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INTERSECTION-THEORETICAL COMPUTATIONS ON Mg CAREL FABER Faculteit der Wiskunde en Informatica, Universiteit van Amsterdam Plantage Muidergracht 24, 1018 TV Amsterdam, Netherlands E-mail: [email protected]

Introduction. In this paper we explore several concrete problems, all more or less related to the intersection theory of the moduli space of (stable) curves, introduced by Mumford [Mu 1]. In Section 1 we only intersect divisors with curves. We find a collection of necessary conditions for ample divisors, but the question whether these conditions are also sufficient is very much open. The other sections are concerned with moduli spaces of curves of low genus, but we use the ring structure of the Chow ring. In Sections 2, 3 we find necessary conditions for very ample divisors on M2 and M3 . The intersection numbers of the kappa-classes are the subject of the Witten conjecture, proven by Kontsevich. In Section 4 we show how to compute these numbers for g = 3 within the framework of algebraic geometry. Finally, in Section 5 we compute λ9 on M4 . This also gives the value of λ3g−1 (for g = 4), which is relevant for counting curves of higher genus on manifolds [BCOV]. Another corollary is a different computation of the class of the Jacobian locus in the moduli space of 4-dimensional principally polarized abelian varieties; in a sense this gives also a different proof that the Schottky locus is irreducible in dimension 4. Acknowledgement. I would like to thank Gerard van der Geer for very useful discussions in connection with Section 5. This research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences. 1. Necessary conditions for ample divisors on Mg . Let g ≥ 2 be an integer and put h = [g/2]. Cornalba and Harris [C-H] determined which divisors on Mg of the form aλ − bδ are ample: this is the case if and only if a > 11b > 0. Divisors of this form are numerically effective (nef) if and only if a ≥ 11b ≥ 0. (More generally, the ample cone is the interior of the nef cone and the nef cone is the closure of the ample cone ([Ha], Ph p. 42)). Here δ = i=0 δi with δi = [∆i ] for i 6= 1 and δ1 = 21 [∆1 ]. 1991 Mathematics Subject Classification: Primary 14C15, 14H10; Secondary 14H42. The paper is in final form and no version of it will be published elsewhere. [71]

72

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Arbarello and Cornalba [A-C] proved that the h + 2 divisors λ, δ0 , δ1 , . . . , δh form for g ≥ 3 a Z-basis of Pic(Mg ) (the Picard group of the moduli functor), using the results of Harer and Mumford (we work over C). As pointed out in [C-H] it would be interesting to determine the nef cone in Pic(Mg ) for g ≥ 3. (For g = 2 the answer is given by the result of [C-H], because of the relation 10λ − δ0 − 2δ1 = 0.) In [Fa 1], Theorem 3.4, the author determined the nef cone for g = 3. The answer is: aλ − b0 δ0 − b1 δ1 is nef on M3 if and only if 2b0 ≥ b1 ≥ 0 and a − 12b0 + b1 ≥ 0. That a nef divisor necessarily satisfies these inequalities, follows from the existence of one-dimensional families of curves for which (deg λ, deg δ0 , deg δ1 ) equals (1, 12, −1) resp. (0, −2, 1) resp. (0, 0, −1). Such families are easily constructed: for the first family, take a simple elliptic pencil and attach it to a fixed one-pointed curve of genus 2; for the second family, take a 4-pointed rational curve with one point moving and attach a fixed two-pointed curve of genus 1 to two of the points and identify the two other points; for the third family, take a 4-pointed rational curve with one point moving and attach two fixed one-pointed curves of genus 1 to two of the points and identify the two other points. That a divisor on M3 satisfying the inequalities is nef, follows once we show that λ, 12λ − δ0 and 10λ − δ0 − 2δ1 are nef. It is well-known that λ is nef. Using induction on the genus one shows that 12λ − δ0 is nef: on M1,1 it vanishes; for g ≥ 2, writing Ph 12λ − δ0 = κ1 + i=1 δi one sees that 12λ − δ0 is positive on every one-dimensional family of curves where the generic fiber has at most nodes of type δ0 ; if on the other hand the generic fiber has a node of type δi for some i > 0, one partially normalizes the family along a section of such nodes and uses the induction hypothesis (cf. the proof of Proposition 3.3 in [Fa 1], which unfortunately proves the result only for g = 3). Finally, the proof that 10λ − δ0 − 2δ1 is nef on M3 is ad hoc (see the proof of Theorem 3.4 in [Fa 1]). All we do in this section is come up with a couple of one-dimensional families of stable curves for which we compute the degrees of the basic divisors. The naive hope is that at least some of these families are extremal (cf. [C-H], p. 475), but the author hastens to add that there is at present very little evidence to support this. The method of producing families is a very simple one: we start out trying to write down all the families for which the generic fiber has 3g − 4 nodes. This turns out to be a bit complicated. However, the situation greatly simplifies as soon as one realizes that the only one-dimensional moduli spaces of stable pointed curves are M0,4 and M1,1 : for the computation of the basic divisor classes on these families, one only needs to know the genera of the pointed curves attached to the moving 4-pointed rational curve resp. the moving one-pointed curve of genus 1 as well as the types of the nodes one gets in this way. In other words, the fixed parts of the families can be taken to be general. We now consider the various types of families obtained in this way and compute on each family the degrees ofPthe basic divisor classes. Each family gives a necessary h condition for the divisor aλ − i=0 bi δi to be nef. In order to write this condition, it will be convenient to define δi = δg−i and bi = bg−i for h < i < g. A) In the case of M1,1 , there is very little choice: we can only attach a (general) one-pointed curve of genus g − 1. Taking a simple elliptic pencil for the moving part, we get—as is well-known—the following degrees: deg λ = 1, deg δ0 = 12, deg δ1 = −1 and deg δi = 0 for 1 < i ≤ h. This gives the necessary condition a − 12b0 + b1 ≥ 0. B) The other families are all constructed from a 4-pointed smooth rational curve with one of the points moving and the other three fixed; when the moving point meets

