aa r X i v : . [ m a t h . AG ] J u l TROPICAL INTERSECTION THEORY FROM TORICVARIETIES
ERIC KATZ
Abstract.
We apply ideas from intersection theory on toric varieties to trop-ical intersection theory. We introduce mixed Minkowski weights on toric va-rieties which interpolate between equivariant and ordinary Chow cohomol-ogy classes on complete toric varieties. These objects fit into the frameworkof tropical intersection theory developed by Allermann and Rau. Standardfacts about intersection theory on toric varieties are applied to show thatthe definitions of tropical intersection product on tropical cycles in R n givenby Allermann-Rau and Mikhalkin are equivalent. We introduce an inducedtropical intersection theory on subvarieties on a toric variety. This gives aconceptional proof that the intersection of tropical ψ -classes on M ,n used byKerber and Markwig computes classical intersection numbers. Introduction
Tropical Geometry is a rapidly developing field of mathematics. In it, algebraicvarieties are replaced by piecewise-linear objects called tropical varieties. Thesetropical varieties capture a great deal of the nature of the original variety. Oneof the original motivations for tropical geometry was that enumerative propertiessurvive tropicalizations and, therefore, enumerative problems in algebraic geometryhave piecewise-linear analogs that give the same answer. To this end, it is desirablethat tropical varieties have a good intersection theory.There have been tropical intersection theories developed for tropical varietiesin R n . The definition of Mikhalkin in [14] uses a stable intersection in that twotropical cycles are perturbed so they intersect transversely. The intersection num-ber produced is independent of the perturbation. Allermann and Rau developed atropical intersection theory that is built out of the operation of intersecting trop-ical cycles with tropical Cartier divisors. This is analogous to the construction ofclassical intersection theory in [4].Tropical intersection theory has been applied in a number of contexts. One suchapplication is the intersection of ψ -classes on M ,n by Kerber and Markwig [12].They define tropical analogs of all the concepts involved and compute the tropicalintersection numbers which are seen to be the same as the classical intersectionnumbers.In this paper, we prove that the Allermann-Rau and the Mikhalkin tropical inter-section theories are equivalent. We also introduce an induced tropical intersectiontheory. For i : V ֒ → X (∆), a subvariety of a toric variety satisfying certain proper-ties, we show that the degree deg( i ∗ ( d ∪ · · · ∪ d k ) ∩ [ V ]) for certain d i ∈ A ( X ) canbe computed using tropical geometry. This induced tropical intersection theory isapplied to show that the tropical ψ -class intersection numbers on M ,n agree with the classical ones. We hope that this induced intersection theory sheds light onwhich other intersection problems can be computed tropically.The method for the equivalence proof is to first prove that the intersection the-ories are equivalent for tropical cycles that are also fans. In this case, the tropicalintersection theory can be related to Chow cohomology on the toric variety de-scribed by the fan. In the case that the toric variety is complete, the intersectiontheory is combinatorial. By a result of Fulton and Sturmfels [5], Chow cohomologyclasses are described by Minkowski weights which are exactly tropical cycles. Thecup product is described by the fan displacement rule given by Fulton and Sturm-fels. Equivariant Chow cohomology classes are described by piecewise polynomialsby a result of Payne [17]. Tropical Cartier divisors which are piecewise-linear func-tions supported on tropical cycles can be interpreted as a hybrid object that wecall mixed Minkowski weights which interpolate between equivariant and ordinaryChow cohomology. We define a map κ from mixed Minkowski weights to Chow co-homology. It turns out that the associated Weil divisor of a tropical Cartier divisor( c, f ) is exactly the Chow cohomology class κ ( c, f ). The Mikhalkin definition ofintersection product becomes the cup product of Chow classes by the fan displace-ment rule . The equivalence of the two definitions of intersection theory followsfrom the identity ( c ∪ c ) ∩ [ X (∆)] = c ∩ ( c ∩ [ X (∆)]) . By a straightforwardargument, the intersection theory on general tropical cycles in R n can be reducedto the fan case.For the result on tropical ψ -classes, we use an embedding of M ,n into a toricvariety discovered by Kapranov [9] and studied in detail by Gibney and Maclagan[7]. The intersection problem on M ,n can be related to an intersection problemon the toric variety where it is then described by tropical intersection theory.Our proof of the equivalence of tropical intersection theories which are bothpurely combinatorial makes use of non-combinatorial facts from the theory of toricvarieties and intersection theory. A combinatorial proof was given by Rau [16].We would like to thank Sean Keel, Michael Kerber, Diane Maclagan, HannahMarkwig, Sam Payne, Johannes Rau, and David Speyer for valuable discussions.2. Toric Varieties and Tropical Geometry
