and Loc( D ) ∋ y , h D ,y = − a y − lin( h D ,y ) . Then define D σ = X y ∈ Y a y [ y ] Remark 3.9.
The definition of D σ makes sense even when σ is marked. However, if D has completelocus, then by ([PS], Proposition 3.1) every invariant Cartier divisor on T V ( D ) is principal. It followsthat deg( D σ ) = 0 for every marked σ . Remark 3.10.
Compare this definition to ([PS], Definition 3.26). In our notation, D σ = − h (cid:12)(cid:12) σ (0). ORUS INVARIANT CURVES 11
With this new definition, we can summarize the discussion above with the following equation forthe intersection theory of a T-invariant divisor with a horizontal curve. h D, C σ i = deg( D σ )By linearity, the same formula applies when D is a T -invariant Q -Cartier divisor.4. The T Cone Theorem
Given a normal variety X , let Z ( X ) be the proper 1-cycles, and define N ( X ) := (CaDiv( X ) / ∼ ) ⊗ Z R N ( X ) := ( Z ( X ) / ∼ ) ⊗ Z R where ∼ denotes numerical equivalence of divisors in the first definition, and numerical equivalence ofcurves in the second. The vector space N ( X ) contains the cone Nef( X ) generated by classes of nefdivisors, and the vector space N ( X ) contains the cone N E ( X ) generated by classes of irreduciblecomplete curves. The Mori cone
N E ( X ) is the closure of N E ( X ). With respect to the intersectionproduct, N ( X ) and N ( X ) are dual vector spaces, and the cones Nef( X ) , N E ( X ) are dual cones.When X is the toric variety of a fan Σ, the closure of the torus orbit corresponding to a wall of Σdefines an element of N E ( X ). The celebrated toric cone theorem ([CLO], Theorem 6.3.20(b)) statesthat N E ( X ) is generated as a cone by these classes. In this section, we prove the corresponding resultfor T -varieties. We continue to assume that all T -varieties are complete complexity-one T -varietiesover a projective curve Y . Theorem 4.1.
Let
T V ( S ) be an n -dimensional T -variety, and let y ′ ∈ Y be any point for which S y ′ = tail( S ) . Then (9) N E ( T V ( S )) = X y ∈ Yτ a wall of S y dim(tail( τ )) Any Cartier divisor D on a T -variety T V ( S ) is linearly equivalent to a T -invariant Cartier divisor. Different authors have different definitions of concavity; to us, a function ϕ : N Q → Q is concaveif ϕ ( tv + (1 − t ) w ) ≥ tϕ ( v ) + (1 − t ) ϕ ( w ) for all v, w ∈ N Q and all t ∈ [0 , Proposition 4.3. ( [PS] , Corollary 3.29) A T-invariant Cartier divisor D ∈ T − CaDiv ( S ) withCartier support function { h D ,y } is nef iff all h y are concave and deg( D σ ) ≥ for every maximalcone σ of the tailfan. Toward our goal of proving Theorem 4.1, we will use Proposition 4.3 to show that a Cartierdivisor is nef if it intersects all vertical and horizontal curves nonnegatively. The proof of this factrequires a combinatorial lemma. Given a Cartier support function { h D ,y } and any D ∈ S , y ∈ Y such that dim( D y ) = dim( N Q ), define e h D ,y : N Q → Q to be the unique affine function that extends h D ,y : |D y | → Q . Lemma 4.4. Let { h D ,y } be a Cartier support function. The following are equivalent • h y : N Q → Q is concave. • For every wall τ = D y ∩ D ′ y of S y , there is some v ∈ D ′ y \D y with h D ′ ,y ( v ) ≤ e h D ,y ( v ) . Proof. This is a straightforward extension of ([CLO], Lemma 6.1.5 ( a ) ⇐⇒ ( d )) (where it is provedfor Cartier support functions on a fan). (cid:3) Proposition 4.5. A Cartier divisor D ∈ CaDiv( T V ( S )) is nef iff h D, C i ≥ for all vertical andhorizontal curves C .Proof. The forward direction follows from the definition of nef. To prove the reverse direction, let D ∈ CaDiv( S ) satisfy the condition that h D, C i ≥ D with a linearly equivalent