aa r X i v : . [ m a t h . AG ] A ug The homology of tropical varieties
Paul HackingOctober 31, 2018
Given a closed subvariety X of an algebraic torus T , the associated tropicalvariety is a polyhedral fan in the space of 1-parameter subgroups of the toruswhich describes the behaviour of the subvariety at infinity. We show thatthe link of the origin has only top rational homology if a genericity conditionis satisfied. Our result is obtained using work of Tevelev [T] and Deligne’stheory of mixed Hodge structures [D].Here is a sketch of the proof. We use the tropical variety of X to constructa smooth compactification X ⊂ X with simple normal crossing boundary B . We relate the link L of the tropical variety to the dual complex K of B ,that is, the simplicial complex with vertices corresponding to the irreduciblecomponents B i of B and simplices of dimension j corresponding to ( j + 1)-fold intersections of the B i . Following [D] we identify the homology groupsof K with graded pieces of the weight filtration of the cohomology of X .Since X is an affine variety, it has the homotopy type of a CW complex ofreal dimension equal to the complex dimension of X . From this we deducethat K and L have only top homology.The link of the tropical variety of X ⊂ T was previously shown to haveonly top homology in the following cases: the intersection of the Grass-mannian G (3 ,
6) with the big torus T in its Pl¨ucker embedding [SS], thecomplement of an arrangement of hyperplanes [AK], and the space of ma-trices of rank ≤ T = ( C × ) m × n [MY]. We discuss these and otherexamples from our viewpoint in Sec. 4.It has been conjectured that the link of the tropical variety of an arbitrary subvariety of a torus is homotopy equivalent to a bouquet of spheres (so, inparticular, has only top homology). I expect that this is false in general,but I do not know a counterexample. See also Rem. 2.11.We note that D. Speyer has used similar techniques to study the topology1f the tropicalisation of a curve defined over the field C (( t )) of formal powerseries, see [S, Sec. 10]. We work throughout over k = C . Let X ⊂ T be a closed subvariety ofan algebraic torus T ≃ ( C × ) r . Let K = S n ≥ C (( t /n )) be the field ofPuiseux series (the algebraic closure of the field C (( t )) of Laurent series)and ord : K × → Q the valuation of K/ C such that ord( t ) = 1.Let M = Hom( T, C × ) ≃ Z r be the group of characters of T and N = M ∗ .We have a natural map val : T ( K ) → N Q given by T ( K ) ∋ P ( χ m ord( χ m ( P )) ) . In coordinates( K × ) r ∋ ( a , . . . , a r ) (ord( a ) , . . . , ord( a r )) ∈ Q r Definition 2.1. [EKL, 1.2.1] The tropical variety A of X is the closure ofval( X ( K )) in N R ≃ R r . Theorem 2.2. [EKL, 2.2.5] A is the support of a rational polyhedral fanin N R of pure dimension dim X . Let Σ be a rational polyhedral fan in N R . Let T ⊂ Y be the associatedtorus embedding. Let X = X (Σ) be the closure of X in Y . Theorem 2.3. [T, 2.3] X is compact iff the support | Σ | of Σ contains A . From now on we always assume that X is compact. Theorem 2.4. [ST, 3.9][T2] The intersection X ∩ O is non-empty and haspure dimension equal to the expected dimension for every torus orbit O ⊂ Y iff | Σ | = A .Proof. Suppose | Σ | = A . We first show that X ∩ O is nonempty for everyorbit O ⊂ Y . Let Σ ′ → Σ be a strictly simplical refinement of Σ and f : Y ′ → Y the corresponding toric resolution of Y . Let X ′ be the closure of X in Y ′ . Let O ⊂ Y be an orbit, and O ′ ⊂ Y ′ an orbit such that f ( O ′ ) ⊆ O .Then X ′ ∩ O ′ = ∅ by [T, 2.2], and f ( X ′ ∩ O ′ ) ⊆ X ∩ O , so X ∩ O = ∅ asrequired. 2e next show that X ∩ O has pure dimension equal to the expecteddimension for every orbit O ⊂ Y . Let O ⊂ Y be an orbit of codimension c .Let Z be an irreducible component of the intersection X ∩ O with its reducedinduced structure. Let W be the closure of O in Y and Z the closure of Z in W . Then, since Z is compact, the fan of the toric variety W contains thetropical variety of Z ⊂ O by Thm. 2.3. We deduce that dim Z ≤ dim X − c by Thm. 2.2. On the other hand, since toric varieties are Cohen-Macaulay,the orbit O ⊂ Y is cut out set-theoretically by a regular sequence of length c at each point of O . It follows that dim Z ≥ dim X − c , so dim Z = dim X − c as required.The converse follows from [ST, 3.9].Here is the main result of this paper. Theorem 2.5.
