NEF DIVISORS ON M 0,n FROM GIT

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1 NEF DIVISORS ON M 0,n FROM GIT VALERY ALEXEEV AND DAVID SWINARSKI Abstract We introduce and study the GIT cone of M 0,n, which is generated by the pullbacks of the natural ample line bundles on the GIT quotients (P n // SL( As one application, we prove unconditionally that the log canonical models of M 0,n with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of Matt Simpson Introduction The moduli space of smooth pointed genus zero curves M 0,n has many compactifications Among the most important of these is M 0,n, the moduli space of Deligne-Mumford stable curves There are many beautiful results on these spaces; for instance, Keel and Kapranov described them as explicit blowups of (P n 3 and P n 3 ([Kee9, Kap93b], and Kapranov identifed M 0,n with the Chow quotient of the Grassmannian Gr(, n by a torus But the birational geometry of these spaces is still not fully understood, and it is known to be very complicated For example, Faber computed that the nef cone of M 0,6 has 390 extremal rays Another family of compactifications of M 0,n is provided by the GIT quotients (P n // x SL(, where the vector x specifies the linearization O(x,, x n Let L x denote the distinguished polarization on the GIT quotient descending from the linearization By [Kap93a], there are birational morphisms π x : M 0,n (P n // x SL(, and hence the pullbacks L x := π x L x of the distinguished polarizations on the GIT quotients are big and nef, since π x is birational and L x is ample In this paper we consider the subcone generated by the L x inside the nef cone of M 0,n We call this the GIT cone and denote it GC Our first main result (see Section tells how the line bundles L x intersect the vital curves in M 0,n This allows us to describe a finite set of generators for the GIT cone, which we call the GIT complex 0-cells We show that for n 6 GC is strictly smaller than the nef cone The symmetric group S n acts on M 0,n by permuting the points, and symmetric divisors often play a key role Let S x denote the symmetrization of L x, that is, S x := σ S n L σ x We call the cone generated by these the symmetrized GIT cone, and denote it SGC We give a conjectural list of extremal rays of SGC and give an example showing that SGC is strictly smaller than the symmetric nef cone Thus, the L x and S x do not completely describe the Mori theory of M 0,n Nevertheless, we believe these cones are very useful for the study of M 0,n As an illustration, we reprove a result of Simpson and Fedorchuk Smyth Matt Simpson proved the following results in his dissertation: Assume the S n -equivariant Fulton Conjecture (which is known for n [Gib03] If certain divisors A α (defined below on M 0,n are nef, then the log canonical models of the pair (M 0,n, can be identified either as M 0,β for symmetric sets of weights β or as (P n // SL( with the symmetric linearization (cf [Sim08] Corollary 35, Theorem 5 Simpson then proves that these divisors A α are F-nef ([Sim08] Proposition 6 Using the S n -equivariant Fulton conjecture again, this yields as a corollary that the log canonical models are indeed the spaces claimed Date: December 3, 008

2 VALERY ALEXEEV AND DAVID SWINARSKI In Section 3, we express Simpson s divisors A α as explicit effective combinations of a small number of the L x In particular, this proves that Simpson s A α are nef without using the S n -equivariant Fulton Conjecture This removes one of his two uses of the S n -equivariant Fulton Conjecture In Section, we prove that A α is ample on the appropriate M 0,β, removing Simpson s second use of the S n -equivariant Fulton Conjecture Remark Fedorchuk and Smyth have recently given a proof ([FS08] of Simpson s result which is also fully independent of the S n -equivariant Fulton Conjecture Acknowledgements