INTERSECTION-THEORETICAL COMPUTATIONS ON Mg

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one of the fixed points, the curve breaks up into two 3-pointed smooth rational curves glued at one point. We have to examine the various ways of attaching general curves to this 4-pointed rational curve. E.g., one can attach one curve, necessarily 4-pointed and of genus g − 3. All nodes are of type δ0 and the 3 degenerations have an extra such node. Therefore deg δ0 = −4 + 3 = −1, while the other degrees are zero; one obtains the necessary condition b0 ≥ 0. C) Now attach a 3-pointed curve of genus i and a 1-pointed curve of genus j ≥ 1, with i + j = g − 2. One checks deg δ0 = −3 + 3 = 0 and deg δj = −1, the other degrees vanish. One obtains bj ≥ 0 for j ≥ 1. Thus all bi are non-negative for a nef divisor. R e m a r k. If one uses the families above, one simplifies the proof of Theorem 1 in [A-C] a little bit. D) If we attach two-pointed curves of genus i ≥ 1 and j ≥ 1, with i + j = g − 2, we find deg δ0 = −4 + 2 = −2 and deg δi+1 = 1. So for 2 ≤ k ≤ h we find the condition 2b0 − bk ≥ 0. E) Attaching a two-pointed curve of genus i and two one-pointed curves of genus j and k, with i, j, k ≥ 1 and i + j + k = g − 1, we find that two of the degenerations have an extra node of type δ0 while the third has an extra node of type δj+k . Therefore deg δ0 = −2 + 2 = 0. It is cumbersome to distinguish the various cases that occur for the other degrees, but is also unnecessary: one may simply write the resulting necessary condition in the form bj + bk − bj+k ≥ 0, for j, k with 1 ≤ j ≤ k and j + k ≤ g − 2. F) Attaching 4 one-pointed curves of genera i, j, k, l ≥ 1, with i + j + k + l = g, we get the necessary condition bi + bj + bk + bl − bi+j − bi+k − bi+l ≥ 0. G) If we identify two of the 4 points to each other and attach a two-pointed curve of genus g − 2 to the remaining two points, we obtain the necessary condition 2b0 − b1 ≥ 0. H) As in G), but now we attach 1-pointed curves of genera i, j ≥ 1 to the remaining two points, with i + j = g − 1. The resulting condition is bi + bj − b1 ≥ 0. The only other possibility is to identify the first with the second and the third with the fourth point. This gives a curve of genus 2, so this is irrelevant. We have proven the following theorem. Ph Theorem 1. Assume g ≥ 3. A numerically effective divisor aλ− i=0 bi δi in Pic(Mg ) satisfies the following conditions: a) a − 12b0 + b1 ≥ 0; b) for all j ≥ 1, 2b0 ≥ bj ≥ 0; c) for all j, k with 1 ≤ j ≤ k and j + k ≤ g − 1, bj + bk ≥ bj+k ; d) for all i, j, k, l with 1 ≤ i ≤ j ≤ k ≤ l and i + j + k + l = g, bi + bj + bk + bl ≥ bi+j + bi+k + bi+l . Here bi = bg−i for h < i < g, as before. The conditions in the theorem are somewhat redundant. E.g., it is easy to see that condition (c) implies the non-negativity of the bi with i ≥ 1. As we have seen, the conditions in the theorem are sufficient for g = 3. The proof proceeded by determining the extremal rays of the cone defined by the inequalities and analyzing the (three) extremal rays separately. It may therefore be of some interest to

