Table Of Content

ON THE CONE OF EFFECTIVE 2-CYCLES ON M 0,7 LUCA SCHAFFLER 5 1 Abstract. Fulton’squestionabouteffectivek-cyclesonM0,n for1<k<n−4canbeanswered 0 negatively byappropriately lifting to M0,n theKeel-Vermeiredivisors on M0,k+1. In this paper 2 we focus on the case of 2-cycles on M0,7, and we prove that the 2-dimensional boundary strata p together with the lifts of the Keel-Vermeire divisors are not enough to generate the cone of e effective2-cycles. Wedothisbyprovidingexamplesofeffective2-cycleson M0,7 thatcannotbe S written as an effectivecombination of theaforementioned 2-cycles. These examples are inspired 3 by a blow up construction of Castravet and Tevelev. ] Introduction G A An open problem in the birational geometry of M , the moduli space of stable n-pointed 0,n . rational curves, is the F-conjecture. This conjecture claims that the cone Eff (M ) of effective h 1 0,n t curves, is generated by thenumericalequivalence classes of 1-dimensional boundarystrata, which a m are obtained by intersecting boundary divisors. This is known to be true if n 7 (see [KM]). ≤ A similar question (which is known as Fulton’s question) was stated in [KM] also for the cone [ Eff (M ) of effective k-cycles with 1 < k < n 3: k 0,n 3 − v Is the cone Effk(M0,n) generated by the k-dimensional boundary strata? 6 Denote by V (M )theconegenerated bythenumerical equivalence classes ofthek-dimensional 3 k 0,n 7 boundary strata. Then the question is whether or not Effk(M0,n) is equal to Vk(M0,n). As Keel 1 and Vermeire pointed out in thecase of divisors (see [GKM], [V]), thecone V (M ) is strictly n−4 0,n 0 contained in Eff (M ), and one can see that V (M ) ( Eff (M ) for all 1 < k < n 4 by . n−4 0,n k 0,n k 0,n 1 − appropriately lifting to M the Keel-Vermeire divisors on M (see Section 3, in particular 0 0,n 0,k+1 5 Corollary 3.3). So the problem is to understandwhatlies in Effk(M0,n) Vk(M0,n) (see [HT], [C], \ 1 [CT13], [DGJ], [O] for the codimension 1 case). Recently, a lot of work has been done in order to : v understand the cones of effective and pseudoeffective cycles of higher codimension on projective i varieties (see [DELV], [F], [T], [L], [FL] and [CC]). X We work over an algebraically closed field K of any characteristic. The main result of this r a paper (Theorem 5.8) can be synthesized in the following statement Theorem. The2-dimensionalboundarystrata onM togetherwiththeliftsoftheKeel-Vermeire 0,7 divisors on M are not enough to generate the cone Eff (M ). 0,6 2 0,7 The lifts of the Keel-Vermeire divisors are defined as the pushforwards with respect to the natural inclusion D ֒ M of the Keel-Vermeire divisors on the boundary divisor D (which ab 0,7 ab → is isomorphic to M ) for any a,b 1,...,7 . In this way we produce 315 extremal rays of 0,6 { } ⊂ { } 2010 Mathematics Subject Classification. 14H10, 14C25, 14C17. Key words and phrases. Moduli of curves, Effectivecycles. Partially supported bythe Officeof theGraduate School of theUniversity of Georgia. 1 2 LUCASCHAFFLER Figure 1. 7-points arrangement in P2 which gives a special hypertree surface on 1 M . 0,7 Eff (M ) which lie outside of V (M ) (see Proposition 4.4). Denote with VKV(M ) the cone 2 0,7 2 0,7 2 0,7 generated by V (M ) and by these lifts. 