T -invariant divisor and let { h D ,y } be its Cartier support function. Let τ = D y ∩ D ′ y be a wall of S y . Fix any n τ, D ∈ N with h n τ, D , u τ, D i = µ τ and any v τ ∈ relint( τ ). Thenpick ǫ > v := v τ + ǫn τ, D ∈ D y \D ′ y . Then h D ,y ( v ) = h D ,y ( v τ ) + lin h D ,y ( ǫn τ, D ) e h D ′ ,y ( v ) = h D ′ ,y ( v τ ) + lin h D ′ ,y ( ǫn τ, D ) . Because h D ,y and h D ′ ,y agree on τ , e h D ′ ,y ( v ) − h D ,y ( v ) = (lin h D ′ ,y − lin h D ,y )( ǫn τ, D ) ≥ h D, C τ,y i ≥ 0. Becausethis holds for all walls in all slices S y , we conclude by Lemma 4.4 that each h y is concave.To show that deg( D σ ) ≥ σ of the tailfan, observe that if σ is marked,then deg( D σ ) = 0 by Remark 3.9; if σ is unmarked, then deg( D σ ) = h D, C σ i ≥ (cid:3) To put Proposition 4.5 in context, remember that a T -variety has infinitely many distinct verticalcurves. Indeed, if τ is a wall of tail( S ), then for every y ∈ Y there is (by completeness) a verticalcurve C τ ′ ,y where τ ′ is a wall of S y with tailcone τ . The next proposition shows that the classes ofall such curves lie on a single ray of N ( T V ( S )). Proposition 4.6. Let τ = σ ∩ σ ′ be a wall of tail( S ) , where σ, σ ′ are full dimensional cones of tail( S ) . The classes C τ = (cid:26) [ C τ ′ ,y ] (cid:12)(cid:12)(cid:12)(cid:12) τ ′ = D y ∩ D ′ y for some D , D ′ withtail( D ) = σ, tail( D ′ ) = σ ′ (cid:27) ⊆ N ( T V ( S )) are positive multiples of each other. Specifically, [ C τ ,y ] = µ − τ µ τ [ C τ ,y ] .Proof. Let { h D ,y } be the Cartier support funtion of some D ∈ T-CaDiv( T V ( S )). All h D ,y withtail( D ) = σ (respectively σ ′ ) will have the same linear part, say − u σ ∈ M Q (respectively − u σ ′ ∈ M Q ).Then for two classes [ C τ ,y ] , [ C τ ,y ] ∈ C τ , Equation 7 calculates the intersections as h D, C τ ,y i = µ − τ h u σ − u σ ′ , n τ , D i h D, C τ ,y i = µ − τ h u σ − u σ ′ , n τ , D i Since we can choose n τ , D = n τ , D , it follows that h D, C τ ,y i = µ − τ µ τ h D, C τ ,y i for all D . (cid:3) We are finally ready to prove Theorem 4.1. Using the propositions above, the proof is nearlyidentical to the proof of the toric cone theorem in ([CLO], Theorem 6.3.20(b)). Proof. (Theorem 4.1) Let Γ be the rational polyhedral cone in N E ( T V ( S )) defined by the righthand side of Equation 9. By definition, Γ includes the classes of all horizontal curves; by Proposition4.6, it also includes the classes of all vertical curves. Therefore, Proposition 4.5 implies that Γ ∨ =Nef( T V ( S )), so Γ = Γ ∨∨ = N E ( T V ( S )). (cid:3) ORUS INVARIANT CURVES 13 Examples Example 1. Consider the divisoral fan S shown in Figure 5. T V ( S ) is the projectivizedcotangent bundle of the first Hirzebruch surface. All horizontal divisors in g T V ( S ) are contracted.For each vertical divisor D [ y ] ,v and each maximal p -divisor D i ∈ S , we write the Weil divisor P a y [ y ]and an element u ∈ M in Table 1 to encode the Cartier support function { h D i ,y ( w ) = − a y −h u, w i} of D [ y ] ,v . For example, the Cartier support function for D [0] , (0 , includes h D , ∞ ( v ) = 1 − h ( − , − , v i . D D D D D D D D ∞ Figure 5. D D D D D D D D D [0] , (0 , , 1) (0 , 1) (1 , 1) (1 , 1) (0 , 0) (0 , 0) (0 , 0) (0 , D [0] , (0 , [0] − [1] [0] − [ ∞ ] [0] − [ ∞ ] [0] − [1] (1 , − 1) (0 , 0) (0 , 0) ( − , − 1) (0 , 1) (0 , 0) (0 , 0) (1 , D [0] , (0 , − , 0) (0 , 0) (0 , 0) (0 , 0) ( − , − 1) ( − , − 1) (0 , − 1) (0 , − D [1] , (1 , , 0) (1 , 0) (0 , 0) (0 , 0) (0 , 0) (0 , 0) (1 , 0) (1 , D [1] , (0 , [1] − [0] [1] − [0] [1] − [ ∞ ] [1] − [ ∞ ] [1] − [0] [1] − [0] 0(0 , 0) ( − , 1) (1 , 1) ( − , 0) ( − , 0) ( − , − 1) ( − , − 