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SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg
GAVRIL FARKAS
1. INTRODUCTION
The aim of this paper is to describe a systematic way of constructing effectivedivisors on Mg having exceptionally small slope. In particular, these divisors providea string of counterexamples to the Harris-Morrison Slope Conjecture (cf. [HMo]). In aprevious paper [FP], we showed that the divisor K10 on M10 consisting of sections ofK3 surfaces contradicts the Slope Conjecture on M10. Since the moduli spaces Mg areknown to behave erratically for small g and since the condition that a curve of genus g
lie on a K3 surface is divisorial only for g = 10, the question remained whether K10 isan isolated example or the first in a series of counterexamples. Here we prove that anyeffective divisor on Mg consisting of curves satisfying a Green-Lazarsfeld syzygy typecondition for a linear system residual to a pencil of minimal degree, violates the SlopeConjecture. A consequence of the existence of these effective divisors is that variousmoduli spaces Mg,n with g †22, are proved to be of general type.
We recall that the slope s(D) of an effective divisor D on Mg is defined as thesmallest rational number a/b â„ 0 such that the divisor class aλâ b(ÎŽ0 + · · ·+ÎŽ[g/2])â [D]is an effective combination of boundary divisors. The Slope Conjecture predicts thats(D) â„ 6 + 12/(g + 1) for all effective divisors D on Mg (cf. [HMo]). More generally,
the question of finding a good lower bound for the slope of Mg
sg := infs(D) : D â Eff(Mg)
is of great interest for a variety of reasons, for instance it would provide a new geometricsolution to the Schottky problem. In a different direction, since s(KMg
) = 13/2 (cf.
[HM]), to prove that Mg is of general type it suffices to exhibit a single effective divisor
D on Mg of slope s(D) < 13/2.
The Slope Conjecture is true for all Mg with g †9 but in [FP] we proved that
on M10, we have the equality s(K10) = 7 < 6 + 12/11 (in fact K10 is the only effectivedivisor on M10 having slope < 6 + 12/11). In [FP] we also showed that K10 has fourincarnations as a geometric subvariety of M10. In particular, K10 can be thought of aseither
(1) the locus of curves [C] â M10 for which the rank 2 Mukai type Brill-Noether locus
SU2(C, KC , 6) := E â SU2(C, KC) : h0(C, E) â„ 7
is not equal to â , or
Research partially supported by the NSF Grant DMS-0450670 and by the Sloan Foundation.
1
2 G. FARKAS
(2) the locus of curves [C] â M10 carrying a pencil A â W 16 (C) such that the multipli-
cation map ”A : Sym2H0(C, KC â Aâš) â H0(C, (KC â Aâš)â2) is not surjective.
The geometric conditions (1) and (2), unlike the original definition of K10, canbe extended to other genera. We fix g = 2k â 2 and denote by Ï : G1
k â Mg theHurwitz scheme of k-sheeted coverings of P1 of genus g parametrizing pairs (C, A) withA â W 1
k (C). For each i â„ 0 we introduce the cycle Ug,i consisting of pairs (C, A) â G1k
such that KC â Aâš fails the Green-Lazarsfeld property (Ni). By setting Zg,i := Ï(Ug,i)we obtain an induced geometric stratification of Mg
Zg,0 â Zg,1 â . . . â Zg,i â . . . â Mg.
If for each (C, A) â G1k we consider the map C
|KCâAâš|ââ Pkâ2 induced by the residual
linear system, we can define two vector bundles A and B on G1k such that
A(C, A) = H0(ΩiPkâ2(i + 2)) and B(C, A) = H0(Ωi
Pkâ2(i + 2) âOC).
For g = 6i + 10 (hence k = 3i + 6), it turns out that rank(A) = rank(B) and Ug,i is thedegeneracy locus of the natural vector bundle map Ï : A â B, that is, Zg,i is a virtualdivisor on Mg, with virtual class Ïâc1(B â A). The main result of this article is the
computation of the compactification inside Mg of this degeneracy locus (see Theorem4.1 for a more precise statement):
Theorem A. If Ï : G13i+6 â M6i+10 is the compactification of the Hurwitz stack by limit
linear series, then there is a natural extension of the vector bundle map Ï : A â B over G13i+6
such that Zg,i is the degeneracy locus of Ï. Moreover the class of the pushforward to Mg of thevirtual degeneracy locus of Ï is given by
Ïâ(c1(B) â c1(A)) ⥠a λ â b0 ÎŽ0 â b1 ÎŽ1 â · · · â b3i+5ÎŽ3i+5,
where a, b0, . . . , b3i+5 are explicitly determined coefficients such that bj â„ b0 for j â„ 1 and
a
b0=
3(4i + 7)(6i2 + 19i + 12)
(i + 2)(12i2 + 31i + 18)< 6 +
12
g + 1.
Corollary. If the degeneracy locus Z6i+10,i is an actual divisor on M6i+10, then we have that
s(Z6i+10,i) =3(4i + 7)(6i2 + 19i + 12)
(i + 2)(12i2 + 31i + 18)< 6 +
12
g + 1,
thus contradicting the Slope Conjecture on M6i+10.
The idea of the proof is to define a whole host of vector bundles Ga,b and Ha,b overG1
k for a, b â„ 0, having fibres
Ga,b(C, A) = H0(ΩaPkâ2(a + b) âOC) and Ha,b(C, A) = H0(Ωa
Pkâ2(a + b)).
These bundles are related to one another by certain exact sequences (4) and (6) over G1k
(see Section 3). After an analysis over each boundary divisor Ïâ1(âj), where 0 †j â€3i + 5, we find in Section 3 a unique way of extending Ga,b and Ha,b to vector bundles
over G1k such that (4) and (6) continue to make sense and be exact and then use this to
compute the Chern numbers of A = Hi,2 and B = Gi,2.
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 3
We expect Z6i+10,i to be always a divisor on M6i+10 but we are able to check thisonly for small i. To verify this in general one would have to prove that if [C] â M6i+10
is a general curve, then one (or equivalently all) of the finitely many linear systems
g3i+49i+12 = KC(âg1
3i+6) satisfies property (Ni). This is a direct generalization of Greenâs
Conjecture to the case of curves and line bundles L with h1(L) = 2 (see Section 2 formore on this analogy).
For g = 10 we recover of course the results from [FP]. In the next two cases, g = 16and 22 we have complete results:
Theorem 1.1. The following subvariety of M16
Z16,1 := [C] â M16 : âL â W 721(C) such that C
|L|â P
7 is not cut out by quadrics,
is an effective divisor, and the class of its compactification is given by the formula
Z16,1 ⥠286(407λ â 61 ÎŽ0 â 325 ÎŽ1 â b2 ÎŽ2 â · · · â b8 ÎŽ8
),
where bj ℠b1 for all 2 †j †8. In particular s(Z16,1) = 407/61 = 6.6721... < 6 + 12/17 =
6.705..., hence Z16,1 provides a counterexample to the Slope Conjecture on M16.