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find (generators for) the extremal rays of the cone in the theorem. We have done this for low genus: λ 12λ − δ0 g=4: 10λ − δ0 − 2δ1 10λ − δ0 − 2δ1 − 2δ2 21λ − 2δ0 − 3δ1 − 4δ2 λ 12λ − δ0 g=5: 10λ − δ0 − 2δ1 − δ2 10λ − δ0 − 2δ1 − 2δ2 32λ − 3δ0 − 4δ1 − 6δ2 λ 12λ − δ0 10λ − δ0 − 2δ1 − 2δ2 g=6: 10λ − δ0 − 2δ1 − 2δ3 10λ − δ0 − 2δ1 − 2δ2 − 2δ3 32λ − 3δ0 − 4δ1 − 6δ2 − 6δ3 98λ − 9δ0 − 10δ1 − 16δ2 − 18δ3 Unfortunately, we have not been able to discover a general pattern. (There are 10 extremal divisors for g = 7, 20 extremal divisors for g = 8 and 21 extremal divisors for g = 9.) It is easy to see that λ, 12λ − δ0 and 10λ − 2δ + δ0 are extremal in every genus. It should be interesting to know the answer to the following question. Question. a) Is 10λ − 2δ + δ0 nef for all g ≥ 4? b) Are the conditions in the theorem sufficient? Note that an affirmative answer to the first question implies the result of [C-H] mentioned above, since 12λ − δ0 is nef. Note also that a divisor satisfying the conditions in the theorem is non-negative on every one-dimensional family of curves whose general member is smooth. This follows easily from [C-H, (4.4) and Prop. (4.7)]. (I would like to thank Maurizio Cornalba for reminding me of these results.) 2. Necessary conditions for very ample divisors on M2 . We know which divisors on M2 are ample: it is easy to see that λ and δ1 form a Z-basis of the functorial Picard group Pic(M2 ); then aλ + bδ1 is ample if and only if a > b > 0, as follows from the relation 10λ = δ0 + 2δ1 and the fact that λ and 12λ − δ0 are nef. Therefore it might be worthwhile to study which divisors are very ample on the space M2 . Suppose that D = aλ + bδ1 is a very ample divisor. Then for every k-dimensional subvariety V of M2 the intersection product Dk · [V ] is a positive integer, the degree of [V ] in the embedding of M2 determined by |D|. We work this out for the subvarieties that we know; we use Mumford’s computation [Mu 1] of the Chow ring (with Q-coefficients) of M2 . The result may be formulated as follows: A∗ (M2 ) = Q[λ, δ1 ]/(δ1 (λ + δ1 ), λ2 (5λ − δ1 )). The other piece of information we need is on p. 324 of [Mu 1]: λ3 =

1 2880 p.

However, one

INTERSECTION-THEORETICAL COMPUTATIONS ON Mg

75

should realize that the identity element in A∗ (M2 ) is [M2 ]Q = 21 [M2 ], which means that λ3 · [M2 ] =

1 . 1440

Therefore a3 + 15a2 b − 15ab2 + 5b3 . 1440 One of the requirements is therefore that the integers a and b are such that the expression above is an integer. It is not hard to see that this is the case if and only if D3 · [M2 ] =

60|a

and

12|b.