Suppose that | Σ | = A and the following condition is satisfied: ( ∗ ) For each torus orbit O ⊂ Y , X ∩ O is smooth and is connected if ithas positive dimension.Then the link L of ∈ A has only top reduced rational homology, i.e., ˜ H i ( L, Q ) = 0 for i < dim L = dim X − .Example . Let Y be a projective toric variety. Let X ⊂ Y be a completeintersection. That is, X = H ∩ · · · ∩ H c where H i is an ample divisor on Y .Assume that H i is a general element of a basepoint free linear system foreach i . Let Y ⊂ Y be the open toric subvariety consisting of orbits meeting X and Σ the fan of Y . Then | Σ | = A by Thm. 2.4 and X ⊂ Y satisifes thecondition ( ∗ ) by Bertini’s theorem [H, III.7.9, III.10.9].If Σ is the (complete) fan of Y , the fan Σ is the union of the cones of Σof codimension ≥ c . So it is clear in this example that the link L of 0 ∈ A has only top reduced homology. Indeed, let r = dim Y . Then the link K of 0 ∈ Σ is a polyhedral subdivision of the ( r − L is the( r − c − K , hence ˜ H i ( L, Z ) = ˜ H i ( S r − , Z ) = 0 for i < r − c − ∗ ) is given by the following lemma. Lemma 2.7.
Assume that | Σ | = A . Then the following conditions areequivalent.(1) X ∩ O is smooth for each orbit O ⊂ Y .(2) The multiplication map m : T × X → Y is smooth. roof. The fibre of the multiplication map over a point y ∈ O ⊂ Y isisomorphic to ( X ∩ O ) × S , where S ⊂ T is the stabiliser of y . Now m is smooth iff it is flat and each fibre is smooth. The map m is surjectiveand has equidimensional fibres by Thm. 2.4. Finally, if W is integral, Z isnormal, and f : W → Z is dominant and has reduced fibres, then f is flatiff it has equidimensional fibres by [EGA4, 14.4.4, 15.2.3]. This gives theequivalence. Definition 2.8. [T, 1.1,1.3] We say X ⊂ Y is tropical if m : T × X → Y is flat and surjective. (Then in particular X ∩ O is non-empty and has theexpected dimension for each orbit O ⊂ Y , so | Σ | = A by Thm. 2.4.) We say X ⊂ T is sch¨on if m is smooth for some (equivalently, any [T, 1.4]) tropicalcompactification X ⊂ Y . Example . Here we give some examples of sch¨on subvarieties of tori. (Formore examples see Sec. 4.)(1) Let Y be a projective toric variety and X ⊂ Y a general completeintersection as in Ex. 2.6. Let T ⊂ Y be the big torus and X = X ∩ T .Then X ∩ O is either empty or smooth of the expected dimension forevery orbit O ⊂ Y by Bertini’s theorem. Hence X ⊂ T is sch¨on.(2) Let Y be a projective toric variety and G a group acting transitively on Y . Let X ⊂ Y be a smooth subvariety. Then, for g ∈ G general, gX ∩ O is either empty or smooth of the expected dimension for every orbit O ⊂ Y by [H, III.10.8]. Let T ⊂ Y be the big torus and X ′ = gX ∩ T .Then X ′ ⊂ T is sch¨on for g ∈ G general. Example . Here is a simple example X ⊂ T which is not sch¨on. Let Y be a projective toric variety and X ⊂ Y a closed subvariety such that X meets the big torus T ⊂ Y and X is singular at a point which is containedin an orbit O ⊂ Y of codimension 1. Let X = X ∩ T . Then X ⊂ T is not sch¨on. Indeed, suppose that m : T × X ′ → Y ′ is smooth for sometropical compactification X ′ ⊂ Y ′ . We may assume that the toric birationalmap f : Y ′ Y is a morphism by [T, 