In a similar manner we have the following example of a geometric divisor on M22
of very small slope:
Theorem 1.2. The following subvariety of M22
Z22,2 := [C] â M22 : âL â W 1030 (C) such that C
|L|â P
10 fails property (N2),
is an effective divisor on M22 and the class of its compactification is given by the formula
Z22,2 ⥠25194(1665 λ â 256 ÎŽ0 â 1407 ÎŽ1 â b2ÎŽ2 â · · · â b11 ÎŽ11
),
where bj ℠b1 for all 2 †j †11. In particular s(Z22,2) = 1665/256 = 6.50390... <
6 + 12/23 = 6.52173..., and Z22,2 gives a counterexample to the Slope Conjecture on M22.
We note that while Theorem A gives a sequence of (virtual) examples of divisorscontradicting the Slope Conjecture, for any given g one can construct other (actual)divisors on Mg of slope < 6+12/(g +1) also defined in terms of syzygies. For instance,
the locus consisting of curves [C] â M20 for which there exists L â W 420(C) such that
C|L|â P4 is not cut out by quartics (or equivalently, the map I4(L) â H0(L) â I5(L) is
not bijective), is a divisor on M20 violating the Slope Conjecture.
We record the following consequence of Theorems 1.1 and 1.2:
Theorem 1.3. (1) The moduli space of n-pointed curves M22,n is of general type for all n â„ 2.
(2) The moduli space M21,n is of general type for all n â„ 4.
(3) The moduli space M16,n is of general type for all n â„ 9.
(4) The moduli space M20,n is of general type for n â„ 6.
4 G. FARKAS
It is also possible to give examples of divisors of small slope on the moduli spacesMg,n of n-pointed stable curves. We have only pursued this for genera g †22 with the
goal of proving that various moduli spaces Mg,n are of general type (this is automaticfor g â„ 23). One of our examples is the following:
Theorem 1.4. The following subvariety of M14,1
Z114,0 := [C, p] â M14,1 : âL â W 6
18(C) such that C|L(âp)|ââ P
5 fails property (N0)
is an effective divisor on M14,1. The class of its compactification Z114,0 is given by the formula
Z114,0 ⥠33
(237 λ + 14 Ï â 35 ÎŽ0 â 169ÎŽ1 â 183ÎŽ13 â b2ÎŽ2 â · · · â b12ÎŽ12),
where bj ℠15 + 27j for 3 †j †12, j 6= 4, b2 ℠325 and b4 ℠271. In particular [Z114,0]
lies outside the cone of Pic(M14,1) spanned by pullbacks of effective divisors from M14, the
boundary divisors ÎŽ0, . . . , ÎŽ13 and the Weierstrass divisor W on M14,1.
An equivalent formulation of the Slope Conjecture on Mg for g such that g + 1is composite (which we learned from S. Keel), is to say that the Brill-Noether divisors
Mrg,d consisting of curves with a gr
d when gâ (r +1)(gâ d+ r) = â1, lie on a face of the
effective cone Eff(Mg). It is well-known that s(Mrg,d) = 6 + 12/(g + 1) (cf. [EH3]), so
the Slope Conjecture singles out these divisors as being of minimal slope. One can ask a
similar question on Mg,1. Are the pullbacks Ïâ(Mrg,d) of the Brill-Noether divisors from
Mg and the Weierstrass divisor W := [C, p] â Mg,1 : p â C is a Weierstrass point on
a face of the effective cone Eff(Mg,1)? The question makes sense especially since in
[EH2] it is proved that the class of any generalized Brill-Noether divisor on Mg,1 (that
is, any codimension 1 locus of curves (C, p) â Mg,1 having a linear series with special
ramification at p) lies inside the cone of Pic(Mg,1) spanned by by [Ïâ(Mrg,d)] and [W].
Theorem 1.4 shows that at least for g â„ 14 the answer to the question raised above isemphatically negative.
Acknowledgments: I had useful conversations with many people on subjectsrelated to this paper. I especially benefitted from discussions with J. Harris, S. Keel, D.Khosla and M. Popa.
2. SYZYGIES OF ALGEBRAIC CURVES
In this paragraph we review a few facts about the resolution of the ideal of acurve embedded in a projective space. As a general reference for syzygies and Koszulcohomology we recommend [L] or [GL].
Suppose that C is a smooth curve of genus g and L is a very ample line bundleon C giving an embedding C â Pr = P(V ), where V = H0(L). We denote by IC/Pr theideal of C in Pr and consider its minimal resolution of free S = Sym(V )-modules
0 â Fr+1 â · · · â F2 â F1 â IC/Pr â 0.
Then one can write Fj = âlâZS(âj â l)bj,l(C), where bj,l(C) = dimCTorSj
(S(C), C
)j+l
is
the graded Betti number of C that comes on the l-th row and j-th column in the Betti
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 5
diagram of C. Following Green and Lazarsfeld we say that the pair (C, L) satisfy theproperty (Ni) for some integer i â„ 1, if Fj = âS(âj â 1) for all j †i (or equivalentlyin terms of graded Betti numbers, bi,l(C) = 0 for all l â„ 2). Using the computation ofbj,l(C) in terms of Koszul cohomology, there is a well-known cohomological interpreta-tion of property (Ni): we denote by M = ΩPr(1) and ML = M âOC , hence we have anexact sequence
0 â ML â H0(L) âOC â L â 0.
By taking exterior powers, for each i â„ 0 we obtain the exact sequence:
(1) 0 â â§i+1ML â â§i+iH0(L) âOC â â§iML â L â 0.
If L is a normally generated line bundle on C, then (C, L) satisfies property (Ni) if andonly if for all j â„ 1, the natural map
ui,j : â§i+1H0(L) â H0(Lâj) â H0(â§iML â Lâ(j+1))
obtained by tensoring the sequence (1) and taking global sections, is surjective (cf. e.g.[GL], Lemma 1.10).
We will be interested in the vector bundle ML in the case when C is a curve ofgenus g = 2k â 2 and L = KC â Aâš is residual to a base point free pencil A â W 1
k (C).We start by establishing a more general technical result used throughout the paper:
Proposition 2.1. Let C be a curve of genus g and L a globally generated grd on C. If p â„ 1 is
an integer such that g + minr, p †d, then for all j â„ 2 and 0 †i †p + 2 â j, we have thevanishing H1(â§iML â Lâj) = 0.
Proof. We use a filtration argument due to Lazarsfeld [L]. We choose general pointsx1, . . . , xrâ1 â C such that h0
(C, L â OC(âx1 â · · · â xrâ1)
)= 2 and then there is an
exact sequence
0 ââ Lâš(x1 + · · · + xrâ1) ââ ML ââ ârâ1j=1OC(âxj) ââ 0.