We would like to thank Angela Gibney and David Smyth for helpful discussions regarding this work The main line bundles and their intersection theory Let x be a vector in Q n such that 0 < x i for all i and n i= x i = Then x specifies a linearization on (P n, and hence a distinguished ample line bundle L x on the GIT quotient (P n // x SL( On the other hand, x gives a linearization for the action of T = G n m/ diag G m on Gr(, n, and by [Kap93a] one has the Gelfand-MacPherson correspondence (P n // x SL( = Gr(, n// x T (see [HMSV05] for a nice exposition There is a birational morphism π x : M 0,n (P n // x SL( ([Kap93b], [Kap93a], [Has03] Definition L x := π x L x The line bundle L x is nef, since it is the pullback of an ample bundle under a morphism If all x i < then the morphism π x is birational, and then L x is also big We can extend the definition to cover the case when some coordinates of x are zero Start with n = Then if any x i = 0, L x is the trivial line bundle If n and x i = 0, then define L x = π i Lˆx, where π i : M 0,n M 0,n is the morphism which forgets the i th marked point and ˆx is the vector obtained from x by omitting the i th coordinate We also formally set L x = 0 if any x i = The following lemma allows us to compute intersections of the L x with vital curves: Lemma Let {a, b, c, d} be a partition of the set {,, n} into four nonempty subsets, and let [C(a, b, c, d] be the corresponding vital curve class Let x be a set of weights as above Write x a = i a x i, x b = i b x i, etc We abbreviate min := min{x a, x b, x c, x d }, max := max{x a, x b, x c, x d } Then 0 if max, ( L x [C(a, b, c, d] = ( max if max and max + min, min if max and max + min Proof We reduce the calculation to M 0, and exploit the fact that M 0, = P is in a natural way a conic in the toric variety P For simplicity of notation, we assume that the partition {a, b, c, d} is ordered, ie a = {,, #a}, b = {#a +,, #a + #b}, etc Recall ([Kee9],[KM96] that a vital curve in M 0,n parametrizes nodal curves whose dual graph is a trivalent tree except at one vertex, which is -valent As the cross ratio of the four nodes on the corresponding component varies, a P in M 0,n is swept out The -valent vertex partitions in the n leaves into groups, and this is recorded by a partition {a, b, c, d} Two vital curves are linearly equivalent if and only if they have the same partition, so the partition specifies a class [C(a, b, c, d] Now consider the map (P (P n which is the product of the diagonal map a : P (P a and the similar maps b, c, d The pullback of the linearization x on (P n to (P is x = (x a, x b, x c, x d Therefore, the pullback of L x to (P // x SL( is L x This gives the following

3 NEF DIVISORS ON M 0,n FROM GIT 3 commutative diagram: M 0,n π x (P n // x SL( i i a,b,c,d M 0, = (P // x SL( in which the lower arrow is an isomorphism because both varieties are isomorphic to P The class of i [M 0, ] in M 0,n is [C(a, b, c, d] Thus, to complete the proof we just need to compute the degree of the sheaf L x on P By the Gelfand-MacPherson correspondence (P // x SL( = Gr(, // x, T where T = G m/diagg m The Plücker embedding descends to a map of torus quotients Gr(, // x T P 5 // x T The GIT quotient P 5 // x T, together with the polarization given by the GIT construction, is the polarized toric variety corresponding to the fiber over the point x = (x a, x b, x c, x d of the polytopal map from the simplex σ 6 R 6 with 6 vertices to the hypersimplex (, to R Denote the coordinates in R 6 by z ij = z ji with i j, and the coordinates in R by x i Then the map is given by x i = j i z ij Assume that x a x b x c x d and that x a + x b x c + x d Then an easy explicit computation shows that if x a > then the fiber over x is empty, and if x a then it is the triangle with the vertices (x, (x x + x 3 x /, (x x x 3 + x /, 0, 0, ( x + x + x 