2.5]. Now X ∩ O is singular byconstruction, and f : Y ′ → Y is an isomorphism over O because O ⊂ Y hascodimension 1, hence X ′ ∩ f − O is also singular, a contradiction. Remark . It has been suggested that the link L of the tropical varietyof an arbitrary subvariety of a torus is homotopy equivalent to a bouquetof top dimensional spheres (so, in particular, has only top homology). Iexpect that this is false in general, but I do not know a counterexample.4owever, there are many examples where the hypothesis ( ∗ ) of Thm. 2.5 isnot satisfied but the conclusion is still valid. For example, let X ⊂ Y be acomplete intersection in a projective toric variety such that X ∩ O has theexpected dimension for each orbit O ⊂ Y and let X = X ∩ T ⊂ T where T ⊂ Y is the big torus. Then X ⊂ T is not sch¨on in general but L is abouquet of top-dimensional spheres, cf. Ex. 2.10, 2.6. See also Ex. 4.4 foranother example. Construction . [T, 1.7] We can always construct a tropical compactifi-cation X ⊂ Y as follows. Choose a projective toric compactification Y of T . Let X denote the closure of X in Y . Assume for simplicity that S = { t ∈ T | t · X = X } ⊂ T is trivial (otherwise, we can pass to the quotient X/S ⊂ T /S ). Consider theembedding
T ֒ → Hilb( Y ) given by t t − [ X ]. Let Y be the normalisationof the closure of T in Hilb( Y ). (So Y is a projective toric compactificationof T .) Let X be the closure of X in Y , and Y ⊂ Y the open toric subvarietyconsisting of orbits meeting X . Let U ⊂
Hilb( Y ) × Y denote the universalfamily over Hilb( Y ) and U = U ∩ (Hilb( Y ) × T ). One shows that thereis an identification T × X ∼ / / m " " FFFFFFFFF U | Y } } {{{{{{{{ Y (1)given by ( t, x ) ( tx, t ) [T, p. 1093, Pf. of 1.7]. In particular, m is flat. Remark . We note that, in the situation of 2.12, we can verify thecondition ( ∗ ) using Gr¨obner basis techniques. Let O ⊂ Y be an orbit. Let σ be the cone in the fan of Y corresponding to O , and w ∈ N an integralpoint in the relative interior of σ . We regard w as a 1-parameter subgroup C × → T of T . Then, by construction, the limit lim t → w ( t ) lies in the orbit O . Let X w be the flat limit of the 1-parameter family w ( t ) − X as t → U →
Hilb( Y ) over O are the translates of X w . Let y ∈ O be a point and S ⊂ T the stabiliser of y . The fibre of m over y is isomorphicto both ( X ∩ O ) × S and X w ∩ T (by the identification (1)). Hence X ∩ O issmooth (resp. connected) iff X w ∩ T is so. Suppose now that Y ≃ P N , andlet I ⊂ k [ X , . . . , X N ] be the homogeneous ideal of X ⊂ P N . Then X w isthe zero locus of the initial ideal of I with respect to w .5 The stratification of the boundary and the weightfiltration
Let X be a smooth projective variety of dimension n , and B ⊂ X a simplenormal crossing divisor. We define the dual complex of B to be the CWcomplex K defined as follows. Let B , . . . , B m be the irreducible componentsof B and write B I = T i ∈ I B i for I ⊂ [ m ]. To each connected component Z of B I we associate a simplex σ with vertices labelled by I . The facetof σ labelled by I \ { i } is identified with the simplex corresponding to theconnected component of B I \{ i } containing Z . Theorem 3.1.