Taking exterior powers, we can write the exact sequence
0 â â§iâ1(ârâ1
j=1OC(âxj))âLâš(x1 + · · ·+xrâ1) ââ â§iML ââ â§i
(ârâ1
j=1OC(âxj))ââ 0.
In order to conclude that H0(â§iML â Lâj) = 0, it suffices to show that for all 1 †i â€minr â 1, p we have the following:
(1) H0(Lâj â OC(âDi)) = 0 for each effective divisor Di â Ci with support in the setx1, . . . , xrâ1, and
(2) H0(Lâ(jâ1) â OC(Drâi)
)= 0, for any effective divisor Drâi â Crâi with support
contained in x1, . . . , xrâ1.
Statement (2) follows because of degree reasons. The only way (1) could fail is whenj = 2 and h0(KC â Lâš) â„ r â i + 1, which by the Riemann-Roch theorem implies thatg + i â„ d + 1 and this contradicts our numerical assumption.
€
Remark 2.2. (1) For p ℠r, the condition g+minp, r †d is equivalent with H1(L) = 0.(2) Since we work in characteristic 0, the same filtration argument from the proof ofProposition 2.1 can be used to show that ML is a stable vector bundle on C.
6 G. FARKAS
We fix g = 2k â 2 where k â„ 3. We denote by M0g the open substack of Mg con-
sisting of curves C of genus g which carry no g1kâ1âs and which have no automorphisms
(clearly the complement of M0g inside Mg has codimension â„ 2). Over M0
g we have a
universal curve Ï : C â M0g and we also consider the Hurwitz stack Ï : G1
k â M0g
parametrising pairs (C, A), where [C] â M0g and A â W 1
k (C) is a (necessarily base point
free) g1k on C. It is a classical result that G1
k is a smooth irreducible stack which is fi-
nite over M0g. The isomorphism G1
k 3 (C, A) 7â (C, KC â Aâš) â Gkâ23kâ6 will be used
throughout the paper.
Let us consider now an element (C, A) â G1k and set L := KCâAâš. The genericity
assumption on C implies that L is very ample and gives an embedding C â Pkâ2 ofdegree 3k â 6. We denote by IC the ideal sheaf of C in this embedding.
Proposition 2.3. Fix integers g = 2k â 2 with k â„ 3 and i â„ 0 with 2k â„ 3i + 7. If(C, A) â G1
k and L = KC â Aâš, then (C, L) fails to satisfy property (Ni) if and only if
h0(Pkâ2,â§iMP
kâ2 â IC(2)) â„
(k â 2
i + 1
)(k â 3i â 6) + 2
(k â 1
i + 1
)â
(k â 1
i + 2
)+ 1.
Proof. We use the following diagram of exact sequences for the embedding C|L|â Pkâ2:
0 0 0yy
yâ§i+1MPkâ2 â IC(1) ââ â§i+1H0(OPkâ2(1)) â IC(1) ââ â§iMPkâ2 â IC(2)y
yy
â§i+1MPkâ2(1) ââ â§i+1H0(OPkâ2(1)) âOPkâ2(1) ââ â§iMPkâ2(2)ya
yb
yc
â§i+1ML â L ââ â§i+1H0(L) â L ââ â§iML â L2yy
y0 0 0
For simplicity we denote by the same symbol the map on global sections corresponding
to a sheaf morphism and the morphism itself. Since H1(Pkâ2,â§i+1MPkâ2(1)) = 0 (be-cause of Bott vanishing, see [OSS], pg. 8), we can apply the snake lemma to the diagramobtained by taking global sections in the two lower rows. The map induced by b is anisomorphism, which gives an isomorphism
H0(Pkâ2,â§iMPkâ2 â IC(2)) âŒ= H1(Pkâ2,â§i+1MPkâ2 â IC(1)).
From Proposition 2.1 we have that H1(â§iML â L2) = 0, and thus C satisfies (Ni) if andonly if the map
H1(â§i+1ML â L) â â§i+1H0(L) â H1(L)
is an isomorphism, or equivalently h1(â§i+1ML â L) †h1(â§i+1H0(L) â L) = 2(kâ1i+1
).
From the Riemann-Roch theorem, this is the same thing as having
dim(Coker(a)) â€
(k â 2
i + 1
)(k â 3i â 6) + 2
(k â 1
i + 1
)â
(k â 1
i + 2
).
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 7
But Coker(a) âŒ= H0(â§iMPkâ2 â IC(2)), so failure of (Ni) is equivalent to the map chaving a kernel of dimension at least
(k â 2
i + 1
)(k â 3i â 6) + 2
(k â 1
i + 1
)â
(k â 1
i + 2
)+ 1 =
=
(k â 2
i + 1
)(k â 3i â 6)
(1 â
1
(k â i â 2)(i + 2)
)+ 1.
€
Remark 2.4. An immediate consequence of this proof is that when k â 3i â 6 < 0 (orequivalently g †6i + 8), the pair (C, A) always fails property (Ni). When k = 3i + 6
we have that (C, A) fails (Ni) if and only if h0(Pkâ2,â§iMPkâ2 â IC(2)) â„ 1, which weexpect to be a divisorial condition on G1
k.
This proposition allows us to define a determinantal substack of G1k consisting of
pairs (C, A) for which L fails to satisfy property (Ni). On the fibre product C ĂM0g
G1k
there is a universal Poincare bundle L whose existence is guaranteed by the universalproperty of G1
k. If p1 : C ĂM0g
G1k â C and p2 : C ĂM0
gG1
k â G1k are the natural
projections, then E := p2â(pâ1ÏÏâLâš) is a vector bundle of rank kâ1 with fibre E(C, A) =
H0(C, KC âAâš) over each point (C, A) â G1k. We have a tautological embedding of the
pullback of the universal curve C ĂM0gG1
k into the projective bundle u : P(E) â G1k and
we denote by J â OP(E) the ideal sheaf of the image. Next, we define the vector bundleN on P(E) by the sequence
0 ââ N â uâ(E) ââ OP(E)(1) ââ 0,
and we further introduce two vector bundles A and B over G1k by setting
A := uâ
(â§iN âOP(E)(2)
), and B := uâ
(â§iN âOCĂ
M0gG1
k(2)
).
If C â Pkâ2 is the embedding given by L = KC â Aâš, then
A(C, A) = H0(Pkâ2,â§iMPkâ2(2)) and B(C, A) = H0(C,â§iML â Lâ2)
and there is a natural vector bundle morphism Ï : A â B. From Grauertâs Theorem wesee that both A and B are vector bundles over G1
k (for B use that H1(C,â§iML âL2) = 0because of Proposition 2.1). Moreover from Bottâs Theorem (see again [OSS]) and theRiemann-Roch Theorem respectively, we obtain that
rank(A) = (i + 1)
(k
i + 2
)and rank(B) =
(k â 2
i
)(4k â 9 â 3i).