2 0,7 Examplesofeffective2-cyclesonM whosenumericalequivalenceclassesdonotlieinthecone 0,7 VKV(M ) are produced using the following blow up construction of Castravet and Tevelev (see 2 0,7 [CT12, Theorem 3.1]): take seven labeled points in P2 which do not lie on a (possibly reducible) conic. Then the blow up of P2 at these points can be embedded in M as an effective 2-cycle. 0,7 Using this construction and considering particular arrangements of seven labeled points in P2, we define what we call special hypertree surfaces on M (see Definition 5.6), which are related 0,7 to Castravet and Tevelev hypertrees (see [CT13]). In Theorem 5.8 we prove that the numerical equivalence class of a special hypertree surface does not lie in the cone VKV(M ). This implies 2 0,7 that VKV(M ) (Eff (M ), which is our main result. An example of 7-points arrangement in 2 0,7 2 0,7 P2 which gives rise to a special hypertree surface on M is the one shown in Figure 1. 0,7 All the other special hypertree surfaces are obtained by permuting the labels of the points arrangement in Figure 1. In Section 5.3 we show that there are 210 (resp. 30) distinct numerical equivalence classes of special hypertree surfaces on M if the characteristic of the base field is 0,7 different from 2 (resp. equal to 2). Summing up, if we denote with VKV+CT(M ) the cone generated by VKV(M ) and by 2 0,7 2 0,7 the numerical equivalence classes of the embedded blow ups of P2 at seven points, we have the following chain of containments V (M )( VKV(M )( VKV+CT(M ) Eff (M ). 2 0,7 2 0,7 2 0,7 ⊆ 2 0,7 The second main result of this paper is an explicit description of the intersection theory of the 2-dimensional boundary strata on M . In Proposition 2.4 and Proposition 2.5 we give formulas 0,7 1All the figures in this paper were realized using the software GeoGebra, Copyright (cid:13)cInternational GeoGebra Institute,2013. ON THE CONE OF EFFECTIVE 2-CYCLES ON M0,7 3 that compute the intersection number of two 2-dimensional boundary strata on M . Then we 0,7 study the numerical equivalence classes of these 2-cycles (see Propositions 2.7 and 2.8), and this, together with some recent results of Chen and Coskun in [CC], allows us to give a complete description of the cone V (M ) (see Corollary 2.10). We also fully describe the bilinear form 2 0,7 N (M ) N (M ) R given by the intersection product (see Propositions 2.11 and 2.12). 2 0,7 2 0,7 × → In Section 1 we recall some basic facts and notations about M that are used in this paper. 0,n Section 2 contains the formulas for the intersection of two 2-dimensional boundary strata on M , and the complete study of the cone V (M ). In Section 3 there is a detailed description 0,7 2 0,7 of the lifting technique, which is immediately applied in Section 4 to describe the lifts to M 0,7 of the Keel-Vermeire divisors on M . Section 5 is where we discuss the embedded blow ups of 0,6 P2 in M and where we prove our main theorem. In Section 6 we generalize the construction 0,7 of the two cones VKV(M ) and VKV+CT(M ) to any M for n > 7. We also state some 2 0,7 2 0,7 0,n questions that will be the object of further investigation. Acknowledgements. I would like to express my gratitude to my advisor, Valery Alexeev, for hisinsightfulcomments andhelpfuldiscussions. Iamalso gratefultoAngelaGibneyforhergreat suggestions and support. Many thanks to Noah Giansiracusa, Daniel Krashen, Dino Lorenzini and Robert Varley for interesting discussions and for their helpful feedback. My gratitude also goes to Ana-Maria Castravet, Dawei Chen, Izzet