We review some notions from the theory of toric varieties. A toric variety X (∆)is determined by a rational fan ∆ in a vector space N R . The torus T acting on X hasa one-parameter subgroup lattice canonically isomorphic to a lattice N ⊂ N R . Thecharacter lattice of T is denoted by M = N ∨ . The properties of the toric variety arereflected in those of the fan. X (∆) is complete if and only if | ∆ | = N R while X (∆)is smooth if and only if ∆ is unimodular, that is, if every cone in ∆ is generated bya subset of an integral basis for N . Any fan ∆ can be refined to a unimodular fan e ∆ which induces a resolution of singularities X ( e ∆) → X (∆) [3]. This resolutionalgorithm can be executed so that cones that are already unimodular do not needto be subdivided. If τ is a cone in ∆, the orbit closure V ( τ ) is a toric variety givenby a fan ∆ τ whose cones are σ = ( σ + N τ ) /N τ for σ ⊇ τ where N τ is the linearspan of τ in N .Let X = X (∆) be a complete n -dimensional toric variety with torus T . Boththe Chow cohomology, A ∗ ( X ) and the equivariant Chow cohomology, A ∗ T ( X ) havecombinatorial descriptions. ROPICAL INTERSECTION THEORY FROM TORIC VARIETIES 3
The description of A ∗ ( X ) is given by Fulton and Sturmfels [5]. Let ∆ ( k ) denotethe set of all cones in ∆ of codimension k . If τ ∈ ∆ ( k +1) is contained in a cone σ ∈ ∆ ( k ) , let v σ/τ ∈ N/N τ be the primitive generator of the ray σ in ∆ τ . Definition 2.1.
A function c : ∆ ( k ) → Z is said to be a Minkowski weight if itsatisfies the balancing condition , that is, for every τ ∈ ∆ ( k +1) , X σ ⊃ τ c ( σ ) v σ/τ = 0 in N/N τ . The main result of [5] is that A k ( X ) is canonically isomorphic to the space ofcodimension k Minkowski weights. The usual operations on Chow cohomologycan be described combinatorially. Let c ∈ A k ( X ) , c ∈ A k ( X ). The Cartesianproduct c × c ∈ A k + k ( X (∆ × ∆)) is given by ( c × c )( σ × σ ) = c ( σ ) c ( σ ).The cup product is described by the fan displacement rule. Let v be a generic (asdescribed in [5]) element of N .( c ∪ c )( γ ) = X ( σ ,σ ) ∈ ∆ ( k × ∆ ( k m γσ ,σ c ( σ ) c ( σ )where m γσ ,σ are defined by m γσ ,σ = ( [ N : N σ + N σ ] if γ ⊂ σ , γ ⊂ σ , σ ∩ ( σ + v ) = ∅ c ) of c ∈ A n ( X (∆)) is defined to be c (0), the value of c on theunique 0-dimensional cone 0. Example 2.2.
Let c , c be the codimension 1 Minkowski weights below where theweight on each ray is 1 except where indicated: ρ ρ ρ ν ν ν They are weights on some complete fan ∆ in R that refines both of the fansindicated. ν is in the direction h− , i The Minkowski weight c ∪ c is determinedby ( c ∪ c )(0). Let v = h , i . The non-zero constants m ρν correspond to theintersection of the first fan with the h , i -translate of the second:Then m ρ ν = [ Z : h , i + h , − i ] = 1, m ρ ν = [ Z : h− , i + h , i ] = 2.Therefore, ( c ∪ c )(0) = m ρ ν c ( ρ ) c ( ν ) + m ρ ν c ( ρ ) c ( ν ) = 1 + 2 = 3.Payne [17] gave a combinatorial description of A ∗ T ( X ) even in the case that X isnot complete. The ring of integral piecewise polynomial functions PP ∗ (∆) consistsof continuous functions f : | ∆ | → R such that the restriction f σ of f to each cone KATZ σ ∈ ∆ is a polynomial in Sym ∗ ( M/ ( σ ⊥ )). There is a canonical non-equivariantrestriction map η ∗ : A ∗ T ( X ) → A ∗ ( X )that is described combinatorially by Payne and the author [11]. The kernel ofthat map contains ( M ), the ideal generated by linear functions. The case where X is smooth was proven by Brion [2]. Elements of A T ( X ) are given by T -Cartierdivisors which are described by piecewise-linear functions on ∆, that is continuousfunctions f : | ∆ | → R such that the restriction f σ of f to each maximal cone σ ∈ ∆is an integer linear function in M . If X is smooth, the canonical non-equivariantrestriction map η ∗ : A T ( X ) → A ( X ) ∼ = A n − ( X )has a much simpler combinatorial description as η ∗ f = X ρ i − f ( v i )[ V ( ρ i )]where the sum is over rays ρ i in ∆ and v i is the primitive integer vector along ρ i [3].If V ( τ ) is an orbit closure in X (∆) that is pointwise-fixed by the sub-torus T ( τ ),then there is a natural restriction map A T ( X ) → A T ( V ( τ ))A map A ∗ T ( V ( τ )) → A ∗ T/T ( τ ) ( V ( τ )) is not canonically defined but requires a split-ting of T ( τ ) ֒ → T . Once a splitting is chosen, the image of a piecewise polynomialon ∆ is a piecewise polynomial on ∆ τ , well-defined up to an element of M ( τ ) = τ ⊥ .The following diagram commutes A ∗ T ( X ) (cid:15) (cid:15) η ∗ / / A ∗ ( X ) (cid:15) (cid:15) A ∗ T/T ( σ ) ( V ( τ )) η ∗ / / A ∗ ( V ( τ )) . Let ∆ be a complete unimodular fan. For c ∈ A k ( X (∆)), one can give a descrip-tion of η ∗ f ∪ c ∈ A k +1 ( X (∆)). Lemma 2.3. η ∗ f ∪ c ∈ A k +1 ( X (∆)) is the Minkowski weight given as follows, for τ ∈ ∆ ( k +1) , ( η ∗ f ∪ c )( τ ) = − X σ ∈ ∆ ( k ) | σ ⊃ τ f ( v σ/τ ) c ( σ ) Proof.