Then Proposition 2.3 can be restated as follows:
Proposition 2.5. The cycle
Ug,i := (C, A) â G1k : (C, KC â Aâš) fails property (Ni),
can be realized as the degeneracy locus of rank (iâ1)(kâ1i+1
)+(i+2)
(kâ1i+2
)â
(kâ2i+1
)(kâ3iâ6)â1
of the vector bundle map Ï : A â B over G1k.
8 G. FARKAS
In this way we obtain a stratification of M0g with strata Zg,i := Ï(Ug,i) consisting
of those curves C for which there exists a pencil A â W 1k (C) such that (C, KC âAâš) fails
property (Ni). Note that when g = 6i + 10 then rank(A) = rank(B) and Ug,i is simplythe degeneracy locus of Ï and we expect Zg,i to be a divisor on M0
g.
It is of course very natural to compare this newly defined stratification
Zg,0 â Zg,1 â . . . â Zg,i â . . . â M0g
with the more classical stratification of Mg defined in terms of syzygies of KC . If foreach i â„ 0 we set
Ng,i := [C] â Mg : (C, KC) fails property (Ni),
then Greenâs Conjecture for generic curves of fixed gonality (cf. [V3], [V2]) can be readas saying that for all 0 †i †(g â 2)/2, the locus Ng,i coincides with the (i + 2)-gonallocus M1
g,i+2 of curves with a g1i+2, while Ng,i = Mg for all i â„ (g â 1)/2. One of the
morals of this paper is that the stratification Zg,i is very different in nature from theone given by gonality: whereas Ng,0 is the locus of hyperelliptic curves of genus g, thesmallest stratum Zg,0 is intimately related to the locus of curves lying on K3 surfaceswhich is well-known to be transversal to any Brill-Noether locus (a curve of genus glying on a general K3 surface satisfies the Brill-Noether Theorem, cf. [L]).
We have already seen that Zg,i = M0g when g †6i + 8 (cf. Remark 2.4). A result
due to Mukai and Voisin (cf. [V1], Proposition 3.2) states that if C is a smooth curve ofgenus g = 2k â 2 sitting on a K3 surface S with Pic(S) = Z [C], then for all A â W 1
k (C)
the multiplication map Sym2H0(KC â Aâš) â H0((KC â Aâš)â2) is not surjective, inother words, [C] â Zg,0. In a forthcoming paper we will show that the converse alsoholds, that is, the closure in Mg of the smallest stratum Zg,0 coincides with the locus
Kg := [C] â Mg : C lies on a K3 surface.
The possibility of such an equality of cycles has already been raised in Voisinâs paper(cf. [V1], Remarques 4.13). Its main appeal lies in the fact that it gives an intrinsiccharacterization of a curve lying on a K3 surface which makes no reference to the K3surface itself! We now make the following:
Conjecture 2.6. For an even genus g â„ 6i + 10, the stratum Zg,i is a proper subvarietyof M0
g. In particular, when g = 6i + 10, the stratum Zg,i is a divisor on M0g.
For g †8 the conjecture is trivially true because Kg = Mg, that is, every curveof (even) genus g †8 sits on a K3 surface. The first interesting case is g = 10 whenthe conjecture holds and the identification K10 = Z10,0 is part of a more general picture(see [FP] for details). We have checked Conjecture 2.6 for all genera g †24 sometimesusing the program Macaulay. We will describe this in detail in the most interestingcases, g = 16 and g = 22, when the strata Z16,1 and Z22,2 are divisors on M16 and M22
respectively.
Theorem 2.7. Let C be a general curve of genus 16. Then every linear series A â W 19 (C) gives
an embedding C|KCâAâš|ââ P
7 of degree 21 such that the ideal of C is generated by quadrics (thatis, it satisfies property (N1)). In particular Z16,1 is a divisor on M16.
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 9
Proof. From the irreducibility of the Hurwitz scheme of coverings of P1 it follows thatthe variety G1
9 = G116,9 parametrizing pairs (C, A) with [C] â M16 and A â W 1
9 (C) is
irreducible. To prove that Z16,1 is a divisor it is enough to exhibit an element (C, A) â G19
with KC â Aâš very ample, such that the embedded curve C|KCâAâš|
â P7 is cut out byquadrics.
We consider 13 general points in P2 denoted by p1, p2, q1, . . . , q7 and r1, . . . , r4
respectively, and define the linear system
H ⥠8h â 3(Ep1 + Ep2) â 27â
i=1
Eqi â4â
j=1
Erj
on the blow-up S = Bl13(P2). Here h denotes the pullback of the line class from P2.
Using the program Macaulay it is easy to check that S|H|â P7 is an embedding and the
graded Betti diagram of S is
1 â â ââ 7 â ââ â 35 56
Thus S is cut out by quadrics. To carry out this calculation we chose the 13 pointsin P2 randomly using the Hilbert-Burch theorem so that they satisfy the Minimal Res-olution Conjecture (see [SchT] for details on how to pick random points in P2 usingMacaulay). Next we consider a curve C â S in the linear system
(2) C ⥠14h â 5(Ep1 + Ep2) â 46â
i=1
Eqi â 3Eq7 â 23â
j=1
Erj â Er4 .
By using Macaulay we pick C randomly in its linear system and then check that C issmooth, g(C) = 16 and deg(C) = 21. To show that C is cut out by quadrics one cancompute directly the Betti diagram of C. Otherwise, since S is cut out by quadrics, toconclude the same thing about C, it suffices to show that the map
m : H0(S, H) â H0(S, 2H â C) â H0(S, 3H â C)
is surjective (or equivalently injective). Since h0(S, 2H â C) = 2, from the base pointfree pencil trick we get that Ker(m) = H0(S, C âH) = 0, because C âH ⥠6hâ 2Ep1 â
2Ep2 â 2â6
i=1 Eqi âEq7 ââ3
j=1 Erj is clearly not effective for a general choice of the 13points. €
Remark 2.8. A possible objection to this proof could be that while we defined the Hur-witz space G1
9 â M016 over the open part of M16 consisting of curves carrying no g1
8âs,the curve C we constructed in the proof of Theorem 2.7 might lie outside M0
16. How-ever, this does not affect our conclusions because for every g and l < (g + 3)/2, theHurwitz space of pairs (C, A) with [C] â Mg an A â W 1
l (C), is an irreducible variety,that is, there can be no component of the Hurwitz space mapping to a proper subvarietyof the l-gonal locus M1
g,l (This statement can be easily proved with the methods from
[AC1]).
10 G. FARKAS
Remark 2.9. For later use we are going to record a slight generalization of Theorem 2.7.The conclusion of the theorem holds for the image in P7 of any plane curve C that issufficiently general in the linear system given by (2). In particular one can choose Cto have a cusp at a general extra point y â P2, in which case we obtain that a generalone-cuspidal curve C â P7 of arithmetic genus 16 and degree 21 is cut out by quadrics(this again is quite easily checked with Macaulay). Since the Hilbert scheme of thesecurves is irreducible (use again the Hurwitz scheme), it follows that for a general (C, y) âM15,1 and for an arbitrary linear system L â W 7
21(C) having a cusp at y, the ideal of the
cuspidal image curve C|L|â P7 is generated by quadrics.