Coskun and Jenia Tevelev for great discussions related to this paper. A special thanks to Ana-Maria Castravet for pointing out a mistake in the first version of this paper, and to Jenia Tevelev for a helpful discussion on Lemma 5.7. I am also grateful to the referees for their careful reading of the paper and their comments. Contents Introduction 1 1. Preliminaries: boundary strata on M 3 0,n 2. The cone of boundary 2-strata on M 4 0,7 3. Lift of effective cycles 11 4. Lifts to M of the Keel-Vermeire divisors on M 13 0,7 0,6 5. Embedded blow ups of P2 in M 15 0,n 6. Generalization to M for any n > 7 and further questions 21 0,n References 21 1. Preliminaries: boundary strata on M 0,n In this section we review some of the main definitions and facts about the boundary strata on M . For a more detailed discussion, see for example [KM]. Equivalence between k-cycles 0,n on M refers to numerical equivalence, which is the same as rational equivalence and algebraic 0,n equivalence by [Ke]. Definition 1.1. The irreducible components of the locus of points on M parametrizing stable 0,n n-pointed rational curves with at least n 3 k nodes, have dimension k and are called boundary − − k-strata. Codimension 1 (resp. 1-dimensional) boundary strata are also called boundary divisors (resp. F-curves). 4 LUCASCHAFFLER Definition 1.2. Given n 3 and 0 k n 3, define V (M ) to be the cone generated by k 0,n ≥ ≤ ≤ − the equivalence classes of the boundary k-strata on M (V stands for “vital cycles”, as they 0,n were called in [KM]). Notation. If n is a positive integer, then [n] denotes the set 1,...,n . { } Combinatorial description of boundary divisors. There is a bijection between boundary divisors and partitions I Ic = [n], with 2 I n 2. DI = DIc denotes the boundarydivisor ∐ ≤ | | ≤ − corresponding to the partition I Ic = [n]. δI = δIc denotes the equivalence class of DI. For ∐ simplicity, the equivalence class of a boundary divisor will be called just boundary divisor. Combinatorialdescription ofequivalenceclassesofF-curves. Thereisabijectionbetween equivalence classes of F-curves and partitions of [n]= 1,...,n into four nonempty subsets (see { } [KM,Lemma4.3]). GivenapartitionI I I I = [n],wedenotebyF theequivalence 1∐ 2∐ 3∐ 4 I1,I2,I3,I4 class of the F-curves corresponding to that partition. Every boundarystratumon M can berealized as thecomplete intersection of all thebound- 0,n ary divisors containing it as follows. Let B be a boundary stratum and let C(B) be the stable n-pointed rational curve corresponding to the generic point of B (C(B) has as many nodes as the codimension of B). If Sing(C(B)) denotes the set of singular points of C(B), given p Sing(C(B)) let T be the set of markings that are over one of the two connected components p ∈ of the normalization of C(B) at p. Then we have that B = D . \ Tp p∈Sing(C(B)) Moreover, since the boundary of M has normal crossings, we have that the equivalence class 0,n of B is the product of all the δ as p varies among the nodes of C(B). Tp The last thing we want to recall is [Ke, Fact 4]: given two boundary divisors D , D on M , I J 0,n then D D = I J, which by definition means I J ∩ 6 ∅⇔ ∗∗ I J or I Jc or I J or I Jc. ⊆ ⊆ ⊇ ⊇ 2. The cone of boundary 2-strata on M 0,7 The main object of our study is Eff (M ), which is a subcone of the real vector space 2 0,7 N (M )(inSection 2.5weshowthatdim N (M )= 127). Westartbyanalyzingthesubcone 2 0,7 R 2 0,7 V (M ) Eff (M ). The first thing we want to do is to give a combinatorial description of 2 0,7 2 0,7 ⊆ the boundary 2-strata on M . After this, we study their intersections and their equivalence 0,7 classes. 