EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES

Transcription

1 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES GAVRIL FARKAS ABSTRACT. We discuss the transition from uniruledness to being of general type for moduli spaces of curves and Hurwitz spaces of covers. First, we construct and calculate the class of a virtual divisor on the moduli space M 23 of stable curves of genus 23, having slope less than that of the canonical divisor K M23. Furthermore, we calculate the class of a series of virtual divisors on the moduli space M 2s 2 +s+1, with s 3, having slope less that that of the Brill-Noether divisors. Combined with the recent proofs of the Strong Maximal Rank Conjecture obtained independently by Liu, Osserman, Teixidor, Zhang and by Jansen and Payne respectively, this completes the proof that both moduli spaces M 22 and M 23 are varieties of general type. Finally, we prove the effectiveness of the canonical bundle of several Hurwitz spaces H g,k of degree k admissible covers of P 1 from curves of genus 14 g 19. Following a principle due to Mumford, most moduli spaces that appear in algebraic geometry classifying curves, abelian varieties, K3 surfaces are of general type, with a finite number of exceptions, which are unirational, or at least uniruled. Understanding the transition from negative Kodaira dimension to being of general type is usually quite difficult. With one exception the moduli space of spin curves [FV], for all these moduli spaces there are notorious open cases, when the Kodaira dimension is not known. The aim of this paper is to shed some light on this change of the birational nature of two highly related moduli spaces of curves. In a series of landmark papers [HM], [H], [EH2] published in the 1980s, Harris, Mumford and Eisenbud proved that M g is a variety of general type for g > 23. This contrasts with the classical result of Severi [Sev] that M g is unirational for g 10 see [AC] for a beautiful modern treatment and with the more recent results of Chang-Ran [CR1], [CR2], [CR3], Sernesi [Ser], Verra [Ve] and Schreyer [Sch], which summarized, amount to the statement that M g is uniruled for g 16. The Slope Conjecture of Harris and Morrison [HMo] predicted that the Brill-Noether divisors are the effective divisors on M g having minimal slope g+1. This led people to expect that M g changes from uniruledness to being of general type precisely at genus g = 23. However the Slope Conjecture turned out to be false and there are instances of effective divisors on M g for infinitely many genera g 10 having slope less than g+1, see [FP], [F2], [F3], [Kh], [FR]. In view of these examples it is to be expected that there should be an effective divisor of slope less than 13 2 = on M 23 as well, which would imply that M 23 is of general type. The best known result on M 23 is the statement κm 23 2, proven in [F1] via a study of the relative position of the three Brill-Noether divisors on M 23. Effective divisors on M g One central aim of this paper is to address this transition case and reduce the calculation of the Kodaira dimension of M 23 to a transversality statement for generic 1

2 2 G. FARKAS curves of genus 23. In two very recent breakthrough papers, this transversality statement, also known as the Strong Maximal Rank Conjecture has been established independently by Liu, Osserman, Teixidor, Zhang [LOTZ1] and Jensen, Payne [JP2] respectively. We begin by describing our construction of an effective divisor on M 23. By Brill- Noether theory, a general curve C of genus 23 carries a two dimensional family of linear series L W 6 26 C, all satisfying h1 C, L = 3. Each of these linear series is complete and very ample. Consider the multiplication map φ L : Sym 2 H 0 C, L H 0 C, L 2. By Riemann-Roch h 0 C, L 2 = 30, whereas dim Sym 2 H 0 C, L = 28. Imposing the condition that φ L be non-injective, one expects a codimension 3 locus inside the parameter space of pairs [C, L]. Since this parameter space has 2-dimensional fibres over M 23, by projection, one expects a divisor inside the moduli space M 23. Theorem 0.1. The following locus consisting of curves of genus 23 { } D := [C] M 23 : L W26C 6 with Sym 2 H 0 C, L φ L H 0 C, L 2 not injective is a virtual divisor on M 23. The virtual class of its compactification inside M 23 equals [ D] virt = λ δ δ where b j b 1 for j 2. In particular, s [ D] virt = = < j=2 b j δ j CH 1 M 23, For the precise definition of the virtual class [ D] virt and which coincides with the class of the closure of D inside a partial compactification of M 23, we refer to Definition 1.3. Recall that the slope of an effective divisor D on the moduli space M g not containing any boundary divisor i in its support is defined as the quantity sd := a inf i 0 b i, where [D] = aλ g 2 i=0 b iδ i CH 1 M g, with a, b i 0. The canonical class of the moduli space M g can be expressed as K Mg = 13λ 2 g 2 i=0 δ i δ 1, see [HM]. Since s [ D] virt < sk Mg = 13 2, Theorem 0.1 implies the following conditional result on the Kodaira dimension of M 23 : Corollary 0.2. Assuming D is a genuine divisor on M 23, that is, the multiplication map φ L : Sym 2 H 0 C, L H 0 C, L 2 is injective for every linear system L W 6 26 C on a general curve C of genus 23, then M 23 is a variety of general type. The construction of D is inspired by results in [FP], where it is established that any divisor on M g having slope less that g+1 must necessarily contain the locus K g M g of curves lying on a K3 surface. Finding geometric divisors on M g which verify this condition has proven to be difficult, for curves on K3 are notorious for behaving generically with respect to most interesting geometric properties, like those of Brill- Noether nature. A construction similar to that in Theorem 0.1 can be carried out for

3 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 3 infinitely many genera using linear series having Brill-Noether number equal to 1. We fix s 2 and set g := 2s 2 + s + 1, d := 2s 2 + 2s + 1 and r := 2s. A general curve [C] M 2s 2 +s+1 carries a one dimensional family of linear systems L W2s 2s 2 +2s+1 C. For each such L we consider the multiplication map φ L : Sym 2 H 0 C, L H 0 C, L 2. Observe that h 0 C, L 2 Sym 2 H 0 C, L = 2d + 1 g s + 12s + 1 = 1, therefore, the condition that the map φ L be non-injective is expected to lead to a codimension two locus in the parameter space of pairs [C, L], where L W 2s 2s 2 +2s+1 C. Theorem 0.3. Fix s 2. The following subvariety inside M 2s 2 +s+1 { } D s := [C] M 2s 2 +s+1 : L W2s 2s 2 +2s+1 C with φ L not injective is a virtual divisor on M 2s 2 +s+1. The slope of the virtual class of its compactification D s inside M 2s 2 +s+1 equals s [ D s ] virt = 348s8 56s s 6 90s s s s s s 8 28s s 6 5s s s s s g s 6 140s 5 162s s s s s 2 + s + 124s 8 28s s 6 5s s s s s In particular, s [ D s ] virt < g+1, for all s 3. The Brill-Noether divisors M d g,r of curves [C] M g such that Wd r C, where ρg, r, d := g r + 1g d + r = 1, have slope g+1, see [HM] and [EH2]. Theorem 0.3 shows that if, as expected, D s is a genuine divisor on M 2s 2 +s+1, its slope is smaller than that of g+1 for all s 3. For s = 2, applying Theorem 0.3 we obtain the divisor D 2 on M 11, which has already been interpreted in [BF] and earlier in [FO], as the failure locus of the Mercat Conjecture for curves of genus 11. Precisely, D 2 consists of curves [C] M 11 carrying a stable rank two vector bundle E with h 0 C, E 4 and dete W13 4 C. In this case D 2 is known to be a genuine divisor on M 11 and Theorem 0.3 yields s [ D 2 ] = 7, which agrees with the conclusions of [BF] Theorem 2. The case s = 3 of Theorem 0.3 leads to the virtual divisor D 3 on M 22. In this case s [ D 3 ] = = < 6.5, thus if D 3 is a genuine divisor, then M 22 is of general type. The calculation of the class [ D 3 ] virt, that is, the case s = 3 of Theorem 0.3 has already been carried out in [F4] Theorem 7.1. In loc.cit. it was also announced that D 3 is an effective divisor on M 22, but unfortunately the intended proof has not materialized. One can generalize the construction of the virtual divisors D s and that of D and consider for every a 1 and s 2, the locus D s,a of curves [C] M 2s 2 +s+a such that there exists a linear system L W2s 2s 2 +2s+a C with the multiplication map φ L : Sym 2 H 0 C, L H 0 C, L 2 being non-injective. A parameter count shows that D s,a is indeed a virtual divisor. In this way, D s,1 = D s, whereas setting s = 3 and a = 2, we recover the divisor D = D 3,2 on M 23. However, finding a closed formula for the virtual class [ D s,a ] virt of the closure of D s,a in M 2s 2 +s+a is a daunting task. One =