In a somewhat similar manner we are going to show that for g = 22 the locusZ22,2 is a divisor.
Theorem 2.10. Let C be a general curve of genus 22. Then for every linear series A â W 112(C)
we have that |KC â Aâš| is very ample and the resulting embedding A|KCâAâš|
â P10 satisfies
property (N2). In particular, the locus of curves C of genus 22 carrying a linear series g1030 =
KC(âg112) which fails to satisfy property (N2), is a divisor on M22.
Proof. Ideally we would like to construct a rational surface S â P10 which satisfiesproperty (N2) and then consider a suitable curve C â S with g(C) = 22 and deg(C) =30. For numerical reasons this will turn out not to be possible but we will be able toconstruct a smooth curve C âČ â S with g(C âČ) = 20 and deg(C âČ) = 28 which satisfies(N2). The desired curve C will be a smoothing in P10 of C âȘ L âȘ LâČ, where L and LâČ aregeneral chords of C âČ.
We start by choosing random points p, p1, . . . , p7, q1, . . . , q5 â P2. Then the linearsystem
H = 11h â 4Ep â 37â
i=1
Epi â 25â
j=1
Eqj
yields an embedding S â P10 of Bl13(P2). Using Macaulay we get that the upper left
corner of the Betti diagram of S is
1 â â â ââ 29 98 72 ââ â â 264 â
that is, S satisfies property (N2), hence H1(S,â§3MS(1)) = 0. If Π⥠H is a generalhyperplane section of S, then g(Î) = 13, deg(Î) = 22 and an argument very similarto the one in Proposition 2.1 shows that the vector bundle MÎ is stable. In particularH1(Î,â§2MÎ(2)) = 0. Since MS|Î
âŒ= MÎ âOÎ one can write the exact sequence
0 ââ â§iMS(1) ââ â§iMS(2) ââ â§iMÎ(2) â â§iâ1MÎ(2) ââ 0,
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 11
which leads to the vanishing H2(S,â§3MS(1)) = 0. Suppose now that C â S is a smoothnon-degenerate curve. We have a commutative diagram:
H1(C,â§3MC(1)
)ââ â§3H0
(OC(1)
)â H1(OC(1))y=
y=
H1(â§2MS(2H â C)
)â H2
(â§3MS(H â C)
)â â§3H0(OS(1)) â H2(H â C),
where the left vertical map is isomorphic because H1(S,â§3MS(1)) = H2(S,â§3MS(1)) =0. To conclude that C satisfies (N2), it suffices to show that H1
(S,â§2MS(2H â C)
)= 0,
or equivalently, that the map f : â§2H0(OS(1)) â H0(2H â C) â H0(MS(3H â C))obtained from the Koszul complex, is surjective. If
(deg(C), g(C)
)= (30, 22), then one
can check easily that h0(S, 2HâC) = 3 and h0(S, MS(3HâC)) = 207, hence f cannot besurjective for dimensional reasons. The closest we can get to these numerical invariantsis when (deg(C), g(C)) = (28, 20) and this is the type of curve on S we will be lookingfor. Take
C âČ âĄ 17h â 6Ep â 57â
i=
Epi â 35â
j=1
Eqj ,
and consider the embedding C âČ â P10 given by |H|. Using Macaulay we check that C âČ
is a smooth curve of genus 20 and degree 28 with graded Betti diagram
1 â â ââ 27 80 ââ â â 432
Hence C âČ satisfies (N2). Now choose two general chords L and LâČ of C and defineC := C âČ âȘ L âȘ LâČ. Clearly g(C) = 22, deg(C) = 30 and one last check with Macaulayshows that b2j(C) = 0 for j â„ 2, that is, C satisfies (N2).
€
Remark 2.11. Just like in the case g = 16 we have a slight variation of the last Theo-rem. The same proof shows that for a general (C, y) â M21,1 and for an arbitrary linear
series L â W 1030 (C) that has a cusp at y, the one-cuspidal image curve C
|L|â P10 satisfies
property (N2).
3. INTERSECTION THEORY CALCULATIONS ON Mg
Recall that we have realized Z6i+10,i as the image of the degeneracy locus U6i+10,i
of a vector bundle morphism Ï : A â B over G1k. To compute the class of the compact-
ification Z6i+10,i we are going to extend the determinantal structure of Z6i+10,i over the
boundary divisors in M6i+10.
We set g := 6i + 10, k := 3i + 6 and denote by Mg := M0g âȘ
(âȘ3i+5
j=0 â0j
)the locally
closed substack of Mg defined as the union of the locus M0g of smooth curves carrying
no linear systems g1kâ1 to which we add the open subsets â0
j â âj for 1 †j †3i + 5
consisting of 1-nodal genus g curves C âȘy D, with [C] â Mgâj and [D, y] â Mj,1 beingBrill-Noether general curves, and the locus â0
0 â â0 containing 1-nodal irreduciblegenus g curves C âČ = C/q ⌠y, where [C, q] â Mgâ1 is a Brill-Noether general pointed
12 G. FARKAS
curve and y â C, together with their degenerations consisting of unions of a smoothgenus g â 1 curve and a nodal rational curve. One can then extend the finite covering
Ï : Gkâ23kâ6 â M0
g to a proper, generically finite map
Ï : Gkâ23kâ6 â Mg
by letting Gkâ23kâ6 be the variety of limit gkâ2
3kâ6âs on the treelike curves from Mg (see [EH1],Theorem 3.4 for the construction of the space of limit linear series).
We will be interested in intersecting the divisors Z6i+10,i on Mg with test curves
in the boundary of M6i+10 which are defined as follows: we fix a Brill-Noether generalcurve C of genus 2k â 3 = 6i + 9, a general point q â C and a general elliptic curve E.We define two 1-parameter families
(3) C0 := C
y ⌠q: y â C â â0 â M2kâ2 and C1 := CâȘyE : y â C â â1 â M2kâ2.
It is well-known that these families intersect the generators of Pic(M2kâ2) as follows:
C0 · λ = 0, C0 · ÎŽ0 = â(4k â 6), C0 · ÎŽ1 = 1 and C0 · ÎŽa = 0 for a â„ 2, and
C1 · λ = 0, C1 · ÎŽ0 = 0, C1 · ÎŽ1 = â(4k â 8), C1 · ÎŽa = 0 for a â„ 2.
Next, we fix an integer 2 †j †k â 1, a general curve C of genus 2k â 2 â j anda general curve pointed curve (D, y) of genus j. We define the 1-parameter familyCj := C âȘy D : y â C â âj â M2kâ2. We have that
Cj · λ = 0, Cj · ÎŽa = 0 for a 6= j and Cj · ÎŽj = â(4k â 6 â 2j).