2.1. Combinatorial description of the boundary 2-strata on M . According to Defini- 0,7 tion 1.1, a boundary 2-stratum on M is the closure of the locus of points parametrizing stable 0,7 7-pointed rational curves of the shape shown in Figure 2, where I J K is a given partition of ∐ ∐ [7]. ON THE CONE OF EFFECTIVE 2-CYCLES ON M0,7 5 Figure 2. Stable 7-pointed rational curve parametrized by the generic point of a boundary 2-stratum. Stability imposes that 2 I 4, 1 J 3 and 2 K 4. Therefore ≤ | | ≤ ≤ | | ≤ ≤ | |≤ there is a bijection between set-theoretically distinct boundary 2-strata, and partitions I J K ∐ ∐ of [7], with 2 I 4, 1 J 3 and 2 K 4, modulo the equivalence relation ≤ | | ≤ ≤ | | ≤ ≤ | | ≤ I J K K J I. ∐ ∐ ∼ ∐ ∐ Withs M wedenotetheboundary2-stratumcorrespondingtothepartitionI J K of I,J,K 0,7 ⊂ ∐ ∐ [7]. The equivalence class of s is denoted by σ . Obviously, we have that σ = δ δ . I,J,K I,J,K I,J,K I K · An easy combinatorial count tells us that there are 490 set-theoretically distinct boundary 2- strata s . A similar description applies for codimension 2 boundary strata on M for n 8. I,J,K 0,n ≥ For general results about boundary strata of codimension 2 on M , see [CC, Section 6]. 0,n 2.2. Intersection of two distinct boundary 2-strata. Given σ and σ , our goal is I,J,K L,M,N to compute the intersection σ σ = δ δ δ δ . This intersection is clearly zero, I,J,K L,M,N I K L N · · · · unless we require that the condition defined here below is satisfied. Definition 2.1. Consider two boundary 2-strata s and s . Assume that I,J,K L,M,N I L and I N and K L and K N. ∗∗ ∗∗ ∗∗ ∗∗ If this condition is satisfied, we write s s . I,J,K L,M,N ∗∗ Lemma 2.2. Let D ,D and D be three distinct boundary divisors on M such that I I I1 I2 I3 0,7 a∗∗ b for all a,b 1,2,3 . Then D D D is an F-curve. { } ⊂ { } I1 ∩ I2 ∩ I3 Proof. Assume without loss of generality that I I = . We know that I I and I I , 1 2 3 1 3 2 ∩ ∅ ∗∗ ∗∗ therefore (I I or I Ic or I I or I Ic) and 3 ⊂ 1 3 ⊂ 1 3 ⊃ 1 3 ⊃ 1 (I I or I Ic or I I or I Ic). 3 ⊂ 2 3 ⊂ 2 3 ⊃ 2 3 ⊃ 2 Among these 16 cases, the only possible are (I I and I Ic) or (I Ic and I I ) or 3 ⊂ 1 3 ⊂ 2 3 ⊂ 1 3 ⊂ 2 (I Ic and I Ic) or (I Ic and I I ) or 3 ⊂ 1 3 ⊂ 2 3 ⊂ 1 3 ⊃ 2 (I I and I Ic) or (I I and I I ) or 3 ⊃ 1 3 ⊂ 2 3 ⊃ 1 3 ⊃ 2 (I I and I Ic) or (I Ic and I I ). 3 ⊃ 1 3 ⊃ 2 3 ⊃ 1 3 ⊃ 2 6 LUCASCHAFFLER Up to changing I with Ic, we just need to consider 3 3 (I I and I Ic) or (I Ic and I I ) or 3 ⊂ 1 3 ⊂ 2 3 ⊂ 1 3 ⊂ 2 (I Ic and I Ic) or (I Ic and I I ). 3 ⊂ 1 3 ⊂ 2 3 ⊂ 1 3 ⊃ 2 Now, inspecting each oneof these four cases, it is easy to see that the intersection D D D is an F-curve. I1∩ I2∩ (cid:3)I3 Lemma 2.3. Let s and s be two distinct boundary 2-strata on M satisfying the I,J,K L,M,N 0,7 condition s s . Then we can write σ σ = δ δ δ δ where, either I,J,K ∗∗ L,M,N I,J,K · L,M,N I1 · I2 · I3 · I4 the four boundary divisors δ ,δ ,δ and δ are pairwise distinct, or exactly two of them are I1 I2 I3 I4 equal. In the latter case, we assume that I = I . In any case, we assume that I I = and 3 4 1 2 ∩ ∅ I I . 