3 + x / ((x + x x 3 x /, x 3, (x x x 3 + x /, 0, ( x + x + x 3 + x /, 0 ((x + x x 3 x /, (x x + x 3 x /, x, ( x + x + x 3 + x /, 0, 0 This polytope is a standard triangle, shifted and dilated by a factor λ = x a = max Thus, (P 5, O(// x T = (P, O(λ The quotient Gr(, //T is a conic in this P and L x = O P (λ This proves the formula in the first two cases The third case holds by symmetry Remark Let (x a, x b, x c, x d be a point in (, Define d(x a, x b, x c, x d to be the distance from (x a, x b, x c, x d to the boundary of (, Then L x [C(a, b, c, d] is a multiple of d(x a, x b, x c, x d Notation for symmetrized line bundles It is natural to work with symmetrizations of the L x We use the following notation for these: Definition 3 We denote the symmetrization of any L x by S x : S x := σ S n L σ x For some special sets of weights, we have additional notation Let a be a rational number such that n a n, and write ā = (n a This condition on a ensures that 0 a ā We write L(a, i for L x where x j = a if j i and x i = ā, and call the i th entry the odd entry We write n V (a, n := L(a, i Thus, V (a, n is a reduced symmetrization: V (a, n = i= (n! S x Note that when a = /n, we have a = ā; our convention in this case is that V (a, n = L n (a,,a (that is, we do not further reduce the symmetrization

4 VALERY ALEXEEV AND DAVID SWINARSKI Results on the GIT cone and symmetrized GIT cone Suppose x (, n with x i 0 for any i The GIT stability criterion for points on a line is well-known: (P,, P n (P n is SL(-stable (resp semistable with respect to the linearization x if and only if for every subset I {,, n} for which the points {P i i I} coincide, i I x i < (resp Thus, for a generic linearization, stability and semistability coincide; strictly semistable points only occur for those x which lie on a hyperplane of the form i I x i = for some subset I {,, n} This is the familiar wall-and-chamber decomposition of the space of linearizations, due to Brion-Procesi, Dolgachev Hu and Thaddeus We call the subdivision of (, n given by these hyperplanes the GIT complex The GIT cone is generated by the 0-cells of the GIT complex because L x [C(a, b, c, d] is linear on each chamber For n = 5, we can easily check that the GIT cone is the nef cone The 0-cells for n = 5 are of the form (/3, /3, /3, /3, (/, /, /, /, 0, (,, 0, 0, 0, or the permutations of these The line bundles of type (,, 0, 0, 0 are trivial The line bundles of type (/3, /3, /3, /3 each contract four disjoint curves, giving P The line bundles of type (/, /, /, /, 0 give morphisms to P But for n 6 we see that the GIT cone is strictly contained in the nef cone Indeed, when n = 6, Faber computes that the nef cone has 390 extremal rays ([Fab00] We computed the 0-cells of the GIT complex by a search implemented in M The interior 0-cells are (/3, /3, /3, /3, /3, /3, (/, /, /, /, /, 3/, (/, /, /, /, /, /, (3/5, /5, /5, /5, /5, /5 and their symmetric images, and the boundary 0-cells are (,, 0, 0, 0, 0, (/, /, /, /, 0, 0, and (/3, /3, /3, /3, /3, 0 and their symmetric images There are only 9 in total, so there are not nearly enough to generate the nef cone We have a more precise conjectural description of the symmetrized GIT cone: Conjecture SGC = V (a, n a = t, t Z, n t n For n = 6, we can check the conjecture For symmetric divisors, the shape of the partition {a, b, c, d} determines the intersection D [C(a, b, c, d], so we need only record the intersections with C(3,,, and C(,,, These are as follows: V (/3, 6 V (/, 6 C(,,, C(3,,, 0 3/ Then if (S x C(,,,, S x C(3,,, = (a, b, we can write S x c V (/3, 6 + c V (/, 6, where ( ( ( ( c a = = a 6 b c 0 3/ b 3 b 3 b is automatically nonnegative, since S x is nef So the conjecture will be true if a 6 b 0 for any x But F ( x = a 6 b = (S x C(,,, 6 (S x C(3,,, is a piecewise linear function which breaks only along the hyperplanes i I x i = Hence F attains its minimum on a 0-cell of the GIT complex We check all the 0-cells for n = 6 and see that F 0 for all of them We give an example showing that SGC is strictly contained in the symmetric nef cone for n = 6 The F-conjecture is known for n = 6 ([Fab00], [KM96], [GKM0], so any line bundle which intersects the vital curves nonnegatively is nef Suppose D := r D + r 3 D 3 is a symmetric divisor on M 0,6 Then by [KM96] Corollary, (D C(,,,, D C(3,,, = (r 3 r, 3r r 3, and D is nef provided r 3 r 0 and 3r r 3 0, or r 3 r (/3r 3 But there are values of