During our calculations we will often need the following enumerative result (cf. [HM],pg. 71):
Proposition 3.1. Let C be a general curve of genus j, y â C a general fixed point and aninteger 0 †α †j/2.
(1) The number of pencils A â W 1jâα+1(Y ) satisfying h0(Y, A(â(j + 1 â 2α)y)) â„ 1 is
equal to
a(j, α) =j + 1 â 2α
j + 1 â α
(j
α
).
(2) The number of pencils A â W 1jâα(C) satisfying h0(Y, A(â(j â 2α)q)) â„ 1 for some
unspecified point q â Y is equal to
b(j, α) = (j â 2α â 1)(j â 2α)(j â 2α + 1)
(j
α
).
Definition 3.2. (1) For a fixed general curve C of genus 2k â 3 we denote by ÎŽ(k, 1) the
(finite) number of linear series L â W kâ23kâ6(C) for which there exists a point y â C with
h0(L(â2y)) â„ k â 2 and h0(L(âky)) â„ 1.(2) We fix 2 †j †k â 1, 0 †α †j/2 and a general curve C of genus 2k â 2 â j. We
denote by ÎŽ(k, j, α) the number of linear series LC â W kâ23kâjâ4(C) for which there exists
a point y â C such that h0(L(â(α+1)y)
)â„ kâ1âα, h0
(L(â(jâα+2)y)
)â„ kâ1+αâj
and h0(L(â(k + 1)y)
)â„ 1.
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 13
The numbers Ύ(k, 1) and Ύ(k, j, α) can be computed by degeneration. Since we
do not need their actual values we skip this calculation. Next we describe Gkâ23kâ6 set
theoretically. To set notation, if X is a treelike curve and l is a limit grd on X , for a
component Y of X we denote by lY = (LY , VY â H0(LY )) the Y -aspect of l. For a pointy â Y we denote by alY
s (C)s=0...r the vanishing sequence of l at y and by Ï(lY , y) :=
Ï(g, r, d) ââr
i=0(alYi (y) â i) the adjusted Brill-Noether number with respect to y
Proposition 3.3. (1) Let C1y = C âȘy E be an element of â0
1. If (lC , lE) is a limit gkâ23kâ6 on C1
y ,
then VC = H0(LC) and LC â W kâ23kâ6(C) has a cusp at y, that is, h0(C, L(â2y)) = k â 2.
If y â C is a general point, then lE =(OE((3k â 6)y), (2k â 5)y + |(k â 1)y|
), that is,
lE is uniquely determined. If y â C is one of the finitely many points for which there exists
LC â W kâ23kâ6(C) such that Ï(LC , y) = â1, then lE(â(2k â 6)y) is a gkâ2
k with vanishingsequence at y being â„ (0, 2, 3, . . . , k â 2, k). Moreover, at the level of 1-cycles we have theidentification Ïâ(C1) ⥠X + ÎŽ(k, 1) T , where
X := (y, D) â C Ă Ckâ2 : h0(C, D + 2y) â„ 2
and T is the curve consisting of gkâ2k âs on E with vanishing â„ (0, 2, . . . , k â 2, k) at the fixed
point y â E.
(2) Let C0y = C/y ⌠q be an element of â0
0. Then limit linear series of type gkâ23kâ6 on C0
y are
in 1:1 correspondence with complete linear series L on C of type gkâ23kâ6 satisfying the condition
h0(C, LâOC(âyâq)) = h0(C, L)â1. Thus there is an isomorphism between the cycle Ïâ(C0)
of gkâ23kâ6 on all curves C0
y with y â C and the smooth curve
Y := (y, D) â C Ă Ckâ2 : h0(C, D + y + q) â„ 2.
Proof. We only prove (1) the remaining case being similar. Suppose l = (lC , lE) is a limit
gkâ23kâ6 on C1
y . Using the additivity of the Brill-Noether number (see e.g. [EH1], Lemma3.6), we get that 0 = Ï(2k â 2, k â 2, 3k â 6) â„ Ï(lC , y) + Ï(lE , y). Since the vanishingsequence at y of the E-aspect of l is
alE (y) †(2k â 5, 2k â 4, . . . , 3k â 8, 3k â 6),
it follows that we also have the inequality alC (y) â„ (0, 2, . . . , k â 1). If y â C is general,then Ï(lC , y) = 0 and alC (y) = (0, 2, . . . , k â 1). Moreover, lC corresponds to a completelinear series |LC | on C such that h1(C, LC) = 1 and h1(LC â OC(â2y)) = 2. If y â Cis one of the points such that Ï(lC , y) = â1, then since C is Brill-Noether general we
must have alC0 (y) = 0, which implies that alC (y) = (0, 2, . . . , k â 2, k), hence alE (y) â„
(2k â 6, 2k â 4, . . . , 3k â 8, 3k â 6), or equivalently, lE = (2k â 6)y + |VE |, where VE â
H0(OE(ky)) is a gkâ2k with vanishing sequence â„ (0, 2, 3, . . . , k â 2, k) at y. An easy
argument shows that the variety Y of such VE âs is isomorphic to P(OE(ky)|2y). €
Proposition 3.4. Let Cjy := C âȘy D be an element from â0
j , where 2 †j †k â 1, g(C) =
2k â 2 â j and g(D) = j. Suppose (lC , lD) is a limit gkâ23kâ6 on Cj
y . If y â C is a general point,then lC has the divisor (j â 2)y as base locus and there exists an integer 0 †α †[j/2] such
that lC(â(j â 2)y) = |LC |, with LC â W kâ23kâjâ4(C) satisfying the inequality
aLC (y) â„ (0, 1, . . . , α â 1, α + 1, α + 2, . . . , j â α, j â α + 2, j â α + 3, . . . , k â 1, k).
14 G. FARKAS
Moreover lD is one of the a(j, α) linear systems on D of type (2k â j â 4)y + |LD|, with
LD â W kâ2kâ2+j(D) such that h0(LD(â(k+1âα)y)) = α and h0(LD(â(k+αâj)y)) = jâα.
If for some integer 0 †α †[j/2], y â C is one of the b(2kâ 2â j, kâ 1 + αâ j) pointsfor which there exists LC â Pic3kâjâ3(C) such that h0(LC(â(α + 2)y)) = k â 1 â α andh0(LC((j â α + 2)y)) = k â j + α, then there are two possibilities:
(1) The aspect lC is of the form (jâ2)y + |LC(ây)|, whereas lD = (2kâ jâ4)y + |LD|, where
LD â W kâ2kâ2+j(D) satisfies h0
(LD(â(k + 1â α)y)
)℠α and h0
(LD(â(k + 1 + αâ j)y)
)â„
j â α â 1.