1 2 | |≤ | | Proof. Write σ σ = δ δ δ δ . Obviously δ = δ and δ = δ . If two boundary I,J,K L,M,N I K L N I K L N · · · · 6 6 divisors amongδ ,δ ,δ andδ areequal, assumewithoutloss of generality that δ = δ . Then I K L N K L we must have that δ = δ , or we would have s = s . Also, δ = δ = δ . This proves N I I,J,K L,M,N N K L 6 6 that there can be at most two boundary divisors among δ ,δ ,δ and δ that are equal. So, let I K L N us write δ δ δ δ = δ δ δ δ , where I,K,L,N = A,B,I ,I and I = I in I · K · L · N A · B · I3 · I4 { } { 3 4} 3 4 case two boundary divisors among δ ,δ ,δ and δ coincide. Finally, we can obviously rewrite I K L N δ δ = δ δ with I I = (here we use the hypothesis s s ) and I I . (cid:3) A· B I1· I2 1∩ 2 ∅ I,J,K∗∗ L,M,N | 1|≤ | 2| Proposition 2.4. Let s and s be two distinct boundary 2-strata on M such that I,J,K L,M,N 0,7 s s (otherwise, the intersection number σ σ is trivially zero). Write I,J,K L,M,N I,J,K L,M,N ∗ ∗ · σ σ = δ δ δ δ as prescribed by Lemma 2.3 (recall that in this lemma we I,J,K · L,M,N I1 · I2 · I3 · I4 assumed, among other things, that I I ). Then 1 2 | | ≤ | | 1 if δ = δ , I = 2 and I 2,4  − I3 I4 | 1| | 2| ∈{ } σ σ = 1 if δ ,δ ,δ and δ are pairwise distinct I,J,K · L,M,N  I1 I2 I3 I4 0 otherwise.  Proof. Let us make some preliminary observations. We have that σ σ = δ δ δ δ = σ δ δ = [s ] δ δ . I,J,K · L,M,N I1 · I2 · I3 · I4 I1,(I1∪I2)c,I2 · I3 · I4 I1,(I1∪I2)c,I2 · I3 · I4 Define S := s and let i: S ֒ M be the inclusion morphism. Using the projection I1,(I1∪I2)c,I2 → 0,7 formula, we obtain that [S] δ δ = i [S] (δ δ ) = [S] i∗(δ δ ) = i∗(δ δ ) =(i∗δ ) (i∗δ ). · I3 · I4 ∗ · I3 · I4 · I3 · I4 I3 · I4 I3 · I4 Now, for j = 3,4, i∗δ = [D D D ], where D D D is an F-curve by Lemma 2.2. Ij I1 ∩ I2 ∩ Ij I1 ∩ I2 ∩ Ij So i∗δ and i∗δ are two equivalence classes of F-curves on the boundary 2-stratum S. There I3 I4 are two possibilities for S up to isomorphism. (i) If I = 2 and I 2,4 , then S = M . By Kapranov’s blow up construction of M | 1| | 2|∈ { } ∼ 0,5 0,n (see [Ka]), we know that M is isomorphic to the blow up of P2 at four points in general 0,5 linear position. Moreover, the F-curves of M correspond to the exceptional divisors of 0,5 the blow up, and the strict transforms of the lines spanned by the blown up points. (ii) If |I2| = 3 and |I1| ∈ {2,3}, then S ∼= M0,4 ×M0,4, which is isomorphic to P1×P1. An F-curve on S corresponds to a line on P1 P1 in the form p P1 or P1 p for some × { }× ×{ } point p P1. ∈ ON THE CONE OF EFFECTIVE 2-CYCLES ON M0,7 7 Observe that in case (i) (resp. case (ii)) the self-intersection of an F-curve is 1 (resp. 0), and − in both cases two distinct F-curves intersect at one point if and only if their intersection number is 1. Now, let us prove our intersection formula for σ σ . I,J,K L,M,N · I = 2 and I = 2. Up to permuting the labels, we have that 1 2 • | | | | S = M M M = M , ∼ 0,{1,2,x} 0,{x,3,4,5,y} 0,{y,6,7} ∼ 0,{x,3,4,5,y} × × where x and y are the nodes of the stable 7-pointed rational curve corresponding to the generic point of S. If δ = δ , then (i∗δ ) (i∗δ ) is equal to the self-intersection of an I3 I4 I3 · I4 F-curve on S = M , which gives σ σ = 1. So let us assume that ∼ 0,{x,3,4,5,y} I,J,K L,M,N · − δ ,δ ,δ and δ are pairwise distinct. Given j = 3,4, since I I and I I , then I1 I2 I3 I4 1 ∗∗ j 2 ∗∗ j i∗δ is equal to one of the following boundary divisors on M Ij 0,{x,3,4,5,y} δ ,δ ,δ or δ . 