Let i : V ( τ ) ֒ → X (∆) be the inclusion of the torus orbit. After picking asplitting T ( τ ) ֒ → T , the lemma follows from the naturality of η ∗ with respect to i ∗ . (cid:3) There is a connection between tropical varieties and Minkowski weights. Let V be a l -dimensional subvariety of a complete X (∆) defined over C . V is said to intersect orbits properly if and only if dim( V ∩ V ( τ )) = l +dim( V ( τ )) − n for all orbitclosures V ( τ ). The tropicalization of V , Trop( V ) is an l -dimensional polyhedralfan in N [18]. Moreover, the top dimensional cones σ in Trop( V ) are assigneda multiplicity ω ( σ ) that satisfies the balancing condition. Proposition 2.2 of [20]states that V intersects orbits properly if and only if Trop( V ) is a union of cones ROPICAL INTERSECTION THEORY FROM TORIC VARIETIES 5 of ∆. Therefore, Trop( V ) induces a Minkowski weight c ∈ A n − l ( X ) called the associated cocycle . The class c acts as an operational Poincar´e-dual to V in thefollowing sense: Lemma 2.4. [10, Lem 9.5]
Suppose V intersects the torus orbits of X properly. If c is the associated cocycle of V , then c ∩ [ X ] = [ V ] ∈ A l ( X ) . The following example will be useful in the sequel.
Example 2.5.
Let Γ ⊂ ( C ∗ ) n × ( C ∗ ) n be the diagonal. Then a straightforwardcomputation shows that Trop(Γ) is the diagonal in R n × R n with multiplicity 1.Therefore, if ∆ is a complete rational fan in R n × R n so that the diagonal is a unionof cones, then the associated cocycle of Γ is a class d ∈ A n ( X (∆)) such that d ( σ ) = ( σ is contained in the diagonal0 otherwise . The diagonal is useful because if ∆ ′ is a refinement of ∆ × ∆ for which Trop(Γ)is a union of cones and π : X (∆ ′ ) → X (∆) is projection onto the first factor, then( c ∪ c ) ∩ [ X (∆)] = π ∗ (( c × c ) ∩ [Γ]) = π ∗ (( c × c ∪ d ) ∩ [ X (∆ ′ )]) . A subvariety V of T can have additional properties that guarantee that some ofthe geometry of the closure of V in X (∆) can be seen from Trop( V ). Definition 2.6. [20]
The pair ( V, X (∆)) is said to be tropical if the multiplicationmap m : V × T → X (∆) is faithfully flat and proper. ( V, X (∆)) is said to be sch¨on if, in addition, m is smooth. The fact that (
V, X (∆)) is a tropical pair guarantees that V intersects torus or-bits properly. It is proved in [20] that if V is sch¨on for some tropical pair ( V, X (∆)),it is sch¨on for any tropical pair. If in addition V is sch¨on and X (∆) is smooth,then the stratification of X (∆) as a toric variety pulls back to V as a stratificationwith smooth strata. 3. Mixed Minkowski Weights
Definition 3.1.
Let c ∈ A k ( X (∆)) be a Minkowski weight of codimension k . The support of c is Supp( c ) = [ σ | c ( σ ) =0 cl( σ ) , the union of the closures of the cones on which c is non-zero. Note that Supp( c ) is a union of cones of codimension at least k . The support ofan associated cocycle of a subvariety V of X (∆) is Trop( V ). Definition 3.2.