4 4 G. FARKAS stumbling block lies in generalizing Proposition 1.8 and 1.9 to this setting, coupled with the major difficulty that the virtual class in question is the push-forward to M 2s 2 +s+a of a class having codimension a + 1 that is, arbitrarily high in the moduli space of pairs [C, L], when the higher Chow rings of these moduli spaces are poorly understood. The Strong Maximal Rank Conjecture The Maximal Rank Conjecture, originally due to Eisenbud and Harris, predicts that for a pair [C, L], where C is a general curve of genus g and L Wd r C is a general linear system, the multiplication of global sections φ L : Sym 2 H 0 C, L H 0 C, L 2 is of maximal rank. The conjecture has been the focus of much attention, both a couple of decades ago [BE] using embedded degenerations in projective space, as well as recently in [JP1] using tropical geometry, or in [LOTZ1] using limit linear series. A full solution of the Maximal Rank Conjecture, considering the multiplication maps φ k L : Symk H 0 C, L H 0 C, L k for all k 2, has been recently found by Larson [La]. A refined version of the Maximal Rank Conjecture, taking into account every linear series L Wd r C on a general curve rather than the general one, has been put forward in [AF] Conjecture 5.4. The Strong Maximal Rank Conjecture, motivated by applications to the birational geometry of M g, predicts that for a general curve C of genus g and for positive integers r, d such that 0 ρg, r, d r 2, the determinantal variety { } L Wd r C : φ L : Sym 2 H 0 C, L H 0 C, L 2 is not of maximal rank has the expected dimension. The Strong Maximal Rank Conjecture in the cases g = 23, d = 26 and r = 6 and respectively g = 22, d = 25, r = 6 amounts to the statement that the virtual divisors D on M 23 discussed in Theorem 0.1, respectively D 3 on M 22 are genuine divisors. These two important cases of the Strong Maximal Conjecture have been recently proved by Liu, Osserman, Teixidor, Zhang [LOTZ2] and independently by Payne and Jensen [JP2]. Putting together Theorem 0.1, Theorem 0.3 in the case s = 3 and the new results [LOTZ2] and [JP2], one can conclude: Theorem 0.4. The moduli spaces M 22 and M 23 are both of general type. Other interesting cases of the Strong Maximal Rank Conjecture for line bundles L with h 1 C, L = 2 have been proved to be equivalent to Mercat s Conjecture for rank 2 vector bundles on a generic curve and shown to hold in [BF] using K3 surfaces. One can formulate a Strong Maximal Rank Conjecture for the maps φ k L as well, though I am unaware of applications to moduli when k 3. The Kodaira dimension of Hurwitz spaces The second goal of this paper is to discuss the transition from unirationality to being of general type in the case of the Hurwitz space H g,k classifying degree k covers C P 1 with only simple ramification, from a smooth curve C of genus g. Hurwitz spaces provide an interesting bridge between the more accessible moduli spaces of pointed rational curves and the moduli space of curves. We denote by H g,k the moduli space of admissible covers constructed by Harris and Mumford [HM] and studied further in [ACV]. It comes equipped with a finite branch map b : H g,k M 0,2g+2k 2 /S 2g+2k 2,

5 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 5 where the target is the moduli space of unordered 2g + 2k 2-pointed stable rational curves, as well as with a map σ : H g,k M g, obtained by assigning to an admissible cover the stable model of its source. It is a fundamental question to describe the birational nature of H g,k and one knows much less than in the case of other moduli spaces like M g or ca g. From Brill- Noether theory it follows that when k g+2 2, every curve of genus g can be represented as a k-sheeted cover of P 1, that is, σ : H g,k M g is dominant, and thus H g,k is of general type whenever M g is. Classically it has been known that H g,k is unirational for k 5, see again [AC] and references therein. Geiss [G] using liaison techniques showed that most Hurwitz spaces H g,6 with g 45 are unirational. Schreyer and Tanturri [ST] put forward the hypothesis that there exist only finitely many pairs g, k, with k 6, such that H g,k is not of general type. We are particularly interested in the range k g+2 2 when H g,k dominates M g and 14 g 19, so that M g is either unirational/uniruled for g = 14, 15, 16, or its Kodaira dimension is unknown, when g = 17, 18, 19. We summarize our results showing the positivity of the canonical bundle of H g,k and we begin with the cases when M g is known to be uniruled. Theorem 0.5. Suppose H is one of the spaces H 14,9 or H 16,9. Then there exists an effective Q-divisor class E on H contracted under the map σ : H M g, such that the twisted canonical class K H + E is effective. A few comments are in order. Firstly, the boundary divisor E that may be empty is supported on the locus i 1 σ i of admissible covers with source curve being of compact type. Secondly, Theorem 0.5 is optimal. In genus 14, Verra [Ve] established the unirationality of the Hurwitz space H 14,8, which is a finite cover of M 14, and concluded in this way that M 14 itself is unirational. In genus 16, the map σ : H 16,9 M 16 is generically finite. Chang and Ran [CR3] showed that M 16 is uniruled more precisely, they proved that K M16 is not pseudo-effective, which by now, one knows that it implies uniruledness. This is the highest genus for which M g is known to be uniruled. Next, we consider the range when the Kodaira dimension of M g is unknown. Theorem 0.6. Suppose H is one of the spaces H 17,11 or H 19,13. Then there exists an effective Q-divisor class E on H contracted under the map σ : H M g, such that the twisted canonical class K H + E is big. We observe that the maps H 17,11 M 17 and H 19,13 M 19 have generically 3 and respectively 5-dimensional fibres. As in Theorem 0.5, the divisor E is supported on the divisor i 1 σ i. Both Theorems 0.5 and 0.6 are proven at the level of a partial compactification G 1 g,k described in detail in Section 3 and which incorporates only admissible covers with a source curve whose stable model is irreducible. In each relevant genus we produce an explicit effective divisor D g on G 1 g,k such that the canonical class K G1 g,k can be expressed as a positive combination of a multiple of [ D g ] and the pull-back under the map σ of an effective divisor on M g.