It turns out that these conditions imply that D2 · [∆0 ] and D2 · [∆1 ] are integers. Also D2 · 4λ is an integer, but D2 · 2λ is an integer if and only if 8|(a + b). Therefore, if for some integer k the class (4k + 2)λ is the fundamental class of an effective 2-cycle, then a very ample D satisfies 8|(a + b). We do not know whether such a k exists; clearly, 20λ = [∆0 ] + [∆1 ] is effective; the fundamental class of the bi-elliptic divisor turns out to be 60λ + 3∆1 . Turning next to one-dimensional subvarieties, the conditions 60|a and 12|b imply that D · [∆00 ] and D · [∆01 ] are integers as well. Proposition 2. A very ample divisor aλ + bδ1 on the moduli space M2 satisfies the following conditions: a) a, b ∈ Z and a > b > 0; b) 60|a and 12|b. Corollary 3. The degree of a projective embedding of M2 is at least 516. P r o o f. We need to determine for which a and b satisfying the conditions in the proposition the expression 5(b−a)3 +6a3 attains its minimum value. Clearly this happens exactly for b = 12 and a = 60. If 60λ + 12δ1 is very ample, the degree of M2 in the corresponding embedding is (5(b − a)3 + 6a3 )/1440 = 516. R e m a r k. It is interesting to compare the obtained necessary conditions with the explicit descriptions of M2 given by Qing Liu ([Liu]). The computations we have done (in characteristic 0) indicate that 60λ + 60δ1 maps M2 to a copy of X (loc. cit., Th´eor`eme 2), that 60λ + 36δ1 maps M2 to the blowing-up of X with center JQ (loc. cit., Corollaire 3.1) and that 60λ + 48δ1 is very ample, realizing M2 as the blowing-up of X with center the ideal generated by I43 , J10 , H62 and I42 H6 (loc. cit., Corollaire 3.2). 3. Necessary conditions for very ample divisors on M3 . In this section we compute necessary conditions for very ample divisors on the moduli space M3 . As we mentioned in Section 1, a divisor D = aλ − bδ0 − cδ1 ∈ Pic(M3 ) with a, b, c ∈ Z is ample if and only if a − 12b + c > 0 and 2b > c > 0. The necessary conditions for very ample D are obtained as in Section 2: for a k-dimensional subvariety V of M3 , the intersection product Dk · [V ] should be an integer. We use the computation of the Chow ring of M3 in [Fa 1]. The computations are more involved than in the case of genus 2; also, we know the fundamental classes of more subvarieties.

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First we look at the degree of M3 : D6 = (aλ − bδ0 − cδ1 )6 =

1 4 2 1 3 3 1 3 2 35 3 3 1 6 90720 a − 576 a c − 18 a b + 48 a bc + 3456 a c 43 2 3 13 2 4 145 5 3 2 + 58 a2 b2 c2 − 96 a bc + 512 a c + 203 20 ab − 12 ab c 31 149 4103 6 2 3 4 5 4 2 + 25 4 ab c − 48 abc + 7680 ac − 72 b + 55b c 505 3 3 65 2 4 91 5 5 6 − 18 b c + 16 b c − 384 bc + 1024 c ,

as follows from [Fa 1], p. 418. The requirement that this is in Z2 implies, firstly, that 2|c, secondly, that 2|a and 4|c, thirdly, that 2|b. Looking in Q3 we get, firstly, that 3|a, secondly, that 3|b. Modulo 5 we get 5|a or 5|(a + 3b + c). Finally, working modulo 7 we find that 7|a should hold. Writing a = 42a1 , b = 6b1 and c = 4c1 , with a1 , b1 , c1 ∈ Z, the condition D6 ·[M3 ] ∈ Z becomes 5|a1 or 5|(3a1 +2b1 +c1 ). Interestingly, unlike the case of genus 2, these conditions are not the only necessary conditions we find. For instance, the condition D5 · δ0 ∈ Z translates in 3|c1 ; then [∆1 ] = 2δ1 gives no further conditions; but the hyperelliptic locus, with fundamental class [H3 ] = 18λ − 2δ0 − 6δ1 , improves the situation modulo 5: necessarily 5|(3a1 + 2b1 + c1 ). It follows that D5 · λ is an integer, so all divisors have integer-valued degrees. In codimension 2, writing c1 = 3c2 with c2 ∈ Z, the condition D4 ·[∆01a ] ∈ Z translates in 5|a1 or 5|c2 or 5|(a1 + c2 ) or 5|(a1 + 3c2 ). The (boundary) classes [∆00 ], [∆01b ], [∆11 ], [Ξ0 ], [Ξ1 ] and [H1 ] ([Fa 1], pp. 340 sqq.) give no further conditions. In codimension 3, the class [(i)] = 8[(i)]Q forces 2|a1 . Write a1 = 2a2 with a2 ∈ Z. Somewhat surprisingly, the class [H01a ] = 4η0 (loc. cit., pp. 386, 388) gives the condition 5|(a2 + 2c2 ). Consequently, combining the various conditions modulo 5, we obtain 5|a2

and

5|b1

and

5|c2 .