The reduced homology of K is identified with the top gradedpieces of the weight filtration on the cohomology of the complement X = X \ B . Precisely, ˜ H i ( K, C ) = Gr W n H n − ( i +1) ( X, C ) . Corollary 3.2. If X is affine, then ˜ H i ( K, C ) = (cid:26) Gr W n H n ( X, C ) if i = n − otherwiseProof of Thm. 3.1. This is essentially contained in [D], see also [V, Sec. 8.4].Define a filtration ˜ W of the complex Ω · X (log B ) of differential forms on X with logarithmic poles along B by˜ W l Ω kX (log B ) = Ω lX (log B ) ∧ Ω k − lX . The filtration of Ω · X (log B ) yields a spectral sequence E p,q = H p + q ( X, Gr ˜ W − p Ω · X (log B )) = ⇒ H p + q (Ω · X (log B )) = H p + q ( X, C ) . which defines a filtration ˜ W on H · ( X, C ). The weight filtration W on H i ( X, C ) is by definition the shift W = ˜ W [ i ], i.e., W j ( H i ) = ˜ W j − i ( H i ).The spectral sequence degenerates at E [D, 3.2.10], so E p,q = Gr ˜ W − p H p + q ( X, C ) . The E term may be computed as follows. Let ˜ B l denote the disjoint unionof the l -fold intersections of the components of B , and j l the map ˜ B l → X .(By convention ˜ B = X .) The Poincar´e residue map defines an isomorphismGr ˜ Wl Ω kX (log B ) ∼ −→ j l ∗ Ω k − l ˜ B l , (2)6ee [V, Prop. 8.32]. This gives an identification E p,q = H p + q ( X, Gr ˜ W − p Ω · X (log B )) = H p + q ( ˜ B ( − p ) , Ω · ˜ B ( − p ) ) = H p + q ( ˜ B ( − p ) , C ) . The differential d : H p + q ( ˜ B ( − p ) ) → H p +1)+ q ( ˜ B ( − p − )is identified (up to sign) with the Gysin map on components [V, Prop. 8.34].Precisely, write s = − p . Then d : H q − s ( ˜ B ( s ) ) → H q − s − ( ˜ B ( s − ) is givenby the maps ( − s + t j ∗ : H q − s ( B I ) → H q − s − ( B J ) , where I = { i < · · · < i s } , J = I \{ i t } , j denotes the inclusion B I ⊂ B J ,and j ∗ is the Gysin map. Equivalently, identify H q − s ( ˜ B ( s ) ) = H n − q ( ˜ B ( s ) )by Poincar´e duality. Then d : H n − q ( ˜ B ( s ) ) → H n − q ( ˜ B ( s − ) is given bythe maps ( − s + t j ∗ : H n − q ( ˜ B ( s ) ) → H n − q ( ˜ B ( s − ) . So, the E term of the spectral sequence is as follows. H ( ˜ B ( n ) ) → H ( ˜ B ( n − ) → · · · → H ( ˜ B (1) ) → H ( ˜ B (0) ) H ( ˜ B ( n − ) → · · · → H ( ˜ B (1) ) → H ( ˜ B (0) )... ... H n − ( ˜ B (1) ) → H n − ( ˜ B (0) ) H n − ( ˜ B (0) ) H n ( ˜ B (0) )The top row ( q = 2 n ) is the complex · · · → H ( ˜ B ( s +1) ) → H ( ˜ B ( s ) ) → H ( ˜ B ( s − ) → · · · , which computes the reduced homology of the dual complex K of B . Wededuce Gr ˜ Ws H n − s ( X, C ) = ˜ H s − ( K, C ) . Proof of Corollary 3.2. If X is affine then X has the homotopy type of aCW complex of dimension n , so H k ( X, C ) = 0 for k > n .7 roof of Thm. 2.5. By our assumption and Lem. 2.7 the multiplication map m : T × X → Y is smooth. Let Y ′ → Y be a toric resolution of Y givenby a refinement Σ ′ of Σ. Then m ′ : T × X ′ → Y ′ is also smooth — it isthe pullback of m [T, 2.5]. So X ′ is smooth with simple normal crossingboundary B ′ = X ′ \ X (because this is true for Y ′ ). Hence the dual complex K of B ′ has only top reduced rational homology by Cor. 3.2.It remains to relate K and the link L of 0 ∈ A . Recall that the fanΣ of Y has support A . The cones of Σ of dimension p correspond to toricstrata Z ⊂ Y of codimension p . These correspond to strata Z ∩ X ⊂ X ofcodimension p , which are connected (by our assumption) unless p = dim X .We can now construct K from L as follows. Give L the structure of apolyhedral complex induced by the fan Σ. For each top dimensional cell, let Z ⊂ Y be the corresponding toric stratum, and k = | Z ∩ X | . We replacethe cell by k copies, identified along their boundaries. Let ˆ L denote theresulting CW complex. Note immediately that ˆ L is homotopy equivalent tothe one point union of L and a collection of top dimensional spheres. Soˆ L has only top reduced rational homology iff L does. Finally let ˆ L ′ denotethe subdivision of ˆ L induced by the refinement Σ ′ of Σ. Then ˆ L ′ is the dualcomplex K of B ′ . This completes the proof.We note the following corollary of the proof. Corollary 3.3.