6 6 G. FARKAS We describe the construction of these divisors in two instances and refer to Section 3 for the remaining cases and further details. First we consider the space H 16,9 which is a generically finite cover of M 16. Let us choose a general pair [C, A] H 16,9. We set L := K C A W21 7 C. Denoting by I C,L k := Ker { Sym k H 0 C, L H 0 C, L k }, by Riemann-Roch one computes that dim I C,L 2 = 9 and dim I C,L 3 = 72. Therefore the locus where there exists a non-trivial syzygy, that is, the map µ C,L : I C,L 2 H 0 C, L I C,L 3 is not an isomorphism is expected to be a divisor on H 16,9. This indeed is the case and we define the syzygy divisor D 16 := {[C, A] H 16,9 : I C,KC A 2 H0 C, K C A } I C,KC A 3. The ramification divisor of the map σ : H 16,9 M 16, viewed as the Gieseker-Petri divisor { GP := [C, A] H 16,9 : H 0 C, A H 0 C, K C A } H 0 C, K C also plays an important role. After computing the class of the closure D 16 of the Koszul divisor D 16 inside the partial compactification G 16,9 1, we find the following explicit representative for the canonical class 1 K G1 16,9 = 2 5 [ D 16 ] [ GP] CH 1 G 1 16,9. This proves Theorem 0.5 in the case g = 16. We may wonder whether 1 is the only effective representative of K G1 16,9, which would imply that the Kodaira dimension of H 16,9 is equal to zero. To summarize, since the smallest Hurwitz cover of M 16 has an effective canonical class, it seems unlikely that the method of establishing the uniruledness/unirationality of M 14 and M 15 by studying a Hurwitz space covering it can be extended to higher genera g 17. The last case we discuss in this introduction is g = 17 and k = 11. We choose a general pair [C, A] H 17,11. The residual linear system L := K C A W 6 21 C induces an embedding C P 6 of degree 21. The multiplication map φ L : Sym 2 H 0 C, L H 0 C, L 2 has a 2-dimensional kernel. We impose the condition that the pencil of quadrics containing the image curve C L P 6 be degenerate, that is, it intersects the discriminant divisor in P Sym 2 H 0 C, L non-transversally. We thus define the locus { D 17 := [C, A] H 17,11 : P } Kerφ KC A is a degerate pencil. Using [FR], we can compute the class [ D 17 ] of the closure of D 17 inside G 17,11. Comparing this class to that of the canonical divisor, we obtain the relation 2 K G1 17,11 = 1 5 [ D 17 ] σ 7λ δ 0 CH 1 G 1 17,11. Since the class 7λ δ 0 can be shown to be big on M 17, the conclusion of Theorem 0.6 in the case g = 17 now follows.

7 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 7 1. THE CONSTRUCTION OF VIRTUAL DIVISORS ON M g The virtual divisor D is constructed as the push-forward of a codimension 3 cycle inside the universal parameter space of linear series of type g 6 26 on curves of genus 23. We describe the construction of this cycle, then extend this determinantal structure over a partial compactification of M 23. This will suffice to compute the virtual slope to be defined below of D and eventually prove Theorem 0.1. We shall follow the same procedure in the case of the divisors D s on M 2s 2 +s+1, where s 2. As long as the two constructions run parallel, we will treat both Theorems 0.1 and 0.3 unitarily. In what follows we shall be in one of the following numerical situations 3 g = 23, r = 6, d = 26, or 4 g = 2s 2 + s + 1, r = 2s, d = 2s 2 + 2s + 1, where s 2. Note that ρg, r, d = 2 in case 3 and ρg, r, d = 1 in case 4. We denote by M part g the open substack of M g of curves C satisfying both properties Wd 1 r C = and C =. Basic results in Brill-Noether theory imply codimm g M part g, M g 2. W r+1 d C is pure 2-dimensional in case 3 C is complete In particular, for every [C] M part g, the variety Wd r respectively 1-dimensional in case 4, and each linear system L Wd r and base point free. Let p 1 1 M g be the locus of curves [C y E], where E is an arbitrary elliptic curve and C is a smooth curve of genus g 1 that verifies the Brill-Noether Theorem. The point of attachment y C is chosen arbitrarily. Furthermore, let p 0 0 M g be the locus consisting of curves [C yq := C/y q] 0, where [C, q] is a Brill-Noether general curve of genus g 1 in the pointed sense of [EH2] and y C is an arbitrary point, together with their degenerations [C q E ], where E is a rational nodal curve that is, E is a nodal elliptic curve and je =. Observe that points of this form describe the intersection of p 0 and p 1. We introduce the partial compactification M part g := M part g p 0 p 1 M g. Note that M part g and M g 0 1 are isomorphic in codimension one, in particular, we identify their Picard groups. We have that Pic Q M part g = CHQ 1 Mpart g = Q λ, δ 0, δ 1, where λ is the Hodge class, δ 0 := [ p 0 ] and δ 1 := [ p 1 ]. We consider the following desingularized universal Picard variety of degree d whose properties we describe. h : Pic d g M part g, i Over a smooth curve [C] M part g, we have h 1 [C] = Pic d C. ii Over a curve [C E] p 1, we have h 1 [C E] = Pic d C Pic 0 E. iii If [C yq ] p 0, with ν : C C yq being the normalization map, then h 1 [C yq ] = Pic d C yq := P P y P q is the desingularization of the compactified Jacobian Pic d C yq of rank one torsion-free sheaves on C yq. Here, P denotes a Poincaré bundle on C Pic d C and P y respectively P q denote the restrictions of P to {y} Pic d C, respectively {q} Pic d C. Thus a point

8 8 G. FARKAS in Pic d C yq can be though of a pair L, Q, where L is a line bundle of degree d on C and L y L q Q is a 1-dimensional quotient. The map Pic d C yq Pic d C yq assigns to a pair L, Q the sheaf L on C yq, defined by the exact sequence 0 L ν L Q 0. Definition 1.1. We denote by G d r the variety classifying pairs [C, l], where [C] Mpart g and l is a limit linear series on the tree-like curve C in the sense of [EH1]. We set σ : G r d Mpart g to be the proper projection map. One has a morphism G d r P ic d g commuting with h and σ, which over curves from p 1 is obtained by retaining the underlying line bundle of the genus g 1-aspect of the corresponding limit linear series. Remark 1.2. Suppose [C yq ] p 0 and let L be a rank 1 torsion free sheaf on C yq with h 0 L = r + 1. Then there exists a unique line bundle L W r d C such that ν L = L. If L is non-locally free, then to this point will correspond two points in G r d. If L W r d 1 C is such that ν L = L, then these points are Lq, Lq q and Ly, Ly y respectively. We need further notation. Let π : C part g introduce the projection map p 2 : C part g bundle L over C part g M part g M part g M part g G r d be the universal curve and we G d r. We choose a Poincaré line G d r, such that for a curve [C E] p 1 and a limit linear series l = l C, l E W r d C W r d E, we have that L [C E,l] Pic d C Pic 0 E. Both sheaves E := p 2 L and F := p 2 L 2 are locally free and ranke = r + 1 and rankf = 2d + 1 g respectively. There is a natural vector bundle morphism over G d r given by multiplication of sections, φ : Sym 2 E F, and we denote by U G d r its first degeneracy locus. We set D := σ U in case 3, respectively D s := σ U in case 4. Since the degeneracy locus U has expected codimension 3 inside G 26 6 in the case 3, respectively codimension 2 inside G 2s 2s 2 +2s+1 in the case 4, the loci D and D s are virtual divisors on M part g, that is, either they are divisors, or the entire space M part g. Definition 1.3. We define the class of the virtual divisor D to be the quantity [ D] virt := σ c 3 F Sym 2 E CH 1 M part 23. Similarly, for s 2 the virtual class of the divisor D s is defined as [ D s ] virt := σ c 2 F Sym 2 E CH 1 M part 2s 2 +s+1.