1) (0 , D [ ∞ ] , ( − , , 0) (0 , 0) ( − , 0) ( − , 0) ( − , 0) ( − , 0) (0 , 0) (0 , D [ ∞ ] , (0 , [ ∞ ] − [1] [ ∞ ] − [0] [ ∞ ] − [0] [ ∞ ] − [0] [ ∞ ] − [0] [ ∞ ] − [1] (1 , 0) (0 , 1) (2 , 1) (0 , 0) (0 , 0) (0 , − 1) (0 , − 1) (1 , Table 1. Torus invariant divisors on T V ( S )Because every maximal-dimensional cone of tail( S ) is marked, T V ( S ) has no horizontal curves.Let τ i,j,y be the wall of S y realized as the intersection between D i and D j (if such a wall exists).Using Proposition 4.6, we see that the numerical equivalence class of C τ i,j,y ,y only depends on i and j ; to save space, we abbreviate [ C τ i,j,y ,y ] as C i,j .As an example of a calculation, consider C , and the T -invariant divisor D [0] , (0 , with Cartiersupport function { h D ,y } . Using notation from Section 3.1, n τ, D = (0 , See [PS] for a definition and description of horizontal and vertical divisors of the Cartier support function are lin h D , = − (1 , − ∈ M and lin h D , = (0 , ∈ M . Theintersection can then be calculated using Equation 7 h D [0] , (0 , , C , i = 1 − h ( − , − (0 , , (0 , i = 1The complete list of intersections is in Table 2. The canonical divisor is also listed; it can beexpressed as a sum of the vertical divisors using the formula from ([PS], Theorem 3.21). C , C , C , C , C , C , C , C , C , C , C , C , D [0] , (0 , D [0] , (0 , D [0] , (0 , − D [1] , (1 , D [1] , (0 , D [ ∞ ] , ( − , D [ ∞ ] , (0 , K X -2 2 -2 0 -2 -2 -2 0 -4 -4 -4 -4 Table 2. Intersections of divisors and curves on T V ( S )5.2. Example 2. Let σ , σ , σ be the cones σ = Q ≥ · (1 , 0) + Q ≥ · (0 , σ = Q ≥ · (0 , 1) + Q ≥ · ( − , − σ = Q ≥ · (1 , 0) + Q ≥ · ( − , − S be the divisorial fan on P having the following maximal p -divisors D = ((2 / , / 2) + σ )[0] + (( − / , − / 2) + σ )[1] + ∅ [ ∞ ] D = ((2 / , / 2) + σ )[0] + (( − / , − / 2) + σ )[1] + (( − , − 1) + σ )[ ∞ ] D = ((2 / , / 2) + σ )[0] + (( − / , − / 2) + σ )[1] + (( − , − 1) + σ )[ ∞ ] D = ∅ [0] + ∅ [1] + (( − , − 1) + σ )[ ∞ ] D D D D ∞ Figure 6. The divisorial fan S The T -variety corresponding to S is a deformation of P . The T -invariant divisors and theirintersections with T -invariant curves are encoded in Table 3 and 4 respectively, using the samenotation as in the previous example. ORUS INVARIANT CURVES 15 D D D D D [0] , ( / , / ) / [0] / [0] − / [1] − / [ ∞ ] / [0] − / [1] − / [ ∞ ] 0(0 , 0) ( -1 / , 0) (0 , -1 / ) (0 , D [1] , ( -2 / , -1 / ) / [1] / [0]+ / [1] − / [ ∞ ] / [0]+ / [1] − / [ ∞ ] 0(0 , 0) ( -1 / , 0) (0 , -1 / ) (0 , D [ ∞ ] , ( − , − / [0] − / [1] / [0] − / [1] [ ∞ ](0 , 0) ( − , 0) (0 , − 1) (0 , D Q ≥ · (1 , -2 / [0]+ / [1] 0 -1 / [0]+ / [1] [ ∞ ](0 , 0) ( -1 / , 0) (0 , -1 / ) (0 , D Q ≥ · (0 , -1 / [0]+ / [1] / [0] − / [1] 0 [ ∞ ](0 , 1) ( − , 1) (0 , 0) (0 , Table 3. Torus invariant divisors on T V ( S ) C τ , C τ , C τ , C σ D [0] , ( / , / ) / / / / D [1] , ( -2 / , -1 / ) / / / / D [ ∞ ] , ( − , − D Q ≥ · (1 , D Q ≥ · (0 , Table 4. Intersections on T V ( S ) References [AH03] Klaus Altmann and J¨urgen Hausen. Polyhedral divisors and algebraic torus actions , extended version,arXiv:math/0306285v1, 2003.[AHS08] Klaus Altmann, J¨urgen Hausen, and Hendrik S¨uß. Gluing affine torus actions via divisorial fans . Transfor-mation Groups, 13(2):215-242, 2008.[A] Klaus Altmann, Nathan Owen Ilten, Lars Petersen, Robert Vollmert, and Hendrik S¨uß. The geometry of T-varieties ∼ dac/toric.html.[P] Lars Petersen, Line Bundles on Complexity-One T-Varieties and Beyond . Ph.D. Thesis. Freie Universit¨at: Ger-many, 2010[PS] Lars Petersen and Hendrick S¨uß, Torus invariant divisors , arXiv:0811.0517v3, 2008. E-mail address ::