5 NEF DIVISORS ON M 0,n FROM GIT 5 (r, r 3 for which r 3 r (/3r 3 but a 6 b = 8r 3 9r < 0 (For instance, (r, r 3 = (, So for n = 6 the SGC is strictly contained in the symmetric nef cone 3 Simpson s divisors are effective combinations of the V (a, n 3 The divisors A α First, we recall the key divisors A α from Simpson s work Write D j = S =j D S Then one basis for the S n -equivariant divisors on M 0,n is {D,, D n } Simpson establishes a formula for the A α in this basis, which we will use as the definition Definition 3 (cf [Sim08] Def 3 and Lemma 3 Write ( k(α := α Then (3 A α = k j= (( j α j(j n Simpson observes that if α [ A /(k+, and hence: k+, D j + n j=k+ ( (j (n j(j n + α D j k+ ], then A α is a convex combination of A /(k+ and Claim 3 It is sufficient to show that A α is nef for α = k+ where k =,, n We call these the critical α and A α Remark It is easy to check that A /3 = 3 A + 3 A / However, this will not affect any of the analysis below, and it does not appear that any of the other critical A α is a convex or effective combination of the others 3 When n is odd It is convenient to use the following notation Definition 33 Write ( (5 We chose the letter f for floor f := n = n l := f + k Proposition 3 Suppose n is odd, and that α = k+ α = f l+, and l {,, f } Then ( (6 A α c V f +, n + + c f V where (f + i(f l + (7 c i = α (f i(f i + (f i + if i l, for some k in {,, (n / } Hence ( f, n (8 c l = + l(f l α(f, f l + and c i = 0 for l + i f This covers all critical α except α = For α =, we have f + i (9 c i = (f i(f i + (f i + for i f

6 6 VALERY ALEXEEV AND DAVID SWINARSKI Remark Since α is of order /n, we see that for α < all but one of the coefficients are of order /n, while the last coefficient is of order When α =, all of the coefficients are of order /n Remark By Simpson s work, we know M 0,(/k,,/k corresponds to A α where α = /(k + Then the nonzero coefficients in the expression above are V (/f, n, V (/(f +, n,, V (/(n k, n It is convenient to order these through l where l = f + k Proof We choose a set of vital curves C,, C f (see Definition 35 below which give a full rank intersection matrix (C i V ( f+j, n for symmetric divisors on M 0,n Then the c i are the solutions of the system (0 ( ( C i V f + j, n c = (C i A α In Lemmas 36 and 37 below, we show that the system (0 takes on the following form: ( (f+ f+ f+ f+3 f+j f c f 0 f+ f+3 α f c (f 0 0 f+3 f α 0 = α (f j+ 0 f+j 0 f 0 6 c f f (l th coordinate l th coordinate We will refer to the upper triangular matrix on the left hand side of ( as U and write u i,j for its entries The vector shown on the right hand side of ( is for α < /3 For α = /3, the vector (C i A α on the right hand side has no trailing zeroes, and for α =, the right hand side vector is (,, T Thus, to prove Proposition 3, it remains only to show that the c i given in the statement of the proposition are indeed the solutions of the system ( This is done in Lemmas 38 and 39 Definition 35 The set of vital curves we use to define the (f (f system above is as follows: C := C(f, f,, C := C(f +, f,, C i := C(f + i, f i,, ( C f := C(f + + (f, f (f,, = C(n 3,,, where for each i we may choose any partition of {,, n} which has the indicated shape ( The next lemma establishes that the intersection matrix C i V f+j, n is the matrix U shown in (