In the situation of Thm. 2.5, if in addition X ∩ O is con-nected for every orbit O ⊂ Y , then we have an identification ˜ H n − ( L, C ) = Gr W n H n ( X, C ) . We say a variety X is very affine if it admits a closed embedding in analgebraic torus. If X is very affine, the intrinsic torus of X is the torus T with character lattice M = H ( O × X ) /k × . Choosing a splitting of the exactsequence 0 → k × → H ( O × X ) → M → X ⊂ T , and any two such are related by a translation. Example . Let X be the complement of an arrangement of m hyperplanesin P n whose stabiliser in PGL( n ) is finite. Then X is very affine with intrinsictorus T = ( C × ) m / C × , and the embedding X ⊂ T is the restriction of thelinear embedding P n ⊂ P m − given by the equations of the hyperplanes.The embedding X ⊂ T is sch¨on, and a tropical compactification X ⊂ Y
8s given by Kapranov’s visible contour construction, see [HKT1, Sec. 2]. In[AK] it was shown that the link L of 0 ∈ A has only top reduced homology,and the rank of H n − ( L, Z ) was computed using the M¨obius function ofthe lattice of flats of the matroid associated to the arrangement. Thm. 2.5gives a different proof that the link has only top reduced rational homology.Moreover, in this case X ∩ O is connected for every orbit O ⊂ Y , and themixed Hodge structure on H i ( X, C ) is pure of weight 2 i for each i . So wehave an identification˜ H n − ( L, C ) = Gr W n H n ( X, C ) = H n ( X, C )by Cor. 3.3. Example . Let X = M ,n , the moduli space of n distinct points on P .The variety X can be realised as the complement of a hyperplane arrange-ment in P n − , in particular it is very affine and the embedding X ⊂ T in itsintrinsic torus is sch¨on by Ex. 4.1.More generally, consider the moduli space X = X ( r, n ) of n hyperplanesin linear general position in P r − . The Gel’fand–MacPherson correspon-dence identifies X ( r, n ) with the quotient G ( r, n ) /H , where G ( r, n ) ⊂ G ( r, n ) is the open subset of the Grassmannian where all Pl¨ucker coordi-nates are nonzero and H = ( C × ) n / C × is the maximal torus which actsfreely on G ( r, n ). See [GeM, 2.2.2]. Thus the tropical variety A of X ( r, n )is identified (up to a linear space factor) with the tropical Grassmannian G ( r, n ) studied in [SS]. In particular, for r = 2, the tropical variety of M ,n corresponds to G (2 , n ), the so called space of phylogenetic trees. For( r, n ) = (3 , L of 0 ∈ A has only top reduced homology, and thetop homology is free of rank 126 [SS, 5.4]. Jointly with Keel and Tevelev,we showed that the embedding X ⊂ T of X (3 ,
6) in its intrinsic torus issch¨on (using work of Lafforgue [L]) and described a tropical compactifica-tion X ⊂ Y explicitly. So Thm. 2.5 gives an alternative proof that L hasonly top reduced rational homology. Moreover, X ∩ O is connected for eachorbit O ⊂ Y , and the mixed Hodge structure on H i ( X (3 , , C ) is pure ofweight 2 i for each i by [HM, 10.22]. So by Cor. 3.3 we have an identification H d − ( L, C ) = Gr W d H d ( X (3 , , C ) = H d ( X (3 , , C )where d = dim X (3 ,
6) = 4. This agrees with the computation of H · ( X, C )in [HM].We note that it is conjectured [KT, 1.14] that X (3 ,
7) and X (3 ,
8) aresch¨on, but in general the compactifications of X ( r, n ) we obtain by toricmethods will be highly singular by [L, 1.8]. The cases X (3 , n ) for n ≤ xample . [HKT2] Let X = X ( n ) denote the moduli space of smoothmarked del Pezzo surfaces of degree 9 − n for 4 ≤ n ≤