9 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 9 If, as the Strong Maximal Conjecture predicts, D and D s are divisors on M g, then their virtual classes defined above coincide with the actual class of the closure of D and D s inside M part g. We now describe the vector bundle morphism φ in more detail. Proposition 1.4. The vector bundle map φ : Sym 2 E F has the following local description: i For [C, L] Gd r, with [C] Mpart g being a smooth curve, one has EC, L = H 0 C, L and FC, L = H 0 C, L 2 and φc, L is the usual multiplication of sections. ii For a point t = C y E, l C, l E σ 1 p 1, where C is a curve of genus g 1, E is an elliptic curve and l C = L C is the C-aspect of the corresponding limit linear series, such that L C Wd rc verifies h0 C, L C 2y r, one has Et = H 0 C, L C and Ft = H 0 C, L 2 C 2y C u2, where u H 0 C, L C is any section such that ord y u = 0. Assume now L C has a base point at y. Then Et = H 0 C, L C = H 0 C, L C y and the image of the map Ft H 0 C, L 2 C is the subspace H0 C, L 2 C 2y H 0 C, L 2 C. iii Let t = [C yq, L] σ 1 p 0 be a point with q, y C and L W r dc yq be a torsion-free sheaf of rank 1, such that h 0 C, ν L y q r, where ν : C C yq is the normalization map. If L is locally free, we have that Et = H 0 C, ν L and Ft = H 0 C, ν L 2 y q C u 2, where u H 0 C, ν L is any section not vanishing at both points y and q. Assume L = ν A is a non locally free torsion-free sheaf of rank 1 on the curve C yq, where A W r d 1 C, the image of the map Ft H0 C, ν L 2 equals H 0 C, A 2. Proof. Both sheaves p 2 L and p 2 L 2 are reflexive, hence in this case, also locally free of ranks r + 1 and respectively 2d + 1 g. The local description of the fibres of both E and F is straightforward. For E it follows from Grauert s Theorem. In the case of F, part i is immediate, whereas for part ii, keeping the notation above, one always has the inclusion H 0 C, L 2 C 2y C u2 Ft. The conclusion follows comparing dimensions, once we observe that u / H 0 C, L C y, because Wd 2 r C =. Case iii is analogous. Our main result concerns the explicit calculation of [ D] virt and [ D s ] virt. In order to prove Theorems 0.1 and 0.3, we describe the restriction of the morphism φ along the pull-backs of two standard test curves in the boundary of M part g which have been used in many other instances: Choose a general pointed curve [C, q] of genus g 1 and a pointed elliptic curve [E, y]. Then construct the families of stable curves of genus g { } C 0 := C yq := C/y q : y C p 0 M g, and { } C 1 := C y E : y C p 1 M g. The intersection of the test curves with the generators of CH 1 M g is well-known, see for instance [HM]: C 0 λ = 0, C 0 δ 0 = degω Cyq = 2 2g, C 0 δ 1 = 1 and C 0 δ j = 0 for j = 2,..., g 2,

10 10 G. FARKAS respectively C 1 λ = 0, C 1 δ 0 = 0, C 1 δ 1 = degk C = 4 2g and C 1 δ j = 0 for j = 2,..., g 2. Having fixed a general pointed curve [C, q] M g 1,1, we first turn our attention to the pull-back σ C 0 G d r. We consider the determinantal variety { } Y := y, L C Wd r C : h0 C, L y q r, together with the projection π 1 : Y C. Inside Y we introduce the following loci } Γ 1 := {y, Ay : y C, A Wd 1 r C and Γ 2 := {y, Aq : y C, A W r d 1 C }. Let ϑ : Y := Bl Γ Y Y be the blow-up of Y along the intersection Γ of Γ 1 and Γ 2. Proposition 1.5. With notation as above, there is a birational morphism f : σ C 0 Y, which is an isomorphism outside ϑ 1 π 1 1 q. The restriction of f to π 1 ϑ f 1 q forgets the aspect of each limit linear series on the elliptic curve E. Furthermore, E σ C 0 and F σ C 0 are pull-backs under f of vector bundles on Y. Proof. We pick y C \ {q} and, as usual, ν : C C yq be the normalization. Recall that W r dc yq Pic d C yq is the variety of rank 1 torsion-free sheaves on C yq with h 0 C yq, L r + 1. A locally free sheaf L W r dc yq is uniquely determined by its pull-back ν L Wd rc, which has the property h0 C, ν L y q = r note that we have assumed that Wd 2 r C =, thus there must exist a section of L that does not vanish simultaneously at both y and q. In other words, the 1-dimensional quotient Q of L y L q is uniquely determined as ν ν L/L. Assume now, L W d rc yq is not locally free, thus L = ν A, for a linear series A Wd 1 r C. Using Remark 1.2, to this point correspond two points in Y, namely y, Ay and y, Aq. Summarizing, for y C \ {q}, there is a birational morphism π1 1 y W r dc yq which is an isomorphism over the locus of locally free sheaves. More precisely, W r dc yq is obtained from π1 1 y by identifying the disjoint divisors Γ 1 π1 1 y and Γ 2 π1 1 y. Finally, when y = q, then C yq degenerates to C q E, where E is a rational nodal cubic. If l C, l E σ 1 [C q E ], then the corresponding Brill-Noether numbers with respect to q satisfy ρl C, q 0 and ρl E, q 2 in case 3, respectively ρl C, q 1 in case 4. The statement about the restrictions E σ C 0 and F σ C 0 follows, because both restrictions are defined by dropping the information coming from the elliptic tail. We now describe the pull-back σ C 1 G d r. To that end, we define the locus { } X := y, L C Wd r C : h0 L 2y r, and by slight abuse of notation, we denote again by π 1 : X C the first projection.