Finally, we checked that the 12 cycles in codimension 4 and the 8 cycles in codimension 5 (loc. cit., pp. 346 sq.) do not give extra conditions. Proposition 4. A very ample divisor aλ − bδ0 − cδ1 on the moduli space M3 satisfies the following conditions: a) a, b, c ∈ Z with a − 12b + c > 0 and 2b > c > 0; b) 420|a and 30|b and 60|c. Corollary 5. The degree of a projective embedding of M3 is at least 650924662500 = 22 · 32 · 55 · 7 · 826571. P r o o f. We need to minimize the expression given for the degree of M3 while fulfilling the conditions in the proposition. Write a = 420A, b = 30B and c = 60C. One shows that in the cone given by 7A − 6B + C ≥ 0 and B ≥ C ≥ 0 the degree is minimal along the (extremal) ray (A, B, C) = (5x, 7x, 7x) (corresponding to 10λ − δ0 − 2δ1 ). Comparing the value for (A, B, C) = (5, 7, 7) with that for (A, B, C) = (2, 2, 1), one concludes A ≤ 5, B ≤ 7 and C ≤ 7. This leaves only a few triples in the interior of the cone; the minimum degree is obtained for (A, B, C) = (2, 2, 1), corresponding to 840λ − 60δ. R e m a r k. In [Fa 1], Questions 5.3 and 5.4, we asked whether the classes X (resp. Y ) are multiples of classes of complete subvarieties of M3 of dimension 4 (resp. 3) having

77

INTERSECTION-THEORETICAL COMPUTATIONS ON Mg

empty intersection with ∆1 (resp. ∆0 ). We still do not know the answers, but we verified that X and −Y = 504λ3 are effective: X= −Y =

1 11 3 48 40 1 15 δ00 + 6 δ01a + 15 δ01b + 8δ11 + 14 ξ0 + 35 ξ1 + 21 η1 ; 1 11 2 2 [(a)]Q + [(b)]Q + [(c)]Q + 30 [(d)]Q + 5 [(f )]Q + 2[(g)]Q

+ 23 η0 .

(For the notation, see [Fa 1], pp. 343, 386, 388.) 4. Algebro-geometric calculation of the intersection numbers of the tautological classes on M3 . Here we show how to compute the intersection numbers of the classes κi (1 ≤ i ≤ 6) on M3 in an algebro-geometric setting. These calculations were done originally in May 1990 to check the genus 3 case of Witten’s conjecture [Wi], now proven by Kontsevich [Ko]. We believe that there is still an interest, though, in finding methods within algebraic geometry that allow to compute the intersection numbers of the kappa- or tau-classes. For instance, the identity 1 K 3g−2 = hτ3g−2 i = hκ3g−3 i = (24)g · g! (in cohomology) should be understood ([Wi], between (2.26) and (2.27)). In [Fa 1] the 4 intersection numbers of κ1 and κ2 were computed; using the identity κ1 = 12λ − δ0 − δ1 , we can read these off from Table 10 on p. 418: κ61 =

176557 107520 ,

κ41 κ2 =

75899 322560 ,

κ21 κ22 =

32941 967680 ,

κ32 =

14507 2903040 .

To compute the other intersection numbers, we need to express the other kappa-classes in terms of the bases introduced in [Fa 1]. The set-up is as in [Mu 1], §8 (and §6): if C is a stable curve of genus 3, ωC is generated by its global sections, unless a) C has 1 or 2 nodes of type δ1 , in which case the global sections generate the subsheaf of ωC vanishing in these nodes; b) C has 3 nodes of type δ1 , i.e., C is a P1 with 3 (possibly singular) elliptic tails, in which case Γ(ωC ) generates the subsheaf of ωC of sections vanishing on the P1 . (See [Mu 1], p. 308.) Let Z ⊂ C 3 be the closure of the locus of pointed curves with 3 nodes of type δ1 and with the point lying on the P1 . Working over C 3 − Z we get 0 → F → π ∗ π∗ ωC 3 /M3 → I∆∗1 · ωC 3 /M3 → 0 with F locally free of rank 2. Working this out as in [Fa 1], p. 367 we get 0 = c3 (F) = π ∗ λ3 − K · π ∗ λ2 + K 2 · π ∗ λ1 − K 3 − (π ∗ λ1 − K) · [∆∗1 ]Q + i1,∗ (K1 + K2 ) modulo [Z]. Multiplying this with K and using that ω 2 is trivial on [∆∗1 ], we get (1)

0 = K · c3 (F) = K · π ∗ λ3 − K 2 · π ∗ λ2 + K 3 · π ∗ λ1 − K 4 + ∗K · [Z].

It is easy to see that K 2 · [Z] = 0, so we also get (2)

0 = K 2 · π ∗ λ3 − K 3 · π ∗ λ2 + K 4 · π ∗ λ1 − K 5 ,

(3)

0 = K 3 · π ∗ λ3 − K 4 · π ∗ λ2 + K 5 · π ∗ λ1 − K 6 ,

(4)

0 = K 4 · π ∗ λ3 − K 5 · π ∗ λ2 + K 6 · π ∗ λ1 − K 7 .