8. Recall that adel Pezzo surface S of degree 9 − n is isomorphic to the blowup of n pointsin P which are in general position (i.e. no 2 points coincide, no 3 arecollinear, no 6 lie on a conic, etc). A marking of S is an identification of thelattice H ( S, Z ) with the standard lattice Z ,n of signature (1 , n ) such that K S
7→ − e + e + · · · + e n . It corresponds to a realisation of S as a blowupof n ordered points in P . Hence X ( n ) is an open subvariety of X (3 , n )(because X (3 , n ) is the moduli space of n points in P in linear generalposition). The lattice K ⊥ S ⊂ H ( X, Z ) is isomorphic to the lattice E n (withnegative definite intersection product). So the Weyl group W = W ( E n )acts on X ( n ) by changing the marking. The action of the Weyl group W on X induces an action on the lattice N of 1-parameter subgroups of T whichpreserves the tropical variety A of X in N R . The link L of 0 ∈ A is describedin [HKT2, §
7] in terms of sub root systems of E n for n ≤ n ≤ X ⊂ T of X inits intrinsic torus is sch¨on and described a tropical compactification X ⊂ Y explicitly. The intersection X ∩ O is connected for each orbit O ⊂ Y . So L has only top reduced rational homology by Thm. 2.5, and H d − ( L, C ) =Gr W d H d ( X ( n ) , C ) where d = dim X ( n ) = 2 n − Example . [MY] Let ˜ X ⊂ ( C × ) mn be the space of matrices of size m × n and rank ≤ X is the zero locus of the 3 × X ⊂ T be the quotient of ˜ X ⊂ ( C × ) mn by thetorus ( C × ) m × ( C × ) n acting by scaling rows and columns. In [MY] it wasshown that the link L of the origin in the tropical variety A of X ⊂ T ishomotopy equivalent to a bouquet of top dimensional spheres. Here we givean algebro-geometric interpretation of this result.A point of X corresponds to n collinear points { p i } in the big torusin P m − , modulo simultaneous translation by the torus. Let f : X ′ → X denote the space of lines through the points { p i } . The morphism f is aresolution of X with exceptional locus Γ ≃ P m − over the singular point P ∈ X where the p i all coincide. Given a point ( C ⊂ P m − , { p i } ) of X ′ , let q j be the intersection of C with the j th coordinate hyperplane. We obtain apointed smooth rational curve ( C, { p i } , { q j } ) such that p i = q j for all i and j , and the q j do not all coincide. Conversely, given such a pointed curve( C, { p i } , { q j } ), let F j be a linear form on C ≃ P defining q j . Then weobtain a linear embedding F = ( F : · · · : F m ) : C ⊂ P m − which is uniquely determined up to translation by the torus.10e construct a compactification X ⊂ X using a moduli space of pointedcurves. Let X ′ denote the (fine) moduli space of pointed curves ( C, { p i } n , { q j } m )where C is a proper connected nodal curve of arithmetic genus 0 (a unionof smooth rational curves such that the dual graph is a tree) and the p i and q j are smooth points of C such that(1) p i = q j for all i and j .(2) Each end component of C contains at least one p i and one q j , and eachinterior component of C contains either a marked point or at least 3nodes.(3) The q j do not all coincide.