A mixed Minkowski weight ( c, f ) of degree k is a Minkowski weight, c ∈ A k ( X (∆)) together with a continuous function f : Supp( c ) → R that restrictsto each cone as a linear function in M/ ( σ ⊥ ) . Let MMW k (∆) denote the set of degree k mixed Minkowski weights. Lemma 3.3. If ∆ is a unimodular complete fan, and ( c, f ) is a mixed Minkowskiweight then f extends to a piecewise-linear function on ∆ . KATZ
Proof.
Set f to be zero on all rays not in Supp( c ). Extend f linearly on topdimensional cones. Such an extension agrees on overlaps. (cid:3) Definition 3.4.
For ∆ unimodular, define the map κ : MMW k (∆) → A k +1 ( X (∆)) as follows: for ( c, f ) , a mixed Minkowski weight, extend f to a piecewise-linearfunction on ∆ and let κ ( c, f ) = c ∪ η ∗ f . κ ( c, f ) has a combinatorial description given by Lemma 2.3. We can use thiscombinatorial formula to extend the definition to non-unimodular fans. We willthen show that the definition is independent of certain choices made. Definition 3.5.
For complete ∆ , for ( c, f ) ∈ MMW k (∆) and τ ∈ ∆ ( k +1) , refine ∆ τ to unimodular e ∆ τ , and let κ ( c, f )( τ ) = − X ρ ∈ ∆ ( k ) τ f ( v ρ/ ) c (Span + ( τ, ρ )) Lemma 3.6.
For ∆ unimodular, κ ( c, f ) is independent of the extension.Proof. Let f , f ∈ PP (∆) be extensions of f . Then κ ( c, f )( τ ) − κ ( c, f )( τ ) = − X σ ∈ ∆ ( k ) | σ ⊃ τ ( f ( v σ/τ ) − f ( v σ/τ )) c ( σ ) . For σ with c ( σ ) = 0, we have f | σ = f | σ , so the sum is 0. (cid:3) Now, we extend independence to the non-unimodular case.
Lemma 3.7. κ ( c, f )( τ ) is independent of the choice of e ∆ τ .Proof. Suppose we have two refinements e ∆ , e ∆ of ∆ τ with extensions f , f of f .We may pick a common unimodular refinement e ∆. Then f , f are both piecewise-linear on e ∆ and the above lemma shows that κ ( c, f )( τ ) is independent of the choicesinvolved. (cid:3) Tropical Intersection Theory for Fans
In this section, we prove that the Tropical Intersection Theory of [1] agrees withthe fan displacement rule of [5]. The main notions of the Tropical Intersection The-ory of [1] have been introduced here in intersection theoretic guise. A codimension k tropical cycle supported on a fan ∆ is just a codimension k Minkowski weighton ∆. A rational function on a tropical cycle c is a piecewise-linear function f onSupp( c ) such that ( c, f ) is a mixed Minkowski weight. The associated Weil divisorof f is manifestly the Minkowski weight κ ( c, f ).Given a codimension k tropical cycle c supported on ∆ ⊂ N R and a morphism offans h : ∆ → ∆ ′ induced from a homomorphism h : N → N ′ of lattices of dimension n and n ′ , Gathmann, Kerber, and Markwig [6] define a codimension k + n ′ − n pushforward tropical cycle, h ∗ ( c ). If ∆ and ∆ ′ are unimodular, this pushforwardcycle coincides with the image of c under the pushforward in cohomology, h ∗ : A k ( X (∆)) → A k + n ′ − n ( X (∆ ′ )) as explained combinatorially in [5]. ROPICAL INTERSECTION THEORY FROM TORIC VARIETIES 7
Definition 4.1.
Consider Minkowski weights c ∈ A k ( X (∆)) , c ∈ A k ( X (∆)) .Let x i , y i be coordinates on N × N corresponding to a fixed basis of M applied to eachfactor. Let χ i = min(0 , y i − x i ) . Let ∆ ′ be a unimodular fan refining ∆ × ∆ suchthat each χ i is piecewise-linear on ∆ ′ . The Allermann-Rau intersection product isthe Minkowski weight c • c = π ∗ ( κ ( . . . κ ( κ (( c × c ) , χ ) , χ ) , . . . ) χ n ))) where π : R n → R n is projection onto the first factor. By our description of κ , we can rewrite the above as c • c = π ∗ ( η ∗ ( χ ∪ · · · ∪ χ n ) ∪ ( c × c )) . Example 4.2.