Pushing-down to M3 we get (10 ) 0

(2 )

0 = 4λ3 − κ1 λ2 + κ2 λ1 − κ3 + N · [(i)]Q , 0 = κ1 λ3 − κ2 λ2 + κ3 λ1 − κ4 ,

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C. F. FABER

(30 )

0 = κ2 λ3 − κ3 λ2 + κ4 λ1 − κ5 ,

0

0 = κ3 λ3 − κ4 λ2 + κ5 λ1 − κ6 .

(4 ) 0

To get κ3 from (1 ) we use two things. Firstly, one computes Y = −504λ3 , as mentioned at the end of Section 3. This follows since both Y and λ3 are in the one-dimensional subspace of A3 (M3 ) of classes vanishing on all subvarieties of ∆0 . The factor −504 is computed using λ4 = 8λλ3 or λ3 · [(i)]Q = 61 λ3 · [(i)]Q . Secondly, to compute N , one uses that κ3 vanishes on the classes [(b)]Q , [(c)]Q , [(f )]Q , [(g)]Q , [(h)]Q and [(i)]Q . This gives 6 relations in N of which 3 are identically zero; the other 3 all imply N = 1. The formulas above allow one to express the kappa-classes in terms of the bases of the Chow groups given in [Fa 1]. We give the formula for κ3 (from which the other formulas follow): κ3 =

31 19 1 1 1 280 [(a)]Q + 840 [(b)]Q + 420 [(c)]Q + 1260 [(d)]Q + 35 [(e)]Q 19 11 93 11 + 840 [(f )]Q + 29 84 [(g)]Q + 35 [(h)]Q + 35 [(i)]Q + 252 η0 .

This gives the following intersection numbers: κ31 κ3 =

4073 161280 ,

κ1 κ2 κ3 =

κ2 κ4 =

971 2903040 ,

149 40320 ,

κ1 κ5 =

κ23 = 1 5760 ,

131 322560 ,

κ6 =

κ21 κ4 =

2173 967680 ,

1 82944 .

5. A few intersection numbers in genus 4. Kontsevich’s proof of Witten’s conjecture enables one to compute the intersection numbers of the kappa-classes on the moduli space of stable curves of arbitrary genus. There are many more intersection numbers that one would like to know, see e.g. [BCOV], (5.54) and end of Appendix A. As a challenge, we pose the following problem: Problem. Find an algorithm that computes the intersection numbers of the divisor classes λ, δ0 , δ1 , . . . , δ[g/2] on Mg . These numbers are known for g = 2 [Mu 1] and g = 3 [Fa 1]. Note that the problem includes the computation of κ3g−3 . 1 Proposition 6. Denote by hg the intersection number λ2g−1 · [Hg ]Q , where Hg is the closure in Mg of the hyperelliptic locus. Then h1 =

1 ; 96

hg =

g−1 2g − 2 2 X hi hg−i i(i + 1)(g − i)(g − i + 1) 2i − 1 2g + 1 i=1

for

g ≥ 2.

P r o o f. This follows from [C-H], Proposition 4.7, which expresses λ on Hg in terms of the classes of the components of the boundary Hg − Hg . It is easy to see that λ2g−2 ξi = 0 for 0 ≤ i ≤ [(g − 1)/2]. Also, 2g − 2 2g−2 λ δj [Hg ]Q = (2j + 2)(2g − 2j + 2) hj hg−j , 2j − 1 1 ∗ because λ = πj∗ λ + πg−j λ on ∆j ∩ Hg . Normalizing h1 to 96 , which reflects the identity 1 λ = 24 p on M1,1 and the fact that an elliptic curve has four 2-torsion points, we get the formula.

INTERSECTION-THEORETICAL COMPUTATIONS ON Mg

79

1 1 31 This gives for instance h2 = 2880 , h3 = 10080 and h4 = 362880 . So this already gives 6 3 the value of λ on M2 , and the value of λ on M3 follows very easily: we only need that 1 [H3 ]Q = 9λ in A1 (M3 ), because clearly λ5 δ0 = λ5 δ1 = 0. We get λ6 = 90720 .

Proposition 7. λ9 =

1 113400

on M4 .

P r o o f. We need to know the class [H4 ] modulo the kernel in A2 (M4 ) of multiplication with λ7 . We computed this class using the test surfaces of [Fa 2]; of the 14 classes at the bottom of p. 432, only κ2 , λ2 and δ12 are not in the kernel of ·λ7 , and the result is: 27 2 5 δ1

(mod ker(·λ7 )).

60κ2 − 810λ2 + 24δ12 ≡ 0

(mod ker(·λ7 )).