(2) The aspect lC is of the form (j â 3)y + (LC , VC), where VC â H0(LC) is such that
H0(LC(â2y)) â VC , and lD = (2k â j â 4)y + |LD| with LD â W kâ2kâ2+j(D) and
aLD(y) = (0, 1, . . . , kâ 1â j + α, k + 1â j + α, . . . , kâ 1âα, k + 1âα, . . . , kâ 1, k + 1).
Finally, if for some 0 †α †[j/2], y â C is one of the ÎŽ(k, j, α) points for which there
exists LC â Pic3kâ4âj(C) such that
aLC (y) = (0, 1, . . . , α â 1, α + 1, . . . , j â α, j â α + 2, . . . , k â 2, k â 1, k + 1),
then lC = (jâ2)y+ |LC | and lD = (2kâjâ5)y+ lâČ
D, where lâČ
D is a (non-complete) gkâ2kâ1+j on
D satisfying alâČ
D(y) = (0, 2, . . . , kâ1+αâj, k+1+αâj, . . . , kâα, k+2âα, . . . , k, k+1).
Proof. Since by definition [D, y] â Mj,1 is Brill-Noether general, we have that Ï(lD, y) â„
0. The locus of those (C, y) â M2kâ2âj,1 carrying a gkâ23kâ6 with Ï(gkâ2
3kâ6, y) †â2 is ofcodimension â„ 2, hence we must also have that Ï(lC , y) â„ â1. Using Proposition 1.2
from [EH3] we have that alCkâ2(y) †k â 2 + j and alC
0 (y) â„ j â 3. There are two cases toconsider:
(a) If Ï(lC , y) = 0, then Ï(lD, y) = 0. Assume first that alC0 (y) = j â 2. Since we have that
0 †alCs (y)âsâ(jâ2) †2 for all 0 †s †kâ2, there must exist an integer 0 †α †[j/2],
such that if LC denotes the gkâ23kâ4âj obtained from lC by removing (j â 2)y, then the
vanishing at y is aLC (y) = (0, 1, . . . , α â 1, α + 1, α + 2, . . . , j â α, j â α + 2, . . . , k). Bycompatibility, lD has (2kâjâ4)y in its base locus and we write lD = (2kâjâ4)y+|LD|,where aLD(y) = (0, 1, . . . , k â 2 + j â α, k + j â α, . . . , k â 1 â α, k + 1 â α, . . . , k). ByRiemann-Roch, the number of such LDâs equals the number of g1
jâα+1âs on D with a
(jâ2α+1)-fold point at y which is computed by a(j, α). If alC0 (y) = jâ3, we denote by
LC â Pic3kâ3âj(C) the line bundle obtained from lC by subtracting (j â 3)y, and againwe see that there must exist 0 †α †[j/2] such that h0(LC(â(j â α + 2)y)) = k â j + αand h0(LC(â(α + 2)y)) = k â 1 â α. By duality, KC â Lâš
C â OC((j â α + 2)y) is oneof the finitely many g1
kâ1âαâs on C with a (j â 2α)-fold point at y. This number equalsb(2k â 2 â j, k â 1 + α â j) and this explains case (2) of our statement.
(b) If Ï(lC , y) = â1, then Ï(lD, y) = 1. An argument similar to that used above gives twopossibilities for alC (y) and the rest follows because lC and lD are compatible at y. €
Retaining the notation from Proposition 3.4, for each integer 0 †α †[j/2] wedefine the following 1-cycles:
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 15
Xj,α := (y, LC) â C Ă Pic3kâjâ4(C) : aLCα (y) ℠α + 1, aLC
jâα(y) â„ j â α + 2,
YâČ
j,α := (y, l) â C Ă Gkâ23kâjâ3(C) : al
1(y) ℠2, alα(y) ℠α + 2, al
jâαâ1(y) â„ j â α + 2,
YâČâČ
j,α := LD â Pickâ2+j(D) : aLDk+αâj(y) â„ k + α â j + 1, aLD
kâαâ1(y) â„ k â α + 1
and
YâČâČâČ
j,α := l â Gkâ2kâ1+j(D) : al
1(y) â„ 2, alkâ1+αâj(y) â„ k +αâ j +1, al
kâ1âα(y) â„ kâα+2.
Via Proposition 3.4 we view Xj,α, YâČ
j,α, YâČâČ
j,α and YâČâČâČ
j,α as 1-cycles on Gkâ23kâ6. Note that Xj,α
projects onto C while YâČ
j,α is isomorphic to the union of b(2kâ2â j, kâ1+αâ j) copies
of P1. We can then rephrase Proposition 3.4 as follows:
Proposition 3.5. If Ï : Gkâ23kâ6 â Mg is the natural projection and [C] â M2kâ2âj where
2 †j †k â 1, then we have the following numerical equivalence relation between 1-cycles:
Ïâ(Cj) âĄ
[j/2]â
α=0
(a(j, α) Xj,α + ÎŽ(k, 2k â 2 â j, k + α â j) Y
âČ
j,α+
+b(2k â 2 â j, k â 1 + α â j) YâČâČ
j,α + ÎŽ(j, α) YâČâČâČ
j,α
).
To compute the class of the cycles introduced in Propositions 3.3 and 3.5, we in-troduce some notation. Let C be a Brill-Noether general curve of genus g. We denote byCd the d-th symmetric product of C, by U â C Ă Cd the universal divisor, and by L the
Poincare bundle on CĂPicd(C). We also introduce the projections Ï1 : CĂCd â C and
Ï2 : C Ă Cd â Cd (and we use the same notation for the projections from C Ă Picd(C)onto the factors). We define the cohomology classes
η = Ïâ1([point]) â H2(C Ă Cd) and x = Ïâ
2(Dx0) â H2(C Ă Cd),
where x0 â C is an arbitrary point and Dx0 := D â Cd : x0 â D (the definition of x isobviously independent of the point x0). Finally, if ÎŽ1, . . . , ÎŽ2g â H1(C, Z) âŒ= H1(Cd, Z) isa symplectic basis, then we define the class
Îł := â
gâ
α=1
(Ïâ
1(Ύα)Ïâ2(ÎŽg+α) â Ïâ
1(ÎŽg+α)Ïâ2(Ύα)
).
With these notations we have the formula (cf. [ACGH], p. 338) [U ] ⥠dη + Îł + x,corresponding to the Hodge decomposition of [U ]. We also record the formulas Îł3 =γη = 0, η2 = 0 and Îł2 = â2ηÏâ
2(Ξ), where Ξ â H2(Cd, Z) is the pullback of the class of
the theta divisor on Jac(C). Similarly, c1(L) = dη + Îł, where η, Îł â H2(C Ă Picd(C))pull back to η, Îł â H2(C Ă Cd). When d = k â 2 and g = 2k â 3, we set M :=Ïâ
1(KC) âOCĂCkâ2(âU), hence c1(M) = (3k â 6)η â Îł â x.