Let us revisit Example 2.2. Let ∆ , ∆ be the complete fanswhose rays are the support of c , c , respectively. c × c is supported on the9 two-dimensional cones of ∆ × ∆ corresponding to the product of rays. c • c = π ∗ ( κ ( κ ( c × c ) , χ ) , χ ). We refine ∆ × ∆ to e ∆ to ensure that χ ispiecewise-linear. This involves subdividing ρ × ν by adding the ray through ω = ( h , i , h , i ) and subdividing ρ × ν by adding the ray through ω =( h− , − i , h− , i ). Let d = κ ( χ , c × c ). d may be supported on 8 rays, the6 rays of ∆ × ∆ and on ω , ω . ( c • c )(0) = κ ( d, χ ) = P ρ χ ( v ρ/ ) d ( ρ ). χ iszero on all of the rays except for χ ( ρ ×
0) = − , χ (0 × ν ) = − d ( ρ × , d (0 × ν ). For d ( ρ × ρ iscontained in ρ × ν , ρ × ν , and ρ × ν . χ is zero on ρ × ν , ρ × ν and d ( ρ ×
0) = − χ ( v ρ × ν /ρ × )(( c × c )( ρ × ν )) = − ( − . Similarly, d (0 × ν ) = − χ ( v ρ × ν / × ν )(( c × c )( ρ × ν )) = − ( − . It follows that κ ( d, χ ) = − ( χ ( ρ × d ( ρ ×
0) + χ (0 × ν ) d (0 × ν )) = ( − − . This agrees with the computation using the fan displacement rule.
Lemma 4.3.
Let ∆ ′ be a refinement of ∆ × ∆ so that χ , . . . , χ n are piecewise-linearon ∆ ′ . Then η ∗ ( χ ∪ · · · ∪ χ n ) = d where d is the associated cocycle of the diagonal from Example 2.5.Proof. We may suppose that ∆ ′ is unimodular. We first show that η ∗ χ i is theassociated cocycle of V ( w i − z i ) where w k , z k are coordinates on ( C ∗ ) n × ( C ∗ ) n . Bythe naturality of η ∗ with respect to the toric projection ( C ∗ ) n × ( C ∗ ) n → C ∗ × C ∗ onto the i th factor of each ( C ∗ ) n , it suffices to consider the case of n = 1. Itis straightforward to verify that the weight of η ∗ χ i on any ray contained in thediagonal is 1. Since the the varieties V ( w i − z i ) intersect transversely, the classof η ∗ ( χ ∪ · · · ∪ χ n ) is exactly the Poincar´e-dual of their intersection which is thediagonal. (cid:3) Theorem 4.4.
The Allermann-Rau intersection product is equal to the cup producton Minkowski weights: c • c = c ∪ c . KATZ
Proof.
Let [Γ] be the cycle-class corresponding to the diagonal. On X (∆ ′ ), we havethe following equation among cycle classes( η ∗ ( χ ∪ · · · ∪ χ n ) ∪ ( c × c )) ∩ [ X (∆ ′ )] = ( c × c ) ∩ [Γ] . This pushes forward by π ∗ to the Poincar´e-dual of c ∪ c . (cid:3) Tropical Intersection Theory
In this section, we consider the more general tropical intersection theory of trop-ical polyhedral complexes in R n . There are two definitions of tropical intersectionproduct: one given by Mikhalkin in [14] and one given by Allermann-Rau in [1]. Inthis section, we prove that they are equal.A polyhedral complex C in N R is a finite collection of polyhedral cells in N R thatcontains the faces of each of its polyhedra and such that the intersection of twopolyhedra is a common face. A polyhedral complex is said to be rational if all ofits cells are rational with respect to the lattice N . An integer affine linear functionon N is a function of the form f ( v ) = h v, m i + a for m ∈ M = N ∨ and a ∈ R .Let C be an rational polyhedral complex in R n , τ , a cell in C , and w , a point inthe relative interior of τ . Define D to be the set of all polyhedra in C that contain τ as a face. For σ ∈ D , let C σ be the cone C σ = { v ∈ R n | w + ǫv ∈ σ for some ǫ > } . The star of C at τ , is the fan Star C ( τ ) made up of the union of the C σ ’s. Theminimal cone in Star C ( τ ) is τ = Span( τ − w ). The star is independent of the choiceof w . Definition 5.1. A k -dimensional tropical cycle is a purely k -dimensional rationalpolyhedral complex together with a weight function on top-dimensional cells ω : C ( k ) → Z such that for any ( k − -dimensional polyhedron τ , the weight induced by ω is atropical cycle on Star C ( τ ) . The degree of a 0-dimensional tropical cycle is the sum of the weights on itspoints.
Definition 5.2.
Let C and C be tropical cycles of dimension k and l respectively.Define the Mikhalkin intersection product C • C [14] as follows: subdivide C and C such that their intersection C ∩ C is a subcomplex of each; for each ( k + l ) − n -dimensional cell τ of C ∩ C ; let ∆ = Star C ( τ ) , ∆ = Star C ( τ ) ; pick a completefan ∆ that contains refinements of ∆ and ∆ as subcomplexes; Star C ( τ ) , Star C ( τ ) induce Minkowski weights c , c on ∆ ; the weight on τ in C · C is ( c ∪ c )( τ ) . The Allermann-Rau intersection product [1] is developed using tropical divisors.