7 Lemma 36 For any i, j {,, f }, C i V NEF DIVISORS ON M 0,n FROM GIT 7 ( f + j, n = u i,j = if i < j f + j (f j + if i = j f + j 0 if i > j Proof Note that the odd entry of V (/(f + j, n is (n /(f + j = j/(f + j We compute the intersection of C i with each L x constituting V (/(f + j, n and sum these to obtain C i V (/(f + j, n Case : i < j First suppose the odd entry of L x lands in the long tail of C i Then, using the notation of Lemma, x a = (f + i f + j + j f + j >, so C i L x = 0 Next suppose the odd entry of L x is on the second tail of C i Then x b = (f i f + j + j f + j, so C i L x = 0 Thus we see that L x contributes to C i V (/(f + j, n only if the odd entry is on the spine of C i The maximum comes from either the long tail, which has weight x a = (f + i /(f + j, or the odd entry, which has weight x c = j/(f + j It seems that both are possible The minimum is x d = /(f + j However, since both x a < and x c <, and both x a + min and x b + min, we find that C i L x = min = /(f + j Thus, C i V (/(f + j, n = /(f + j = /(f + j Case : i = j If the odd entry is on the long tail, x a = (f i f + j + j f + j = f + 3j >, f + j so C i L x = 0 If the odd entry is on the second tail, x b = (f j f + j + j f + j = f + j < f + j Meanwhile, on the long tail, x a = f + j, f + j so in the notation of Lemma max = x a = x b, and max + min =, so C i L x = min = /(f +j Finally, if the odd entry is on the spine, then the long tail gives the maximum, and max + min =, so C i L x = min = /(f + j Thus, C i V (/(f + j, n = (f i f + j + (f j + = f + j f + j Case 3: i > j If the odd entry is not on the long tail, then x a = (f + i f + j and C i L x = 0 If the odd entry is on the long tail, then x a is even bigger, and C i L x = 0 Thus, C i V (/(f + j, n = 0 In the next lemma we compute the right hand side of (

8 8 VALERY ALEXEEV AND DAVID SWINARSKI Lemma 37 Let α = f l+ for l {,, f } Then the intersection numbers C i A α are as follows: α if i < l, (3 C i A α = α if i = l, 0 if i > l For α =, we have C i A = for all i Proof It is lengthy but straightforward to compute this using Definition 3 and [KM96] Corollary We have now identified the systems (0 and ( Next we prove that the c i given in the statement of Proposition 3 are indeed the solutions of this system We will use the following identity: Lemma 38 The following identity holds for rational functions of a single variable y: ( m p= (y p(y p + (y p + = ym m + m y(y + (y m(y m + Proof By induction on m The induction step is easily verified by hand or with a computer algebra system Lemma 39 The c i defined in (7 and (8 for α <, or in (9 for α =, are the solutions of the system ( Proof We give the proof for α < The proof for α = is similar It is easy to verify that (5 u l,l c l = f l f l + u l,l c l + u l,l c l = α Next we consider any i < l We wish to show (6 u i,i c i + u i,i+ c i+ + + u i,l c l + u i,l c l = α l i u i,i c i + u i,i+p c i+p + u i,l c l = α p= l i u i,i c i + u i,i+p c i+p = α u l,l c l p= l i u i,i c i + u i,i+p c i+p = α, p= where we have used our computation of u l,l c l in (5 above and the definition of α to obtain the last line (6

9 Now we substitute u i,j and c j into (6: (7 (f i+ f +i NEF DIVISORS ON M 0,n FROM GIT 9 l i (f +i(f l+ α (f i(f i+(f i+ + f +i+p α (f +i+p(f l+ (f i p(f i p+(f i p+ = α l i (f l+ (f i(f i+ + (f l+ (f i p(f i p+(f i p+ = f l+ p= p= l i (f i(f i+ + (f i p(f i p+(f i p+ = (f l+(f l+ l i p= p= (f i p(f i p+(f i p+ = We substitute y = f i and m = l i Then (7 becomes m p= (y p(y p + (y p + = ( ( (f l+(f l+ (f i(f i+ (y m(y m + y(y +, which reduces to the identity ( We have now proved all the lemmas used in the proof of Proposition 3 3 Example: n = 9 Here f = The four critical α are /5 (which corresponds to (P 9 // SL(, /, /3, and (which corresponds to M 0,9 We use the three vital curves C(, 3,,, C(5,,,, and C(6,,, and the line bundles V (/5, 9, V (/6, 9, V (/7, 9 to form the intersection matrix: and the four vectors C i A α are This leads to the following equivalences: V (/5, 9 V (/6, 9 V (/7, 9 C(, 3,, /3 /7 C(5,,, 0 /3 /7 C(6,,, 0 0 6/7 α : /5 / /3 C(, 3,, A α : 3/5 / /3 C(5,,, A α : 0 / /3 C(6,,, A α : 0 0 /3 A /5 3 V (/5, 9 0 A / 8 V (/5, V (/6, 9 8 A /3 9 V (/5, V (/6, 9 + V (/7, A V (/5, 9 + V (/6, V (/7, 9 6