(The moduli space X ′ can be obtained from M ,n + m as follows: for eachboundary divisor ∆ I ,I = M ,I ∪{∗} × M ,I ∪{∗} we contract the i th factorto a point if I i ⊆ [1 , n ] or I i ( [ n +1 , n + m ].) Define the boundary B of X ′ tobe the locus where the curve C is reducible. It follows by deformation theorythat X ′ is smooth with normal crossing boundary B . The construction ofthe previous paragraph defines an identification X ′ = X ′ \ B . The desiredcompactification X ⊂ X is obtained from X ′ ⊂ X ′ by contracting Γ ⊂ X ′ .Assume without loss of generality that m ≤ n . Consider the resolution f : X ′ → X of X with exceptional locus Γ ≃ P m − described above. By[GoM, Thm. II.1.1*] since 2 dim Γ ≤ dim X and X is affine it follows that X ′ has the homotopy type of a CW complex of dimension dim X . Henceby Thm. 3.1 the dual complex K of the boundary B has only top rationalhomology, and ˜ H d − ( K, C ) = Gr d H d ( X ′ , C ) where d = dim X ′ = m + n − X of X is a tropical compactification X ⊂ Y of X ⊂ T such that X ∩ O is connected for each orbit O ⊂ Y . This is provedusing the general result [HKT2, 2.10]. The toric variety Y corresponds tothe fan Σ with support A given by [MY, 2.11]. In particular, it follows that K is a triangulation of the link L . Hence we obtain an alternative proof that L has only top reduced rational homology, and a geometric interpretationof the top homology group. Acknowledgements : I would like to thank J. Tevelev for allowing me toinclude his unpublished result Thm. 2.4 and for many helpful comments.I would also like to thank S. Keel, S. Payne, D. Speyer, and B. Sturmfelsfor useful discussions. The author was partially supported by NSF grantDMS-0650052. 11 eferences [AK] F. Ardila, C. Klivans, The Bergman complex of a matroid and phylo-genetic trees, J. Combin. Theory Ser. B 96 (2006), no. 1, 38–49.[D] P. Deligne, Th´eorie de Hodge II, Inst. Hautes ´Etudes Sci. Publ. Math.No. 40, (1971), 5–57.[EKL] M. Einsiedler, M. Kapranov, D. Lind, Non-Archimedean amoebasand tropical varieties, J. Reine Angew. Math. 601 (2006), 139–157.[EGA4] A. Grothendieck, J. Dieudonn´e, ´El´ements de G´eom´etrieAlg´ebrique IV: ´Etude locale des sch´emas et des morphismes desch´emas, Inst. Hautes ´Etudes Sci. Publ. Math. 20 (1964), 24 (1965),28 (1966), 32 (1967).[GeM] I. Gel’fand, R. MacPherson, Geometry in Grassmannians and a gen-eralization of the dilogarithm, Adv. in Math. 44 (1982), no. 3, 279–312.[GoM] M. Goresky, R. MacPherson, Stratified Morse theory, Ergeb. Math.Grenzgeb. (3) 14, Springer (1988).[HKT1] P. Hacking, S. Keel, J. Tevelev, Compactification of the modulispace of hyperplane arrangements, J. Algebraic Geom. 15 (2006), no. 4,657–680.[HKT2] P. Hacking, S. Keel, J. Tevelev, Stable pair, tropical, and log canoni-cal compact moduli of del Pezzo surfaces, preprint arXiv:math/0702505[math.AG] (2007).[H] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer,1977.[HM] R. Hain, R. MacPherson, Higher logarithms, Illinois J. Math. 34(1990), no. 2, 392–475.[KT] S. Keel, J. Tevelev, Geometry of Chow quotients of Grassmannians,Duke Math. J. 134 (2006), no. 2, 259–311.[L] L. Lafforgue, Chirurgie des grassmanniennes, CRM Monogr. Ser. 19,A.M.S. (2003).[MY] H. Markwig, J. Yu, Shellability of the moduli space of n tropicallycollinear points in R d , preprint arXiv:0711.0944v1 [math.CO] (2007).12S] D. Speyer, Uniformizing Tropical Curves I: Genus Zero and One,preprint arXiv:0711.2677v1 [math.AG](2007).[SS] D. Speyer, B. Sturmfels, The tropical Grassmannian, Adv. Geom. 4(2004), no. 3, 389–411.[ST] B. Sturmfels, J. Tevelev, Elimination Theory for Tropical Varieties,preprint arXiv:0704.3471v1 [math.AG] (2007).[T] J. Tevelev, Compactifications of Subvarieties of Tori, Amer. J. Math 129(2007), no. 4, 1087–1104.[T2] J. Tevelev, personal communication.[V] C. Voisin, Hodge theory and complex algebraic geometry I, CambridgeStud. Adv. Math. 76, C.U.P. (2002).Paul Hacking, Department of Mathematics, University of Washington, Box354350, Seattle, WA 98195; [email protected]@math.washington.edu