Definition 5.3.
A tropical Cartier divisor φ on C is the data of ( U i , φ i ) where the U i ’s are open subcomplexes of C such that ∪ U i = C and φ i is an integral piecewise-linear function on U i such that on each component of U i ∩ U j , φ i − φ j is an integralaffine linear function. Note that if τ is any cone in C , a tropical Carter divisor φ induces a tropicalCartier divisor φ Star( τ ) on Star C ( τ ). One takes some U i that contains τ . Therestriction of φ i to the cells containing τ induces a function φ Star C ( τ ) on Star C ( τ ). ROPICAL INTERSECTION THEORY FROM TORIC VARIETIES 9
One subtracts off a constant to ensure that the function is 0 at the origin. Thisfunction is well defined up to an global integer affine-linear function.
Definition 5.4.
The associated Weil divisor φ on C , denoted by φ · C is a tropicalcycle supported on the cones of positive codimension in C . For τ ∈ C ( k − , let ω φ ·C ( τ ) = κ (Star C ( τ ) , φ Star C ( τ ) )( τ ) , the weight of τ in the cycle given by taking theassociated tropical Weil divisor of φ Star C ( τ ) on Star C ( τ ) . The associated Weil divisor is easily seen to be a tropical cycle.If h : N → N ′ is a map of lattices, there is a induced map of tropical cycles.Let C be a tropical cycle in N R and C ′ is an integral polyhedral complex in N ′ such that h : C → C ′ is a surjective map of polyhedral complexes of the samedimension. Subdivide C so that the image of each polyhedron in C ′ is containedin a polyhedron. If τ ′ is a top-dimensional polyhedron in C ′ , let τ , . . . , τ k be itstop-dimensional pre-images in C . The weight on τ ′ is ω h ∗ ( C ) ( τ ′ ) = X ω ( h i ∗ τ j )where ω ( h i ∗ τ j ) is the weight on τ j as the image of the following map of tropicalcycles on fans, h i : Star C ( τ j ) → Star C ′ ( τ ′ ) . Definition 5.5.
Let C , C be tropical cycles in R n , the Allermann-Rau tropicalintersection product [1] is defined to be C • C = π ∗ ( χ · . . . · χ n · ( C × C )) where π : R n × R n → R n is projection onto the first factor. Proposition 5.6.
The Mikhalkin intersection product is equivalent to the Allermann-Rau intersection product.Proof.
Each intersection product is supported on cones in C ∩C . For the Mikhalkinintersection product, this follows by definition. For the Allermann-Rau intersectionproduct, one notes that χ · . . . · χ n · ( C × C ) is supported on Γ ∩ ( C × C ) whereΓ is the diagonal in R n × R n . Such cells push forward to cells in C ∩ C .These definitions both reduce to computing intersection products on the stars ofpolyhedra in the intersection of two tropical cycles. Therefore, the two definitionsare equivalent. (cid:3) Induced Intersection Theory
Let i : V → X (∆) be a smooth k -dimensional subvariety of a complete smoothtoric variety over C . Under certain conditions, intersection computations on V canbe performed using only the combinatorics of Trop( V ). We will apply this in thenext section to the moduli space of rational curves.Suppose V intersects torus orbits properly. Let c ∈ A k ( X (∆)) be the asso-ciated cocycle of V . Then Supp( c ) = Trop( V ). Since c ∩ [ X (∆)] = [ V ], for d ∈ A n − k ( X (∆)), i ∗ d ∩ [ V ] = i ∗ d ∩ ( c ∩ [ X (∆)]) = ( i ∗ d ∪ c ) ∩ [ X (∆)]. There-fore, deg( i ∗ d ∩ [ V ]) depends only on the image of d in A ∗ ( X (∆)) / (ker( ∪ c )). Lemma 6.1.
Let d , d ∈ A ( X (∆) . Suppose that for all τ ∈ ∆ ( k +1) with τ ⊂ Supp( c ) that deg(( d ∪ c ) ∩ [ V ( τ )]) = deg(( d ∪ c ) ∩ [ V ( τ )]) . Then d = d in A ∗ ( X (∆)) / (ker( ∪ c )) . Proof.
It suffices to show d ∪ c = d ∪ c as Minkowski weights. Let τ ∈ ∆ ( k +1) .If τ ⊂ Supp( c ), then ( d ∪ c )( τ ) = ( d ∪ c )( τ ). Now suppose τ Supp( c ). Write d i ∩ [ V ( τ )] = P a i,σ [ V ( σ )] where the sum is over σ ∈ ∆ ( k ) with σ ⊂ τ . Note that c ( σ ) = 0 for all σ ⊂ τ . Then( d i ∪ c ) ∩ [ V ( τ )] = c ∩ ( d i ∩ [ V ( τ )]) = X a i,σ c ([ V ( σ )]) = 0 . (cid:3) Definition 6.2.