10 0 VALERY ALEXEEV AND DAVID SWINARSKI 33 When n is even We obtain similar results by similar methods Once again write f = n/ = n/ since n is even We use the curves C i = C(f +i, f i,, for i =,, f as our basis for the intersection matrix The resulting system is (8 f+ f+ f+j f f 0 f+ f+ (f 0 0 f+ 0 (f j+ 0 f+j f f c c c f = α α α 0 and it has the solution given below Note that more formulas are required when n is even than when n is odd We may attribute this to the fact that C and V (/n, n are more symmetric than any other pair under consideration, and as a result u, does not fit the pattern observed in the other diagonal entries of the intersection matrix when n is even Proposition 30 Suppose n 6 is even If α = f+ (which corresponds to (P n // SL(, then If α = f, then If α = f l+ where (9 A α (f /(f + V (/f, n A α ( f V f, n + for some l {3,, f }, then A α c V ( (f + (f f V f +, n ( ( f, n + c f V f, n, c = αf l + f(f, c i = (f l + (f + i α if i l, (f i(f i + (f i + c l = l(f + l α(f, f l + c i = 0 if i > l If α = and n = 6 (which corresponds to M 0,6, then A α V ( 3, V (, 6 0, If α = and n 8 (which corresponds to M 0,n, then ( A α c V f, n + c f V ( f, n

11 NEF DIVISORS ON M 0,n FROM GIT where c = f(f, c i = (f + i if i f, (f i(f i + (f i + (0 c f = f 3 3 Nefness of A α Since each critical A α is an effective combination of the L x, and every A α is a convex combination of the critical A α, we have the following corollary: Corollary 3 A α is nef for all n+ α Ampleness of A α on M 0,β We use the following result of [Ale08] to prove that A α is ample on M 0,β Below, β = (b,, b n, and β (, n denotes the weighted hypersimplex, the subset of R n given by the equality n i= x i = and the inequalities 0 x i b i Theorem Let { x i } be a finite set of points of β (, n satifsying the following conditions: ( Each L xi is the pullback of a Q-line bundle from M 0,β ( For every matroid polytope P with (Int P β (, n, there exists an x i in Int P Then the sheaf L xi on M 0,n is the pullback of an ample Q-line bundle from M 0,β M 0,/k,,/k corresponds to the critical alpha value α = /(k+, and we write A α as an effective combination of V (/( n/ +, n,, V (/(n k, n Proposition Let k {,, n } Write β = (/k,, /k, f = n/, and l = f+ k as in the previous section Then any combination c V (/f, n+ +c l V (/(f +l, n = c V (/f, n+ + c l V (/(n k, n with all the coefficients c i > 0 is ample on M 0,β In particular, when α = /(k +, A α is ample on M 0,β Proof Suppose P Λ is a matroid polytope We say P Λ is needed or necessary if (Int P Λ β (, n We say that a necessary P Λ is captured if there is an x (Int P Λ β (, n such that L x appears in c V (/f, n + + c l V (/(n k, n Now we show that the above line bundles capture all necessary matroid polytopes, and then apply Theorem Reduction It is enough to consider partitions Λ of width 3 Every necessary matroid polytope contains a necessary matroid polytope of partition width 3: Suppose Λ = I I w with w Recall that P Λ := { x (, n x I,, x Iw } By assumption, P Λ is necessary, so there is some x (Int P Λ β (, n such that x Ij for all j =,, w Reorder I,, I w so that x I x Iw Define j to be the index such that ( x Ij + + x Iw < while ( x Ij + x I j + + x Iw Define j to be the index such that (3 x Ij + + x Ij < while ( x Ij + x I j + + x Ij