11 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 11 In what follows we use notation from [EH1] to denote vanishing sequences of limit linear series on curves from the divisor p 1. In particular, if Y is a smooth curve of genus g and y Y, then for a linear series l G r d C we denote by ρl, y := ρg, r, d w l y the adjusted Brill-Noether number. Here w l := r i=0 αl i y is the weight of the point y with respect to l and α l y : α0 ly... αl ry is the ramification sequence of l at y. Proposition 1.6. The variety X is an irreducible component of σ C 1. In case 3 in case 4. c 3 F Sym 2 E σ C 1 = c 3 F Sym 2 E X, and c 2 F Sym 2 E σ C 1 = c 2 F Sym 2 E X Proof. We deal with the case g, r, d = 23, 6, 26, the case 4 being analogous. By the additivity of the Brill-Noether number, if l C, l E σ 1 [C y E] is a limit linear series of type g 6 26, we have that 2 = ρ23, 6, 26 ρl C, y + ρl E, y. Since ρl E, y 0, we obtain that ρl C, y 2. If ρl E, y = 0, then l E = 19y + O E 7y. This shows that l E is uniquely determined, while the aspect l C G 6 26 C is a complete linear series g6 26 with a cusp at y C. This gives rise to an element from X and shows that X {l E } = X is a component of σ C 1. The other components of σ C 1 are indexed by Schubert indices α := 0 α 0... α 6 20 = 26 6, such that lexicographically α > 0, 1, 1, 1, 1, 1, 1, and 7 6 j=0 α j 9, for we must have 1 ρl C, y 1. For such an index α, we set ᾱ c := 20 α 6,..., 20 α 0 to be the complementary Schubert index, then define X α := { y, l C C G 6 26C : α l C y α } and Z α := { l E G 6 26E : α l E y α c}. Then the following relation holds σ C 1 = X + α>0,1,1,1,1,1,1 m α X α Z α, where the multiplicities m α can be determined via Schubert calculus but play no role in our calculation. Our claim now follows for dimension reasons. Applying the pointed Brill-Noether Theorem [EH2], Theorem 1.1, using that C is general, we obtain the estimate dim Xᾱ = 1 + ρ22, 6, 26 6 j=0 α j < 3, for every index α > 0, 1, 1, 1, 1, 1, 1. In the definition of the test curve C 1, the point of attachment y E is fixed, therefore the restrictions of both E and F are pulled-back from X α and one obtains that c 3 F Sym 2 E Xᾱ Zᾱ = 0, for dimension reasons Chern numbers of tautological classes on Jacobians. Often in our calculations, we use facts about intersection theory on Jacobians. We refer to [ACGH] Chapter VIII for background on this topic and to [HM], [H] and [F3] for further applications to divisor class calculations on M g. We start with a Brill-Noether general curve C of genus g and denote by P a Poincaré line bundle on C Pic d C and by π 1 : C Pic d C C and π 2 : C Pic d C Pic d C

12 12 G. FARKAS the two projections. We introduce the class η = π 1 [x 0] H 2 C Pic d C, Z, where x 0 C is an arbitrary point. After picking a symplectic basis δ 1,..., δ 2g H 1 C, Z = H 1 Pic d C, Z, we consider the class γ := g π1δ α π2δ g+α π1δ g+α π2δ α H 2 C Pic d C, Z. α=1 One has the formula c 1 P = dη + γ, which describes the Hodge decomposition of c 1 P, as well as the relations γ 3 = 0, γη = 0, η 2 = 0 and γ 2 = 2ηπ2 θ, see [ACGH] p Assuming W r+1 d C =, that is, when the Brill-Noether number ρg, r + 1, d is negative, the smooth variety Wd r C admits a rank r + 1 vector bundle M := π 2 P C W r d C. In order to compute the Chern numbers of M we employ repeatedly the Harris-Tu formula [HT]. We write r c i M = 1 + x x r+1, i=0 and then for every class ζ H Pic d C, Z one has the following formula: 5 x i 1 1 x i r+1 r+1 ζ = det θ g+r d+i j j+l g + r d + i j j + l! 1 j,l r+1 ζ. One must interpret this formula the following way. Any top Chern number on W r d C can be expressed as a symmetric polynomial in the variables x 1,..., x r+1. This Chern number can be calculated by applying formula 5 and symmetrizing Top intersection products in the Jacobian of a curve of genus 22. We now specialize to the case of a general curve C of genus 22. By Riemann-Roch W26 6 C = W16 1 C and note that W26 7 C =, so we can consider the rank 8 vector bundle M on W 26 6 C. The vector bundle N := R 1 π 2 P C W 6 26 C has rank 2 and we explain how its two Chern classes determine all the Chern classes of M. Proposition 1.7. For a general curve C of genus 22 we set c i := c i M, for i = 1,..., 8, and y i := c i N, for i = 1, 2. Then the following relations hold in H W26 6 C, Z: c 1 = θ y 1, c 2 = 1 2 θ2 θy 1 + y 2, c 3 = 1 3! θ3 1 2 θ2 y 1 + θ 2, c 4 = 1 4! θ4 θ3 3! θ3 y θ2 y 2, c 5 = 1 5! θ5 1 4! θ4 y ! θ3 y 2, c 6 = 1 6! θ6 1 5! θ5 y ! θ4 y 2, and c 7 = 1 7! θ7 1 6! θ6 y ! θ5 y 2. Proof. Let us fix a divisor D C e of large degree e. We write down the exact sequence 0 M π 2 P Oπ D π 2 P Oπ1D π 1 D R 1 π 2 P C W 6 26 C 0.

13 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 13 Then we use that c tot π 2 P Oπ1 D = e θ, whereas the total Chern class of the vector bundle π 2 P Oπ1 D π1 D is trivial, for the divisor D is fixed. One obtains c tot R 1 π 2 P C W 6 26 C e θ = 8 1 i c i, Hence c i+2 = 1 i! y 2θ i 1 i+1! y iθ i i+2! θi+2 for all i 0, which finishes the proof. Using Proposition 1.7, any Chern number on the smooth 8-fold W26 6 C can be expressed purely in terms of monomials in the classes u 1, u 2 and θ, where u 1 and u 2 are the Chern roots of N, that is, i=0 y 1 = c 1 N = u 1 + u 2 and y 2 = c 2 N = u 1 u 2. We record for further use the following formal identities on H top W26 6 C, Z, which are obtained by applying formula 5 in the case g = 22, r = 1 and d = 16, using the canonical isomorphism H 1 C, L = H 0 C, K C L provided by Serre duality. u 3 1θ 5 = 4 22! 11! 7!, u3 2θ 5 = 2 22! 8! 10!, u2 1θ 6 = 3 22! 10! 7!, u2 2θ 6 = 22! 8! 9!, u 1θ 7 = 2 22! 7! 9!, u 2 θ 7 = 0, u 1 u 4 2θ 3 = 2 22! 9! 11!, u4 1u 2 θ 3 = 4 22! 8! 12!, u2 1u 2 θ 5 = 2 22! 8! 10!, u 1u 2 2θ 5 = 0, u 2 1u 3 2θ 3 = 0, u 3 1u 2 2θ 3 = 2 22! 9! 11!, u2 1u 2 2θ 4 = 22! 9! 10!, u4 1θ 4 = 5 22! 7! 12!, u4 2θ 4 = 3 22! 8! 11!, u 3 1u 2 θ 4 = 3 22! 8! 11!, u 1u 3 2θ 4 = 22! 9! 10!, u 1u 2 θ 6 = 22! 8! 9!, θ8 = 22! 7! 8!. To compute the corresponding Chern numbers on W26 6 C, one uses Proposition 1.7 and the previous formulas. Each Chern number corresponds to a degree 8 polynomial in u 1, u 2 and θ, which is symmetric in u 1 and u Top intersection products in the Jacobian of curve of genus 2s 2 + s. We next turn our attention to the top intersection products on W2s 2s 2 +2s+1 C in case 4, that is, when C is a general curve of genus 2s 2 + s, for s 2. We apply systematically 5. Observe that ρ2s 2 + s, 2s, 2s 2 + 2s = 0 and we denote by C 2s+1 := 2s2 + s! 2s! 2s 1! 2! 1! = # W2s 2s 3s! 3s 1! s + 1! s! 2 +2s, C where the last scheme is reduced and 0-dimensional. We collect the following formulas: Proposition 1.8. Let C be as above and set c i := c i M H 2i W2s 2s 2 +2s+1 C, Z to be the Chern classes of the dual of the tautological bundle on W2s 2s 2 +2s+1 C. The following hold: c 2s+1 = x 1 x 2 x 2s+1 = C 2s+1, c 2s c 1 = x 1 x 2 x 2s+1 + x 2 1x 2 x 2s, c 2s 1 c 2 = x 1 x 2 x 2s+1 + x 2 1x 2 x 2s + x 2 1x 2 2x 3 x 2s 1, c 2s 1 c 2 1 = x 1 x 2 x 2s+1 + 2x 2 1x 2 x 2s + x 2 1x 2 2x 3 x 2s 1 + x 3 1x 2 x 3 x 2s 1,