34 35 45 345 If i∗δ = δ ,δ or δ , then i∗δ = δ because I I and δ = δ . If i∗δ = δ , I3 34 35 45 I4 345 3∗∗ 4 I3 6 I4 I3 345 then i∗δ has to be equal to δ ,δ or δ . In any case, σ σ = 1. I4 34 35 45 I,J,K · L,M,N I = 2 and I = 4. We have isomorphisms 1 2 • | | | | S = M M M = M . ∼ 0,{1,2,x}× 0,{x,3,y}× 0,{y,4,5,6,7} ∼ 0,{y,4,5,6,7} If δ = δ , then again (i∗δ ) (i∗δ ) is equal to the self-intersection of an F-curve on I3 I4 I3 · I4 S = M , which gives σ σ = 1. Let us assume that δ = δ . Given ∼ 0,{y,4,5,6,7} I,J,K · L,M,N − I3 6 I4 j = 3,4, then i∗δ is equal to one of the following boundary divisors on M Ij 0,{y,4,5,6,7} δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ or δ . 45 46 47 56 57 67 456 457 467 567 If i∗δ = δ ,δ ,δ ,δ ,δ or δ , then assumeupto achange of labels that i∗δ = δ . I3 45 46 47 56 57 67 I3 45 In this case, i∗δ = δ ,δ or δ . If i∗δ = δ ,δ ,δ or δ , assume up to a I4 67 456 457 I3 456 457 467 567 change of labels that i∗δ = δ . Then i∗δ has to be equal to δ ,δ or δ . Each one I3 456 I4 45 46 56 of these choices for i∗δ and i∗δ gives σ σ = 1. I3 I4 I,J,K · L,M,N I = 2 and I = 3. In this case we have 1 2 • | | | | S = M M M = M M . ∼ 0,{1,2,x} 0,{x,3,4,y} 0,{y,5,6,7} ∼ 0,{x,3,4,y} 0,{y,5,6,7} × × × If δI3 = δI4, then (i∗δI3)·(i∗δI4) is equal to the self-intersection of an F-curve on S ∼= M M , whichgives σ σ = 0. Now consider thecase δ = δ . 0,{x,3,4,y}× 0,{y,5,6,7} I,J,K· L,M,N I3 6 I4 For j = 3,4, i∗δ is equal to the equivalence class of one of the following divisors on Ij M M 0,{x,3,4,y} 0,{y,5,6,7} × D M ,M D ,M D or M D . 34 0,{y,5,6,7} 0,{x,3,4,y} 56 0,{x,3,4,y} 57 0,{x,3,4,y} 67 × × × × Since I I , the only possibility for i∗δ and i∗δ is to belong to two different rulings 3∗∗ 4 I3 I4 of S. It follows that σ σ = 1. I,J,K L,M,N · I = 3 and I = 3. Then 1 2 • | | | | S = M M M = M M . ∼ 0,{1,2,3,x}× 0,{x,4,y}× 0,{y,5,6,7} ∼ 0,{1,2,3,x}× 0,{y,5,6,7} If δ = δ , then (i∗δ ) (i∗δ ) is equal to the self-intersection of an F-curve on S = I3 I4 I3 · I4 ∼ M M , which gives σ σ = 0. For thecase δ = δ , given j = 0,{1,2,3,x}× 0,{y,5,6,7} I,J,K· L,M,N I3 6 I4 8 LUCASCHAFFLER 3,4, i∗δ is equal to the equivalence class of one of the following divisors on M Ij 0,{1,2,3,x}× M 0,{y,5,6,7} D M ,D M ,D M , 12 0,{y,5,6,7} 13 0,{y,5,6,7} 23 0,{y,5,6,7} × × × M D ,M D or M D . 0,{1,2,3,x} 56 0,{1,2,3,x} 57 0,{1,2,3,x} 67 × × × I I implies that i∗δ and i∗δ belong to two different rulings of S. In particular, 3 ∗∗ 4 I3 I4 σ σ = 1. I,J,K L,M,N · At this point, theclaimed intersection formulasumsupalltheconsiderations wemadesofar. (cid:3) 2.3. Self-intersection of a boundary 2-stratum. Wewanttocomputeσ2 = δ δ δ δ . I,J,K I· K· I· K The idea is to find an appropriate Keel relation (see [Ke, page 569, Theorem 1(2)]) that allows us to replace δ and reduce the calculation to the previous case. I Proposition 2.5. Let σ be the equivalence class of a boundary 2-stratum with I K . I,J,K | | ≤ | | Then 0 if I = 2 and J = 1 σ2 =  2 if |J| = 3 | | I,J,K  | | 1 otherwise.  Proof. Up to relabeling the markings, it is enough to prove that σ2 = 0, σ2 = 12,3,4567 123,4,567 σ2 = 1 and σ2 = 2. 12,34,567 12,345,67 σ2 = δ δ δ δ . Let us use the boundary relation • 12,3,4567 12 · 4567 · 12 · 4567 δ = δ δ = δ +δ +δ +δ +δ +δ + X S X S ⇒ 12 13 135 136 137 1356 1357 1,2∈S 1,3∈S 3,4∈Sc 2,4∈Sc +δ +δ δ δ δ δ δ δ δ . 1367 13567 125 126 127 1256 1257 1267 12567 − − − − − − − Butnow, if δ is one of the boundarydivisors that appear in the expression wejustfound T for δ , then 1,2 T is false or 4,5,6,7 T is false. Hence, σ2 = 0. 