12 VALERY ALEXEEV AND DAVID SWINARSKI Then P Λ contains the matroid polytope P Λ, where Λ = (I I j (I j I j (I j I w Moreover, we will show that P Λ contains x in its interior, so it is a necessary matroid polytope of partition width 3 We show x Int P Λ : By construction, x Ij + + x < and x Ij I j + + x Iw < So it remains only to show that x I + + x Ij < For this, we know (5 x I + + x Ij + x I j + + x Ij + x I j + + x Iw = since x β (, n Adding the inequalities ( and ( and combining the result with (5, we obtain (6 x I + + x Ij x I j Since x Int P Λ, we have x < Ij (We remark further that j = for most x and Λ Otherwise, if j >, then (6 combined with x x Ij I implies that we must have j j = 3 and x I = x I = = x Ij, which will only happen for very special x and Λ This completes our first reduction step It allows us to conclude that if all the necessary matroid polytopes for partitions of width 3 are captured, then all the matroid polytopes are captured So assume from now on that w = 3 Reduction We may assume without loss of generality that n n n 3 Recall that by definition V (a, n is a tensor product of the n line bundles of the form L x where x has n entries equal to a and one entry equal to ā For a given matroid polytope P Λ, it is enough to show that one of these L x captures it, and so by symmetry we may assume n n n 3 Reduction 3 It s enough to consider k There are two reasons for this First, M 0,( n = M 0,(/3 n, and the second space corresponds to k = Second, M 0,( n = M 0,n, and it is well-known that K + is ample on M 0,n With these reductions, this proposition (Proposition now follows from the next proposition (Proposition 3, below Proposition 3 Suppose k n/ Let β = (/k,, /k Suppose Λ = (n, n, n 3 is a partition of n into three nonzero parts such that (Int P Λ β (, n Assume n n n 3 a If n 3 k, then L(/(n n 3 captures P Λ b If n 3 < k, then L(/(n k captures P Λ Proof First, we study the intersection (Int P Λ β (, n If (7 min x I = max (x I + x I3 x β (,n x β (,n then (Int P Λ β (, n = Here, β = (/k,, /k Therefore, (7 becomes (8 (n k + n 3 k k n + n 3 So the hypothesis that (Int P Λ β (, n implies that n + n 3 > k Then we also have n < n k Now we prove part a We need to check that x Ij < for L(/(n n 3 for j =,, 3 First we check x I This is n < n n 3 < n n n 3 = n If n =, then n 3 = also, since n n 3 But then k < n + n 3 =, which contradicts the hypothesis that k Therefore, n > as desired

13 NEF DIVISORS ON M 0,n FROM GIT 3 For j =, we have x I x I < by the previous paragraph For j = 3, we have ( x I3 = (n + (n 3 < n n 3 n n 3 < 0 This completes part a Now we prove part b n < n k, so x I = n n k < Also, since n n, we have x I x I < Finally, x I3 = ( (n n k + (n 3 n k = n + n 3 k, n k and this is less than, since by hypothesis n 3 < k in part b Proposition together with [Sim08] Corollary 35 now yields the following result, which has also recently been proved by Fedorchuk and Smyth Corollary (extending [Sim08] Theorem 5; cf [FS08] Fix n and α a rational number in ( n, ] Let M 0,n(α denote the log canonical model of M 0,n with respect to K + α If α is in the range ( k+, n k+ ] for some k =,,, then M 0,n(α = M 0,(/k,,/k If α is in the range ( n, n +], then M 0,n(α = (P n // SL(, where the linearization is given by symmetric weights The V (a, n do not always give M 0,β s The V (a, n are big and nef, and the previous subsection shows that effective combinations of sufficiently many of them are ample on certain M 0,β Here we present an example to show that a single V (a, n taken by itself may not be ample on any M 0,β Proposition 5 Suppose that S is a semiample S n -equivariant line bundle on M 0,n, and that the image of M 0,n under the linear system S is isomorphic to a weighted moduli space M 0,β Then it is also isomorphic to a weighted moduli space M 0,β, where β is a symmetric set of weights Proof Recall from [Has03] that the space of weights β has a chamber decomposition, again given by hyperplanes of the form i I,#I 3,#I c 3 