14 14 G. FARKAS c 2s θ = x 1 x 2 x 2s θ = 2s + 1s C 2s+1, c 2s 1 c 1 θ = x 1 x 2 x 2s θ + x 2 1x 2 x 2s 1 θ c 2s 2 c 2 θ = x 1 x 2 x 2s θ + x 2 1x 2 x 2s 1 θ + x 2 1x 2 2x 3 x 2s 2 θ c 2s 2 c 2 1 θ = x 1 x 2 x 2s θ + 2x 2 1x 2... x 2s 1 θ + x 2 1x 2 2x 3 x 2s 2 θ + x 3 1x 2 x 3 x 2s 2 θ c 2s 1 θ 2 = x 1 x 2 x 2s 1 θ 2, c 2s 2 c 1 θ 2 = x 1 x 2 x 2s 1 θ 2 + x 2 1x 2 x 2s 2 θ 2 Using Proposition 1.8, any top intersection product on the smooth 2s+1-dimensional variety W2s 2s 2 +2s+1 C has been reduced to a sum of monomials in the variables x i and θ. Next we record the values of these monomials: Proposition 1.9. Keep the notation from above. The following hold in H 4s+2 W 2s 2s 2 +2s+1 C, Z : x 1 x 2 x 2s+1 = C 2s+1, x 2 1x 2 2x 3 x 2s 1 = ss 1s s s3s + 1 C 2s+1 x 2 4ss + 1 1x 2 x 2s = 3s + 1 C 2s+1, x 3 1x 2 x 3 x 2s 1 = s2 s s 12s + 3 3s + 13s + 2 C 2s+1 x 1 x 2 x 2s θ = 2s + 1sC 2s+1, x 2 1x 2 x 2r 1 θ = s s 1 x 1 x 2 x 2s θ, 3s + 1 x 2 1x 2 2x 3 x 2s 2 = 2s 32s + 1s + 12 s + 2 x 1 x 2 x 2s θ 93s + 1 x 3 1x 2 x 3 x 2s 2 θ = s 1s + 12 s + 22s 12s + 3 x 1 x 2 x 2s θ 33s + 13s + 2 x 1 x 2 x 2s 1 θ 2 = 2ss + 12s + 1C 2s+1 x 2 1x 2 x 2s 2 θ 2 4s + 1s 1s + 2 = x 1 x 2 x 2s 1 θ 2 33s + 1 x 1 x 2 x 2s 2 θ 3 = 2s + 12s 1s + 2s + 1s2 C 2s+1. 3 We compute the classes of the loci X and Y appearing in Propositions 1.5 and 1.6: Proposition Let [C, q] be general 1-pointed curve of genus 22 and let M denote the tautological rank 7 vector bundle over W26 6 C and c i := c i M H 2i W26 6 C, Z as before. Then the following hold: i [X] = π2 c 6 6ηθπ2 c η + 2γπ2 c 5 H 12 C W26 6 C, Z. ii [Y ] = π2 c 6 2ηθπ2 c η + γπ2 c 3 H 12 C W26 6 C, Z. Proof. Recall that W26 6 C is smooth of dimension 8. We realize X as the degeneracy locus of a vector bundle morphism over C W26 6 C. For each pair y, L C W 26 6 C, there is a natural map H 0 C, L O 2y H 0 C, L which globalizes to a bundle morphism ζ : J 1 P π2 M over C W26 6 C. Here J 1 P is the jet bundle of P. Then we have the identification X = Z 1 ζ, that is, X is the first degeneracy locus of ζ. The Porteous formula yields [X] = c 6 π2 M J 1 P. To evaluate this class, we write down the exact sequence over C Pic 26 C involving the jet bundle 0 π 1K C P J 1 P P 0.

15 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 15 We can compute the total Chern class of the formal inverse of the jet bundle c tot J 1 P 1 = deglη + γ j j 2gC 2 + deglη + γ = j η + γ + γ η + γ + γ 2 + = η + 2γ 6ηθ, leading to the desired formula for [X]. In order to compute [Y ] we proceed in a similar way. We denote by j 0 µ, ν : C C Pic 26 C C Pic 26 C the two projections, by C C Pic 26 C the diagonal and set Γ q := {q} Pic 26 C. We introduce the rank 2 vector bundle B := µ ν P O +ν Γ q over C W 6 26 C. Note that there is a bundle morphism χ : B π 2 M, such that Y = Z 1 χ. Since we also have that c tot B 1 = 1 + deglη + γ + deglη + γ 2 + we immediately obtain the stated expression for [Y ]. 1 η = η + γ 2ηθ, In case 4, recall that C is a general curve of genus 2s 2 + s and W2s 2s 2 +2s+1 C is smooth of dimension 2s + 1. We record the formulas for the classes of X and Y, the proofs being analogous to those of Proposition Proposition Let [C, q] M 2s 2 +s,1 be a general pointed curve. If c i := c i M are the Chern classes of the tautological vector bundle over W2s 2s 2 +2s+1 C, then one has: i [X] = π2 c 2s 6π2 c 2s 2ηθ + 2 4s 2 + 3sη + γ π2 c 2s 1 H 4s C Wd rc. ii [Y ] = π2 c 2s 2π2 c 2s 2ηθ + 2s 2 + 2sη + γ π2 c 2s 1 H 4s C Wd rc. Remark For future reference we also record the following formulas: 6 c 2s+1 π 2 M J 1 P = π 2c 2s+1 6π 2c 2s 1 ηθ + 2s4s + 3η + 2γ π 2c 2s, 7 c 2s+1 π 2 M B = π 2c 2s+1 2π 2c 2s 1 ηθ + 2ss + 1η + γ π 2c 2s. The following formulas are applications of Grothendieck-Riemann-Roch: Proposition Let C be a general curve of genus g 1 and consider the vector bundles A 2 and B 2 on C Pic d C having fibres A 2 y, L = H 0 C, L 2 2y and B 2 y, L = H 0 C, L 2 y q, respectively. i One has the following formulas in case 3: c 1 A 2 = 4θ 4γ 146η, c 1 B 2 = 4θ 2γ 51η, c 2 A 2 = 8θ ηθ + 16γθ, c 2 B 2 = 8θ ηθ + 8θγ, c 3 A 2 = 32 3 θ3 1072ηθ 2 32θ 2 γ and c 3 B 2 = 32 3 θ3 376ηθ 2 16θ 2 γ. ii In the case 4, that is, when g 1 = 2s 2 + s, the following formulas hold: c 1 A 2 = 4θ 4γ 23s + 12s + 1η, c 1 B 2 = 4θ 2γ 2s η, c 2 A 2 = 8θ s 2 + 5s 2ηθ + 16γθ, c 2 B 2 = 8θ s 2 + 4s 1ηθ + 8θγ.