[H4 ] ≡ 3κ2 − 15λ2 + We also have the relation ([Fa 2], p. 440) Thus [H4 ] ≡

51 2 2 λ

+

21 2 5 δ1 .

We compute λ7 δ12 = 71 (λ · [M1,1 ]Q )(λ6 · (−KM3,1 /M3 ) · [M3,1 ]) −4 1 24 · 90720 −1 77760 .

=7· = Therefore

λ9 =

2 51 (2

·

31 362880

+

21 5

·

1 77760 )

=

1 113400 .

Also λ7 κ2 =

169 1360800 .

The hardest part of this proof is the computation of (three of) the coefficients of the class [H4 ]. We present the test surfaces we need to compute these coefficients. Write [H4 ] = 3κ2 − 15λ2 + cλδ0 + dλδ1 + eδ02 + f δ0 δ1 + gδ0 δ2 + hδ12 + iδ1 δ2 + jδ22 + kδ00 + lδ01a + mγ1 + nδ11 . The class [H4 ] ∈ A2 (M4 ) was computed by Mumford ([Mu 1], p. 314). a) Take test surface (α) from [Fa 2], p. 433: two curves of genus 2 attached in one point; on both curves the point varies. We have [H4 ]Q = 6 · 6 = 36 and δ22 = 8. Thus j = 9. b) Test surface (ζ): curves of type δ12 , vary the elliptic tail and the point on the curve of genus 2. We have [H4 ] = 0, δ0 δ2 = −24 and δ1 δ2 = 2. Thus i = 12g. c) Test surface (µ): curves of type δ02 , vary the elliptic curve in a simple pencil with 3 disjoint sections and vary the point on the curve of genus 2. Then δ0 δ2 = −20 and δ22 = 4. To compute [H4 ] we use a trick. Consider the pencil of curves of genus 3 which we get by replacing the one-pointed curve of genus 2 with a fixed one-pointed curve of genus 1. On that pencil λ = 1, δ0 = 12 − 1 − 1 = 10, δ1 = −1, thus [H3 ]Q = 9λ − δ0 − 3δ1 = 2. So on the test surface we get [H4 ]Q = 2 · 6 = 12. Therefore −20g + 36 = 24 so g = 35 and i = 36 5 . d) This test surface is taken from [Fa 3], pp. 72 sq. We take the universal curve over a pencil of curves of genus 2 as in [A-C], p. 155, and we attach a fixed one-pointed curve of genus 2. As in [Fa 3] we have λ = 3(G − Σ), δ0 = 30(G − Σ), δ2 = −2G + Σ. Since G2 = 2, GΣ = 0 and Σ2 = −2 we have δ0 δ2 = −60 and δ22 = 6. To compute κ2 we use the same trick as above: replacing the fixed one-pointed curve of genus 2 by one of genus 1, we get a test surface of curves of genus 3. This will not affect the

80

C. F. FABER

computation of κ2 ; using the formulas of [Fa 1] we find κ2 = 6. Also δ0 Σ = 2γ1 here, thus γ1 = 30. Since [H4 ] = 0, we get 0 = 18 − 60g + 6j + 30m = 30m + 36 so m = − 56 . e) Test surface (λ) from [Fa 2]: curves of type δ12 , vary both the j-invariant of the middle elliptic curve and the (second) point on it. We have δ0 δ2 = −12, δ1 δ2 = 1, δ22 = 1, κ2 = 1, δ01a = 12 and γ1 = 12. Since [H4 ] = 0, we get 0 = 3−12g +i+j +12l +12m = 1 12l − 12 5 so l = 5 . f) Test surface (κ): curves of type δ12 , vary a point on the middle elliptic curve and vary the elliptic tail. Then δ0 δ2 = −12, δ1 δ2 = 1, δ01a = −12, δ11 = −1. Since [H4 ] = 0, we find 0 = −12g + i − 12l − n so n = − 12 5 . g) The final test surface we need is (γ) from [Fa 2]: we attach fixed elliptic tails to two varying points on a curve of genus 2. Then δ12 = 16, δ22 = −2, κ2 = 2, δ11 = 6. When the two varying points are distinct Weierstrass points, we get hyperelliptic curves. So 27 [H4 ]Q = 6 · 5 = 30 and we get 60 = 6 + 16h − 2j + 6n = 16h − 132 5 so h = 5 , as claimed. This finishes the proof of Proposition 7. We can now evaluate the contribution from the constant maps for g = 4 (cf. [BCOV], §5.13, (5.54)): Corollary 8. λ33 =

1 43545600

on M4 .