β i = If two sets of weights lie in the interior of the same chamber, then the resulting moduli spaces are isomorphic First note that if the moduli spaces M 0,σβ all lie in a single chamber, then M 0,β, where β is the average of all the σβ, lies in this chamber too Now suppose that M 0,β and M 0,σβ lie in two different chambers There is no isomorphism between M 0,β and M 0,σβ commuting with π β and π σβ, so this contradicts S being symmetric To see that no such isomorphism is possible, suppose that β satisfies i I β i while i I σβ i >, and consider the locus T in M 0,n whose generic point is a curve with two components: the first component contains the points labelled by I, and these and the point of attachment are allowed to vary; the second component contains the points of I c, which are fixed along with the point of attachment Then T is isomorphic to M 0, I +, and has dimension I But π β (T is a point, while π σβ (T = T Now we consider a specific example: the line bundle V (/6, 8 when n = 8 We compute the intersections of V (/6, 8 and the critical A α with all the vital curves: A A / A /5 V (/6, 8 C(3, 3,, / /5 /3 C(,,, / 0 /3 C(5,,, 0 0 C(3,,, /5 /3 C(,,, 3/ 8/5 0

14 VALERY ALEXEEV AND DAVID SWINARSKI From this data it is clear that the contraction defined by V (/6, 8 does not give any M 0,β If the image were any M 0,β, then by the proposition, it would be possible to use a symmetric β But we know the discrepancies for all symmetric β: these are precisely the A α V (/6, 8 is zero on [C(,,, ], but none of the A α are zero on this class Concluding remarks In this note we have used only a small number of the L x Many interesting questions remain, including: ( Can we identify log canonical models of M 0,n for nonsymmetric divisors as moduli spaces M 0,β where β is a nonsymmetric set of weights? ( Is there a combinatorial interpretation to our coefficients c i? Do they count something? Or, can we get a counting proof if we also keep track of stabilizers and the number of vital curves having a given partition shape? (3 Can we write all symmetric F-nef divisors as effective combinations of the L x and boundary divisor classes? Does using these line bundles make it easier to check the S n -equivariant F-conjecture? References [Ale08] Alexeev, V Weighted Grassmannians and stable hyperplane arrangements arxiv: v [Fab00] Faber, C The nef cone of M 0,6: a proof by inequalities only Unpublished notes circa 000 [FS08] Fedorchuk, M and D I Smyth Ample divisors on M 0,A (with applications to Log MMP for M 0,n arxiv: [Poly] Gawrilow, E and M Joswig polymake Available at [Gib03] Gibney, A Numerical criteria for divisors on M g to be ample mathag/0307 [GKM0] Gibney, A; Keel, S, and I Morrison Towards the ample cone of M g,n J Amer Math Soc 5 no (00, 73 9 [M] Grayson, D and M Stillman Macaulay, a software system for research in algebraic geometry Available at [Has03] Hassett, B Moduli spaces of weighted pointed stable curves Adv Math 73 no (003, [HMSV05] Howard, B; Millson, J; Snowden, A and R Vakil The projective invariants of ordered points on the line mathag/ v6 [Kap93a] Kapranov, M Chow quotients of Grassmannians, I Adv Sov Math 6 no (993, 9 0 [Kap93b] Kapranov, M Veronese curves and Grothendieck-Knudsen moduli space M 0,n J Alg Geom (993, 39 6 [Kee9] Keel, S Intersection theory of moduli space of stable n-pointed curves of genus zero Trans Am Math Soc 330 no (99, [KM96] Keel, S and J McKernan Contractible extremal rays on M 0,n arxiv mathag/ [GIT9] Mumford, D, Fogarty, J and FC Kirwan Geometric Invariant Theory Third Edition Springer, 99 [Rul06] Rulla, W Effective cones of quotients of moduli spaces of stable n-pointed curves of genus zero Trans Am Math Soc 358 no 7 (006, [Sim08] Simpson, M On Log Canonical Models of the Moduli Space of Stable Pointed Genus Zero Curves PhD dissertation, Rice University, 008