16 16 G. FARKAS Proof. Immediate application of Grothendieck-Riemann-Roch with respect to the map ν : C C Pic 26 C C Pic 26 C defined above. The vector bundle A 2 is realized as a push-forward under the map ν: A 2 = ν! µ P 2 O C C Pic 26 C 2 = ν µ P 2 O C C Pic 26 C 2, and we apply Grothendieck-Riemann-Roch to ν. One finds that ch 2 A 2 = 8ηθ and ch n A 2 = 0, for n 3. Furthermore, ν c 1 P 2 = 2θ. The calculation of B 2 is similar and we skip the details. The case 4 follows along similar lines. 2. THE KODAIRA DIMENSION OF M 23 In this section we complete the calculation of the virtual class [ D] virt. We shall use repeatedly that if V is a vector bundle of rank r + 1 on a stack X, the Chern classes of its second symmetric product can be computed as follows: i c 1 Sym 2 V = r + 2c 1 V, ii c 2 Sym 2 V = rr+3 2 c 2 1 V + r + 3c 2V, iii c 3 Sym 2 V = rr+4r 1 6 c 3 1 V + r + 5c 3V + r 2 + 4r 1c 1 Vc 2 V. We expand the virtual class [ D] virt = σ c 3 F Sym 2 E = aλ b 0 δ 0 b 1 δ 1 CH 1 M part 23. Our task is to determine the coefficients a, b 0 and b 1. As we shall explain at the end of this section, this suffices in order to estimate the other coefficients in the boundary expansion of [ D] virt. We begin with the coefficient of δ 1 : Theorem 2.1. Let C be a general curve of genus 22 and denote by C 1 p 1 M 23 the associated test curve. Then the coefficient of δ 1 in the expansion of [ D] virt is equal to 1 b 1 = 2gC 2 σ C 1 c 3 F Sym 2 E = = Proof. We intersect the degeneracy locus of the map φ : Sym 2 E F with the 3-fold σ C 1 = X + ᾱ m αxᾱ Zᾱ, where recall that dim X α < 3, for all Schubert indices α > 0, 1, 1, 1, 1, 1, 1. Therefore, as already explained in Proposition 1.6, it is enough to estimate the contribution coming from X and we can write σ C 1 c 3 F Sym 2 E = c 3 F X c 3 Sym 2 E X c 1 F X c 2 Sym 2 E X + +2c 1 Sym 2 E X c 2 Sym 2 E X c 1 Sym 2 E X c 2 F X +c 2 1Sym 2 E X c 1 F X c 3 1Sym 2 E X. We now evaluate the terms that appear in the right-hand-side of this expression. In the course of proving Proposition 1.10, we set-up a vector bundle morphism ζ : J 1 P π2 M. The kernel Kerζ is locally free of rank 1. If U is the line bundle on X with fibre H 0 C, L Uy, L = H 0 C, L 2y H0 C, L O 2y over a point y, L X, then one has the following exact sequence over X 0 U J 1 P Kerζ 0.

17 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 17 In particular, c 1 U = 2γ + 94η + c 1 Kerζ. The products of the Chern class of Kerζ with other classes coming from C W26 6 C can be computed from the formula in [HT]: 8 c 1 Kerζ ξ X = c 7 π 2M J 1 P ξ X = π2c 7 6ηθπ2c η + 2γπ2c 6 ξ X, where ξ H 2 C W26 6 C, Z. If A 3 denotes the rank 31 vector bundle on X having fibres A 3 y, L = H 0 C, L 2 and constructed in an obvious way as push-forward of a line bundle on the triple product C C Pic 26 C, then U 2 can be embedded in A 3 /A 2. We consider the quotient G := A 3/A 2 U 2. The morphism U 2 A 3 /A 2 vanishes along the locus of pairs y, L where L has a base point, which implies that G has torsion along the Γ X consisting of pairs q, Aq, where A W 6 25 C.Furthermore, F X is identified as a subsheaf of A 3 with the kernel of the map A 3 G. Summarizing, there is an exact sequence of vector bundles on X 0 A 2 X F X U 2 0. Over a general point of y, L X this sequence reflects the decomposition Fy, L = H 0 C, L 2 2y C u 2, where u H 0 C, L is such that ord y u = 1. Hence c 1 F X = c 1 A 2 X + 2c 1 U, c 2 F X = c 2 A 2 X + 2c 1 A 2 X c 1 U c 3 F X = c 3 A 2 + 2c 2 A 2 X c 1 U. Recalling that E X = π 2 M X, we obtain that: σ C 1 c 3 F Sym 2 E = c 3 A 2 X + c 2 A 2 X c 1 U 2 c 3 Sym 2 π2m X rr + 3 c 1 π 2 2M X +r+3c 2 π2m X c 1 A 2 X +c 1 U 2 2r+2c 1 π2m X r + 2c 1 π2m X c 2 A 2 X r + 2c 1 π2m X c 1 A 2 X c 1 U 2 + +r c 2 1π 2M X c 1 A 2 X + r c 2 1π 2M X c 1 U 2 r c 3 1π 2M X. Here, c i π 2 M X = π 2 c i H 2i X, Z and r = rkm 1 = 6. The coefficient of c 1 Kerζ in the product σ C 1 c 3 F Sym 2 E is evaluated via 8. First we consider the part of this product which does not contain c 1 Kerζ and we obtain 36π 2c 2 θ 148π 2c 2 1θ ηπ 2c π 2c 1 c θ ηθ ηθπ 2c π 2c ηπ 2c θ 2 π 2c π 2c 3 H 6 C W 6 26C, Z. This polynomial of degree 3 gets multiplied by the degree 6 polynomial in θ, η and π 2 c i that gives the class [X] obtained in Proposition We add to ot the contribution coming from c 1 Kerζ. One obtains a homogeneous polynomial of degree 9 in η, θ and