Let d ∈ A ( X (∆)) . A piecewise-linear function f on Trop( V ) issaid to lift d on V if for all τ ∈ ∆ ( k +1) , τ ⊂ Trop( V ) , deg( i ∗ d ∩ [ i − V ( τ )]) = κ ( c, f )( τ ) . In other words, if we extend f to a piecewise-linear function f on ∆ then deg( i ∗ d ∩ [ i − V ( τ )]) = deg(( η ∗ f ∪ c ) ∩ [ V ( τ )]) . Theorem 6.3.
Let V be a k -dimensional sch¨on subvariety of smooth X (∆) andsuppose ( V, X (∆)) is a tropical pair. Let d , . . . , d k ∈ A ( X (∆)) . Let f , . . . , f k bepiecewise-linear functions on Trop( V ) lifting d , . . . , d k . Then deg( f · . . . · f k · Trop( V )) = deg(( i ∗ d ∪ . . . i ∗ d k ) ∩ [ V ]) . Proof.
From V sch¨on, we know V intersects the torus orbits of X (∆) transverselyand that one has the identity of cycle classes on X (∆), i ∗ [ i − V ( τ )] = [ V ∩ V ( τ )].Consequently if f i is extended as a piecewise-linear function on ∆, deg(( η ∗ f i ∪ c ) ∩ [ V ( τ )]) = deg(( d i ∪ c ) ∩ [ V ( τ )]) for all τ ∈ ∆ ( k +1) , τ ⊂ Supp( c ). Therefore, η ∗ f i = d i in A ∗ ( X (∆)) / (ker( ∪ c )). Therefore,deg( f · . . . · f k · Trop( V )) = deg(( η ∗ f ∪ . . . ∪ η ∗ f k ∪ c ) ∩ [ X (∆)]) = deg(( d ∪ . . . d k ) ∩ [ V ]) . (cid:3) Intersection Theory on Moduli of Curves
In [12], Kerber and Markwig define tropical psi-classes as piecewise-linear func-tions on a model of tropical M ,n . They use the Allermann-Rau tropical inter-section theory to compute the top-dimensional intersections of the ψ -classes. Theanswer they obtain is equal to the classical answer. We give a non-computationalproof that these numbers are equal. M ,n is the moduli space of stable genus 0 curves with n marked points. See[13] for an elaboration of the results cited below. M ,n has a stratification bycombinatorial types of dual graphs. The big open set consists of smooth rationalcurves while boundary divisors consist of curves of arithmetic genus 0 with twoirreducible components. The zero-dimensional strata correspond to curves withtrivalent dual graphs while the one-dimensional strata correspond to dual graphswith one quadrivalent vertex and the remaining vertices trivalent. For k ∈ [ n ], let E k be the set of one-dimensional strata for which the leaf-label k is incident to thequadrivalent vertex. M ,n carries natural cohomology classes ψ , . . . , ψ n ∈ A ( M ,n ). Let C ,n bethe universal curve over M ,n . Let ω be the relative dualizing sheaf of C ,n → M ,n which should be thought of as a relative cotangent bundle. Let σ , . . . , σ n : M ,n →C ,n be the sections of the universal curve given by the marked points. Then ψ i = c ( σ ∗ i ω ). It is straightforward to show that deg( ψ k · [ M , ]) = 1. By the ROPICAL INTERSECTION THEORY FROM TORIC VARIETIES 11 naturality of ψ -classes, one can show that if S is the closure of a 1-stratum of M ,n then ψ k ∩ [ S ] = (cid:26) S ∈ E k S / ∈ E k . Kapranov proved that M .n is the Chow quotient of the Grassmannian, G (2 , n ) //T where G (2 , n ) is a particular dense open subset of the Grassmannian of 2-planesin C n and T is the n -torus dilating the coordinates in C n [9]. The Grassmannian, G (2 , n ) is embedded by Pl¨ucker coordinates into P ( n ) − . This embedding is equi-variant with respect to the torus action. G (2 , n ) is the inverse image of the torus( C ∗ )( n ) − . Therefore, the Pl¨ucker embedding induces an embedding into a Chowquotient, i : M ,n ֒ → P ( n ) − //T . Now, P ( n ) − //T is a N = ( (cid:0) n (cid:1) − n )-dimensionaltoric variety which we will denote by X (∆ ′ ).The tropicalization of M ,n in ( C ∗ ) N ⊂ X (∆ ′ ) is well understood. We summa-rize the description given by Kerber and Markwig in [12] which relies on the workof Speyer and Sturmfels [19]. It it given in terms of combinatorial types of tropicalrational curves. An n -marked rational tropical curve is a metric tree with n leaveslabelled by integers in the set [ n ] = { , . . . , n } . All edges of the tree except theleaves are given a length. Let