12 { }∗∗ { }∗∗ 12,3,4567 σ2 = δ δ δ δ . Consider • 123,4,567 123 · 567 · 123 · 567 δ = δ δ = δ +δ +δ +δ +δ +δ + X S X S ⇒ 123 14 143 146 147 1436 1437 1,2∈S 1,4∈S 4,5∈Sc 2,5∈Sc +δ +δ δ δ δ δ δ δ δ . 1467 14367 12 126 127 1236 1237 1267 12367 − − − − − − − After replacing δ with the new expression and distributing, we get σ2 = δ 123 123,4,567 − 123 · δ δ δ = δ δ δ δ = ( 1) = 1. 567 12 567 12 4567 567 567 · · − · · · − − σ2 = δ δ δ δ . We use the following boundary relation • 12,34,567 12 · 567 · 12· 567 δ = δ δ = δ +δ +δ +δ +δ +δ + X S X S ⇒ 12 13 134 136 137 1346 1347 1,2∈S 1,3∈S 3,5∈Sc 2,5∈Sc +δ +δ δ δ δ δ δ δ δ 1367 13467 124 126 127 1246 1247 1267 12467 − − − − − − − ⇒ σ2 = δ δ δ δ = δ δ δ δ = 1. 12,34,567 − 12· 567· 124· 567 − 12· 3567 · 567 · 567 ON THE CONE OF EFFECTIVE 2-CYCLES ON M0,7 9 σ2 = δ δ δ δ . • 12,345,67 12 · 67· 12· 67 δ = δ δ = δ +δ +δ +δ +δ +δ + X S X S ⇒ 12 13 134 135 137 1345 1347 1,2∈S 1,3∈S 3,6∈Sc 2,6∈Sc +δ +δ δ δ δ δ δ δ δ 1357 13457 124 125 127 1245 1247 1257 12457 − − − − − − − ⇒ σ2 = δ δ δ δ δ δ δ δ δ δ δ δ = 12,345,67 − 12· 67· 124 · 67 − 12· 67 · 125 · 67 − 12 · 67 · 1245 · 67 = δ δ δ δ δ δ δ δ δ δ δ δ = 1+1 0 = 2. 12 3567 67 67 12 3467 67 67 12 367 67 67 − · · · − · · · − · · · − (cid:3) Remark 2.6. As one of the referees pointed out, Proposition 2.5 can also be proved using [E, Lemma 3.5]. Say we want to compute σ2 . Then, adopting the same notation used in [E, I,J,K Lemma 3.5], one can take B = s and let X B be the pullback of the universal family on I,J,K → M with respect to the inclusion s ֒ M . Then the intersection number σ2 can be 0,7 I,J,K → 0,7 I,J,K computed using the formula provided at the end of [E, Lemma 3.5]. 2.4. Equivalence classes of boundary 2-strata. So far, we considered set theoretically distinct boundary 2-strata. However, we are interested in studying distinct equivalence classes of boundary 2-strata. Proposition 2.7. Consider σ and σ with I K , L N and s = s . I,J,K L,M,N I,J,K L,M,N | | ≤ | | | | ≤ | | 6 Then σ = σ I J = L M and I J = 3. I,J,K L,M,N ⇔ ∪ ∪ | ∪ | Proof. ( ) Assume a,b,c,d,e,f,g = [7] and let I J = a,b,c . Consider the boundary divisor ⇐ { } ∪ { } D = M M = P1 M . Let π: P1 M P1 be the usual projection morphism. abc,defg ∼ 0,4× 0,5 ∼ × 0,5 × 0,5 → If C is the stable 7-pointed rational curve corresponding to the generic point of D , assume the abc node of C and the labels b,c fixed on the twig which contains a,b and c. So we can think of a as parametrizing P1, and therefore π−1(b) = s , π−1(c) = s . In conclusion, s ab,c,defg ac,b,defg ab,c,defg and s are rationally equivalent. ac,b,defg ( ) Let us prove the contrapositive. We proceed by enumerating all the possible cases. ⇒ (i) J = 3. Then 2 = σ σ = σ σ 1,0,1 σ = σ . I,J,K I,J,K L,M,N I,J,K I,J,K L,M,N | | · 6 · ∈{− } ⇒ 6 (ii) I = J = 2. Up to relabeling, we can assume that σ = σ . The boundary I,J,K 12,34,567 | | | | 2-stratum σ can be in one of the following forms L,M,N σ , σ or σ ab,cd,efg abc,d,efg ab,c,defg (σ is excluded because of what we just discussed in (i)). In any case, we can write ab,cde,fg σ = δ δ with S = 4. Therefore,σ σ = δ δ δ δ canbeequal L,M,N S T 12,34,567 L,M,N 12 1234 S T · | | · · · · to just 0 or 1 (more in detail, if S 1234 , then S = 1,2,3,4 and the intersection − ∗∗{ } { } can be either 0 or 1). However σ σ = 1, so σ = σ . 