P r o o f. As explained in [Mu 1], §5, we have on M4 the identity (∗)

(1 + λ1 + λ2 + λ3 + λ4 )(1 − λ1 + λ2 − λ3 + λ4 ) = 1.

One checks that this implies λ33 =

1 9 384 λ1 ,

which finishes the proof.

Corollary 9 (Schottky, Igusa). The class of M4 in A4 equals 8λ. P r o o f. Since (∗) holds also on the toroidal compactification Ae4 , we get λ10 1 = 384λ1 λ33 = 768λ1 λ2 λ3 λ4 . But it follows from Hirzebruch’s proportionality theorem [Hi 1, 2] that 4 Y |B2i | 1 λ1 λ2 λ3 λ4 = = , 4i 1393459200 i=1 1 hence λ10 = 1814400 on Ae4 . Using Theorem 1.5 in [Mu 2] we see that the class of M4 in A4 is a multiple of λ. Denote by t : M4 → A4 the Torelli morphism and denote by 1 J4 its image, the locus of Jacobians. Proposition 7 tells us that t∗ λ9 = 113400 . Applying 1 9 t∗ we get [J4 ] · λ = 113400 , hence [J4 ] = 16λ, hence [J4 ]Q = 8λ, as claimed. (The subtlety corresponding to the fact that a general curve of genus g ≥ 3 has only the trivial automorphism, while its Jacobian has two automorphisms, appears also in computing 1 λ6 on M3 resp. on Ae3 : we have already seen that t∗ λ6 = 90720 ; applying t∗ we get 1 1 6 6 e [J3 ] · λ = 90720 ; since [J3 ] = 2[A3 ]Q , we get λ = 181440 , which is also what one gets using the proportionality theorem.)

INTERSECTION-THEORETICAL COMPUTATIONS ON Mg

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References [A-C] [BCOV]

[C-H]

[Fa 1] [Fa 2] [Fa 3]

[Ha] [Hi 1]

[Hi 2]

[Ko] [Liu] [Mu 1]

[Mu 2]

[Wi]

E. A r b a r e l l o and M. C o r n a l b a, The Picard groups of the moduli spaces of curves, Topology 26 (1987), 153–171. M. B e r s h a d s k y, S. C e c o t t i, H. O o g u r i and C. V a f a, Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Comm. Math. Phys. 165 (1994), 311–428. M. C o r n a l b a and J. H a r r i s, Divisor classes associated to families of stable varieties, ´ with applications to the moduli space of curves, Ann. Scient. Ecole Norm. Sup. (4) 21 (1988), 455–475. C. F a b e r, Chow rings of moduli spaces of curves I: The Chow ring of M3 , Ann. of Math. 132 (1990), 331–419. C. F a b e r, Chow rings of moduli spaces of curves II: Some results on the Chow ring of M4 , Ann. of Math. 132 (1990), 421–449. C. F a b e r, Some results on the codimension-two Chow group of the moduli space of curves, in: Algebraic Curves and Projective Geometry (eds. E. Ballico and C. Ciliberto), Lecture Notes in Math. 1389, Springer, Berlin, 1988, 66–75. R. H a r t s h o r n e, Ample Subvarieties of Algebraic Varieties, Lecture Notes in Math. 156, Springer, Berlin, 1970. F. H i r z e b r u c h, Automorphe Formen und der Satz von Riemann-Roch, in: Symposium Internacional de Topolog´ıa Algebraica (M´exico 1956), Universidad Nacional Aut´ onoma de M´exico and UNESCO, Mexico City, 1958, 129–144; or: Gesammelte Abhandlungen, Band I, Springer, Berlin, 1987, 345–360. F. H i r z e b r u c h, Characteristic numbers of homogeneous domains, in: Seminars on analytic functions, vol. II, IAS, Princeton 1957, 92–104; or: Gesammelte Abhandlungen, Band I, Springer, Berlin, 1987, 361–366. M. K o n t s e v i c h, Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function, Comm. Math. Phys. 147 (1992), 1–23. Qing L i u, Courbes stables de genre 2 et leur sch´ema de modules, Math. Ann. 295 (1993), 201–222. D. M u m f o r d, Towards an enumerative geometry of the moduli space of curves, in: Arithmetic and Geometry II (eds. M. Artin and J. Tate), Progr. Math. 36 (1983), Birkh¨ auser, 271–328. D. M u m f o r d, On the Kodaira Dimension of the Siegel Modular Variety, in: Algebraic Geometry—Open Problems (eds. C. Ciliberto, F. Ghione and F. Orecchia), Lecture Notes in Math. 997, Springer, Berlin, 1983, 348–375. E. W i t t e n, Two dimensional gravity and intersection theory on moduli space, in: Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 243–310.

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