the coordinates of R ( n ) be indexed by two elementsubsets of [ n ]. Denote the set of all such two element subsets by I . Define thefollowing map Θ : R n → R ( n ) a ( a i + a j ) { i,j }∈ I The one-parameter subgroup lattice of X (∆ ′ ) canonically lies in the vector space R ( n ) / Θ( R n ). The image of a tropical curve C in Trop( M ,n ) ⊂ R ( n ) / Θ( R n ) is thepoint whose coordinates are dist( { i, j } ) { i,j }∈ I where dist is the distance between theleaves in the metric graph C . The image of all tropical curves is all of Trop( M ,n ).The image of tropical curves of a fixed combinatorial type gives a cone in the fanTrop( M ,n ).We state some facts about the embedding that are proved using combinatorialtechniques in a recent paper by Gibney and Maclagan [7]. Their work draws on thestudy of the tropical Grassmannian by Speyer and Sturmfels [19] and the theory oftropical compactifications developed by Tevelev [20]. Proposition 7.1. (1) [7, Prop 5.8]
The tropicalization of the image of the moduli of curves,
Trop( M ,n ) is a union of cones in ∆ ′ . (2) [7, Prop 5.4] If ∆ is the union of cones in ∆ ′ that support Trop( M ,n ) then X (∆) is a smooth toric variety. (3) [7, Lem 5.10] The embedding i : M ,n → X (∆) induces an isomorphism i ∗ : Pic( X (∆)) → Pic( M ,n ) . By the resolution of singularities algorithm for toric varieties as presented in[3], we may find a unimodular fan e ∆ refining ∆ ′ without subdividing any conesin ∆. Therefore, there is an open embedding j : X (∆) → X ( e ∆). Moreover, thecomposition M ,n → X (∆) → X ( e ∆) is a regular embedding. The embedding M ,n → X (∆) has a number of nice properties from the pointof view of tropical compactifications. Proposition 7.2. (1) [9] (see [20, Thm 5.5] ) ( M ,n , X (∆)) is a tropical pair. (2) [8, Thm 1.11] M ,n is sch¨on. This implies by Proposition 2.3 of [20] that M ,n intersects torus orbits in X (∆ ′ )properly. Let c be the associated cocycle of M ,n ⊂ X ( e ∆) so that c ∩ [ X ( e ∆)] =[ M ,n ]. The stratification of X (∆) as a toric variety pulls back to the stratificationof M ,n .We treat ψ i as elements of A ( X (∆)). By [2, Section 2.3] as explained in [11], ψ i can be lifted to an element of A T ( X (∆)) which can be extended as a piecewise-linearfunction to an element of A T ( X ( e ∆)). This gives a lift of ψ i to A ( X ( e ∆)).Mikhalkin introduced a tropical ψ -class in [15]. His definition corresponds toa Minkowski weight on the codimension 1 cones in an embedding of Trop( M ,n )in a fan. Kerber and Markwig define a piecewise-linear function on Trop( M ,n )lifting ψ k . For a subset J ⊂ [ n ] of cardinality 1 < | J | < n −
1, they define a vector ν J ∈ R ( n ) by ( ν J ) { i,j } = ( , if |{ i, j } ∩ J | = 10 , else .ν J is the ray in Trop( M ,n ) corresponding to tropical curves with one finite internaledge with the marked points in J attached one end-point and the marked points in[ n ] \ J attached to the other. For k ∈ [ n ], the set V k is defined to be V k = { ν J | k / ∈ J and | J | = 2 } . The piecewise-linear function f k is defined to be the linear extension on cones ofTrop( M ,n ) of ν J ( ν J ∈ V k . Since f k is a piecewise-linear function on Trop( M ,n ), ( c, f i ) is a mixed Minkowskiweight on X (∆). The associated Weil divisor of f k on Trop( M ,n ) is determinedby the following proposition. Proposition 7.3. [12, Prop 3.5] If σ is a codimension cone of M ,n , then theweight ω σ on σ in the tropical Weil divisor of f k is given by ω ( σ ) = (cid:26) (cid:0) n − (cid:1) σ corresponds to a stratum in E k otherwise . This implies that f k lifts (cid:0) n − (cid:1) ψ k on M ,n . Let Φ k = ( n ) f k . Theorem 6.3implies the following: Proposition 7.4. deg(Φ k · . . . · Φ k l · Trop( M ,n )) = deg( Y l ψ k l ∩ [ M ,n ]) . This is the intersection of tropical ψ -classes computed in [12]. ROPICAL INTERSECTION THEORY FROM TORIC VARIETIES 13
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