12,34,567 12,34,567 12,34,567 L,M,N − · 6 (iii) J = 1 and I = 3. This case uses the same strategy we adopted in (ii). | | | | (iv) J = 1 and I = 2. We can assume σ = σ . Because of what we proved so far, I,J,K 12,3,4567 | | | | we can assume that s = s . By our hypotheses, we also have that s L,M,N ab,c,defg ab,c,defg has to be different from s and s . But now, up to permuting 4,5,6,7 and 13,2,4567 23,1,4567 { } 1,2 (which leave σ unchanged), there are few possibilities for σ , which 12,3,4567 ab,c,defg { } are σ , σ , σ , σ , σ , 12,4,3567 13,4,2567 14,2,3567 14,3,2567 14,5,2367 10 LUCASCHAFFLER σ , σ , σ , σ , σ . 34,1,2567 34,5,1267 45,1,2367 45,3,1267 45,6,1237 In each case, one can compute that σ σ = 0 using Proposition 2.4. But ab,c,defg 45,123,67 · σ σ = 1 again by Proposition 2.4, and therefore σ = σ . 12,3,4567 45,123,67 12,3,4567 ab,c,defg · 6 (cid:3) Now, an easy count tells us that there are 420 distinct equivalence classes of boundary2-strata on M . In addition, these 420 equivalence classes generate distinct rays in Eff (M ) as we 0,7 2 0,7 prove in the next proposition. Proposition 2.8. Distinct equivalence classes of boundary 2-strata on M generate distinct 0,7 rays in the cone Eff (M ). 2 0,7 Proof. We say that a boundary 2-stratum σ is of type (a,b,c) if a,b,c = I , J , K . I,J,K { } {| | | | | |} Let α and β be two distinct boundary 2-strata on M . Assume by contradiction that we can 0,7 find r R , r = 1, such that α =rβ. >0 ∈ 6 There are three cases to discuss. α and β are not of type (2,1,4). Then α2 = r2β2 = 0 by Proposition 2.5, so that • 6 r = α2/β2. Consideringallthepossiblecasesforα2andβ2,weseethatr 1/√2,√2 , p ∈ { } which cannot be because r has to be rational. Exactly one among α and β is of type (2,1,4). This is impossible because one side of the • equality α2 = r2β2 would be zero and the other not. Both α and β are of type (2,1,4). Since α = 0, we can find a boundary 2-stratum γ such • 6 that α γ = 0. According to Propositions 2.4 and 2.5, we must have that α γ = 1 and · 6 | · | β γ 0,1 . In any case, the equality α γ = r β γ gives a contradiction. | · | ∈{ } | · | | · | (cid:3) RecentworkofChenandCoskun(see[CC])showsthatthe420equivalenceclassesofboundary 2-strata on M generate extremal rays of Eff (M ). 0,7 2 0,7 Theorem 2.9 ([CC, Theorem 6.1]). Equivalence classes of boundary strata of codimension 2 on M are extremal in Eff2(M ). 0,n 0,n To conclude, the next corollary completely describes the cone V (M ) and sums up what we 2 0,7 know about Eff (M ) so far. 2 0,7 Corollary 2.10. The cone Eff (M ) has at least 420 extremal rays, which are generated by 2 0,7 the distinct equivalence classes of the boundary 2-strata on M . In particular, the closed cone 0,7 V (M ) has exactly 420 extremal rays. 2 0,7 2.5. The intersection form N (M ) N (M ) R. The real vector space N (M ) is 2 0,7 2 0,7 2 0,7 × → equipped with a symmetric bilinear form Q: N (M ) N (M ) R given by the intersection 2 0,7 2 0,7 × → between equivalence classes of 2-cycles. Since Q is nondegenerate, then Q has rank equal to dim N (M ). R 2 0,7 Proposition 2.11. dim N (M )= 127. R 2 0,7 Proof. Let K be our base field. We know that the equivalence classes of the boundary 2-strata span N (M ) in any characteristic. Moreover, the linear dependence relations between the 2 0,7 equivalence classes of the boundary 2-strata on M , only depend on the combinatorics of the 0,7