18 18 G. FARKAS and π2 c i for i = 1,..., 7. The only non-zero monomials are those containing η. After retaining only these monomials and dividing by η, the resulting degree 8 polynomial in θ, c i H W26 6 C, Z can be brought to a manageable form, by using Proposition 1.7. After lengthy but straightforward manipulations carried out using Maple, one finds σ C 1 c 3 Sym 2 E F = ηπ 2 780c 3 1c 4 θ c 3 1c c 2 1c 4 θ c 2 1c 5 θ 5402c 2 1c 6 384θ 3 c 1 c θ 2 c 1 c θc 1 c 2 c c 1 c 6 θ 7990c 1 c 2 c c 1 c 7 216c 2 c 4 θ c 2 c 5 θ 66c 3 c 4 θ c 3 c c 2 c c 4 θ c 5θ c 6 θ c 7 θ. We suppress η and the remaining polynomial lives inside H 16 W26 6 C, Z. We have indicated how to calculate explicitly using 5 all top Chern numbers on W26 6 C and we eventually find that b 1 = 1 42 σ C 1 c 3 F Sym 2 E = Theorem 2.2. Let [C, q] be a general pointed curve of genus 22 and let C 0 p 0 Mpart 23 be the associated test curve. Then It follows that b 0 = σ C 0 c 3 F Sym 2 E = 44b 0 b 1 = Proof. As already noted in Proposition 1.5, the vector bundles E σ C 0 and F σ C 0 are both pull-backs of vector bundles on Y and we denote these vector bundles E and F as well, that is, E σ C 0 = f E Y and F σ C 0 = f F Y. Following broadly the lines of the proof of Theorem 2.1, we evaluate the terms appearing in σ C 0 c 3 F Sym 2 E. Let V be the line bundle on Y with fibre V y, L = H 0 C, L H 0 C, L y q H0 C, L O y+q over a point y, L Y. There is an exact sequence of vector bundles over Y 0 V B Kerχ 0, where χ : B π2 M is the bundle morphism defined in the second part of Proposition In particular, c 1 V = 25η + γ + c 1 Kerχ, for the Chern class of B has been computed in the proof of Proposition By using again [HT], we find the following formulas for the Chern numbers of Kerχ: c 1 Kerχ ξ Y = c 7 π 2 M B ξ Y = π2c 7 +π2c 6 13η+γ 2π2c 4 ηθ ξ Y, for any class ξ H 2 C W 6 26 C, Z. We have previously defined the vector bundle B 2 over C W 6 26 C with fibre B 2y, L = H 0 C, L 2 y q. One has the following exact sequence of bundles over Y 9 0 B 2 Y F Y V 2 0.

19 EFFECTIVE DIVISORS ON MODULI OF CURVES AND HURWITZ SPACES 19 If B 3 is the vector bundle on Y with fibres B 3 y, L = H 0 C, L 2, we have an injective morphism of sheaves V 2 B 3 /B 2 locally given by v 2 v 2 mod H 0 C, L 2 y q, where v H 0 C, L is any section not vanishing at q and y. Then F Y is canonically identified with the kernel of the projection morphism B 3 B 3/B 2 V 2 and the exact sequence 9 now becomes clear. Therefore c 1 F Y = c 1 B 2 Y + 2c 1 V, c 2 F Y = c 2 B 2 Y + 2c 1 B 2 Y c 1 V and c 3 F Y = c 3 B 2 Y + 2c 2 B 2 Y c 1 V. The part of the intersection number σ C 0 c 3 F Sym 2 E that does not contain c 1 Kerχ equals 36π 2c 2 θ 148π 2c 2 1θ 37ηπ 2c π 2c 1 c θ3 + 8ηθ 2 +32ηθπ 2c π 2c 3 1+9ηπ 2c 2 +64θ 2 π 2c 1 +11π 2c 3 H 6 C W 6 26C, Z. We multiply this expression with the class [Y ] computed in Proposition The coefficient of c 1 Kerζ in σ C 0 c 3 F Sym 2 E equals 2c 2 B 2 Y 2r π 2c 2 1 2r + 2c 1 B 2 Y π 2c 1 + rr + 3π 2c r + 3π 2c 2, where recall that r = rkm 1 = 6. All in all, 44b 0 b 1 = σ C 0 c 3 F Sym 2 E and we evaluate this as explained using 5. The following result follows from the definition of the vector bundles E and F given in Proposition 1.4. It will provide the third relation between the coefficients of [D] virt, and thus complete the calculation of its slope. Theorem 2.3. Let [C, q] be a general 1-pointed curve of genus 22 and R M 23 be the pencil obtained by attaching at the fixed point q C a pencil of plane cubics at one of the base points of the pencils. Then one has the relation a 12b 0 + b 1 = σ c 3 F Sym 2 E R = 0. Proof of Corollary 0.2 and of Theorem 0.1. We assume that the Strong Maximal Conjecture holds on M 23, that is, D is a genuine divisor on M part 23 and let D denote its closure in M 23. To determine its slope, we express its class in terms of the standard generators of CH 1 M 23 and write [D] = aλ 11 j=0 b j δ j CH 1 M 23, where the coefficients a, b 0 and b 1 have already been determined. Since a b 0 = , we are in a position to apply Corollary 1.2 from [FP], which gives the inequalities b j b 0 for 1 j 11. Interestingly, M 23 is the last moduli space where the argument from [FP], based on pencils of curves on K3 surface and which reduces the calculation of the slope of an effective divisor on M g to a bound on its λ and δ 0 -coefficients, can be

20 20 G. FARKAS applied. Therefore sd = a b 0 < Thus assuming the Strong Maximal Conjecture on M 23, we can express the canonical class K M23 = 13λ 2δ 0 3δ j=1 δ j as a positive combination of [D], of the big class λ and further boundary classes δ 2,..., δ 11. One concludes that M 23 is a variety of general type. 3. EFFECTIVE DIVISORS OF SMALL SLOPE ON M 2s 2 +s+1 We now complete the proof of Theorem 0.3. Recall that g = 2s 2 + s + 1 and we have expressed the virtual class [ D s ] virt = σ c2 F Sym 2 E = aλ b 0 δ 0 b 1 δ 1 CH 1 M part g. The determination of the coefficients a, b 0 and b 1 follows largely the proof of Theorem 0.1 and we shall highlight the differences. Recall that C 2s+1 denotes the number of linear systems of type g 2s 2s 2 +2s on a general curve of genus 2sr + s. Theorem 3.1. Let C be a general curve of genus 2s 2 + s and denote by C 1 p 1 M 2s 2 +s+1 the associated test curve. Then the coefficient of δ 1 in the expansion of [ D s ] virt is equal to 1 b 1 = 2gC 2 σ C 1 c 2 F Sym 2 E = C 2s+1 2ss 12s + 1 2s 13s + 13s s 6 40s s s s s + 18 Proof. We intersect the degeneracy locus of the map φ : Sym 2 E F with the surface σ C 1, containing X as an irreducible component. It follows from Proposition 1.6 that X is the only component contributing to this intersection product, that is, 10 σ C 1 c 2 F Sym 2 E = c 2 F X c 2 Sym 2 E X c 1 F X c 1 Sym 2 E X +c 2 1Sym 2 E X. The kernel Kerζ of the vector bundle morphism ζ : J 1 P π2 M defined in the proof of Proposition 1.10 is a line bundle on X. If U is the line bundle on X with fibre H 0 C, L Uy, L = H 0 C, L 2y H0 C, L O 2y over a point y, L X, then one has the following exact sequence over X 0 U J 1 P Kerζ 0. From this sequence it follows, c 1 U = 2γ+24s 2 +3sη+c 1 Kerζ, where the products of c 1 Kerζ with arbitrary classes coming from ξ C W 2s 2s 2 +2s+1 C can be computed using once more the Harris-Tu formula [HT]: 11 c 1 Kerζ ξ X = c 2s+1 π 2 M J 1 P ξ X. The Chern classes on the right hand side of 11 has been evaluated in the formula 6. Using a local analysis identical to the one in Proposition 2.1 in the case g = 23 we conclude that the restriction F X sits in the following exact sequence of vector bundles 0 A 2 X F X U 2 0. We obtain the following intersection product on the surface X: c 2 F X Sym 2 E X = c2 A2 X + 2c 1 A 2 X c 1 J1 P 2s + 3c 2 π 2M +.