We now compute the class of the curves X and Y defined in Proposition 3.3:
16 G. FARKAS
Proposition 3.6. Let C be a Brill-Noether general curve of genus g = 2k â 3 and q â C ageneral point.
(1) The class of the curve X = (y, D) â C Ă Ckâ2 : h0(C, D + 2y) â„ 2 is given by
[X] âĄÎžkâ2
(k â 2)!â
Ξkâ3x
(k â 3)!â
2Ξkâ3Îł
(k â 3)!+
+2xΞkâ4Îł
(k â 4)!â
4(k + 1)
(k â 4)!xΞkâ4η +
4k â 2
(k â 3)!Ξkâ3η.
(2) The class of the curve Y = (y, D) â C Ă Ckâ2 : h0(C, D + y + q) â„ 2 is given by
[Y ] âĄÎžkâ2
(k â 2)!â
Ξkâ3x
(k â 3)!â
Ξkâ3Îł
(k â 3)!+
+xΞkâ4Îł
(k â 4)!â
k + 1
(k â 4)!xΞkâ4η +
k â 1
(k â 3)!Ξkâ3η.
Proof. We will realize both X and Y as pullbacks of degeneracy loci and compute theirclasses using the Thom-Porteous formula. We consider the map Δ : C Ă Ckâ2 â Ck
given by Δ(y, D) := 2y + D. It is easy to check that
Δâ(x) = 2η + x and Δâ(Ξ) = 4gη + Ξ â 2Îł.
Then X = Δâ(C1k) and to compute the class of C1
k = E â Ck : h0(E) â„ 2 we in-troduce the rank k vector bundle P := (Ï2)â(Ï
â1KC â OU ) on Ck having total Chern
class ct(P) = (1 â x)âk+4eΞ
1âx (cf. [ACGH], pg. 240). There is a natural bundle mapÏâš : Δâ(Pâš) â H0(KC)âš â OCĂCkâ2
, and X = Zkâ1(Ïâš). Then by Thom-Porteous, we
can write
[X] =[ 1
ct(Δâ(Pâš))
]kâ2
= Δâ( Ξkâ2
(k â 2)!â
xΞkâ3
(k â 3)!
)=
=(4gη + Ξ â 2Îł)kâ2
(k â 2)!â
(4gη + Ξ â 2Îł)kâ3(2η + x)
(k â 3)!,
which quickly leads to the desired expression for [X].
To compute the class of Y we proceed in a similar manner: we consider the mapÏ : C Ă Ckâ2 â Ck given by Ï(y, D) := y + q + D. Then Ïâ(x) = x + η and Ïâ(Ξ) =gη + Ξ â Îł. For each (y, D) â C Ă Ckâ2 we have a natural map
H0(KC|y+q+D)âš â H0(KC)âš
which globalizes to a vector bundle map Ïâš : Ïâ(Pâš) â H0(KC)âš âOCĂCkâ2and then
it is clear that Y = Zkâ1(Ïâš). Applying Thom-Porteous again, we obtain
[Y ] =[ 1
ct(Ïâ(Pâš))
]kâ2
= Ïâ( Ξkâ2
(k â 2)!â
xΞkâ3
(k â 3)!
)=
=(gη + Ξ â Îł)kâ2
(k â 2)!â
(gη + Ξ â Îł)kâ3(η + x)
(k â 3)!,
which after a straightforward calculation gives the class of Y . €
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 17
We also need the following intersection theoretic result:
Lemma 3.7. For j â„ 1 we denote by Fj := (Ï2)â(Mâj) the vector bundle on Ckâ2 with fibre
Fj(D) = H0(C, jKC â jD) over each D â Ckâ2. Then c1(Fj) = âj2Ξ â j(3j â 2)(k â 2)x.
Proof. We apply the Grothendieck-Riemann-Roch Theorem for the map Ï2 : CĂCkâ2 âCkâ2. We obtain that
c1(Fj) =[(Ï2)â
((1+ j((3kâ6)ηâÎłâx)+
j2
2((3kâ6)ηâÎłâx)2
)·(1â (2kâ4)η
))]1
=
= âj2Ξ â j(3j â 2)(k â 2)x.
€
Next we compute the class of Xj,α when 2 †j †k â 1. It is convenient to stateour result as follows:
Proposition 3.8. Let C be a general curve of genus 2k â 2 â j such that 2 †j †k â 1. Wehave the following relation in H4kâ4â2j
(C Ă Pic3kâjâ4(C)
):
[j/2]â
α=0
a(j, α) Xj,α âĄ2(2k â 3)!
k!(k â 2)!(2k â 3 â j)!
( Ξ2kâ2âj
2k â 2 â j+
(3kj + 12k â 9j â 18)Ξ2kâ3âjÎł
2k â 3+
Ξ2kâ3âjη
(2k â 5)(2k â 3)
(60â64k+16k2+j(60â
67
2kâ10k2+6k3)+j2(15+2kâ5k2)+
3
2kj3
)).
Proof. We fix an integer 0 †α †[j/2] and a divisor D â Ce of degree e â„ kâjâ1. We set
d := 3kâ4âj, Î := DĂPicd(C) and denote by u, v : CĂCĂPicd(C) â CĂPicd(C) the
two projections and by Jα(L) the α-th jet bundle of the Poincare bundle on CĂPicd(C).Then P(D) := (Ï2)â
(L â O
CĂPicd(C)
(Î))
is a vector bundle of rank e + k â 1. For
each α â„ 0 we define the vector bundle Jα(L, D) := uâ
(vâ(L) â O(α+1)â+vâ(Î)
), where
â â C Ă C Ă Picd(C) is the diagonal, and consider the vector bundle map
Ïα : Jα(L, D)âš â (Ï2)â(P(D))âš,
which on fibres is the dual of the map H0(LC âO(D)) â H0(LC âO(α+1)y+D(D)) for
each pair (y, LC) â C Ă Picd(C).
The cycle Xj,α consists of pairs (y, LC) such that dim Ker(Ïα(y, LC)) â„ 1 anddim Ker(Ïjâα+1(y, LC)) â„ 2 and its class is given by the formula (cf. [Fu],Theorem14.3):
Xj,α ⥠cj+1âα(α) ckâ1+αâj(j â α + 1) â ckâα(α) ckâ2+αâj(j â α + 1),
where cl(ÎČ) := cl
((Ï2)
â(P(D))âš â JÎČ(L, D)âš). Since ctot
((Ï2)
â(P(D))
= eΞ and c1(L) =dη + γ (independent of D!), after a short calculation we obtain that
cl(ÎČ) =Ξl
l!+
Ξlâ1
(l â 1)!
((ÎČ + 1)Îł + (ÎČ + 1)(d + ÎČ(2k â 3 â j))η
)â
Ξlâ1
(l â 2)!(ÎČ + 1)(ÎČ + 2)η.
A straightforward computation now leads to the stated formula. €
18 G. FARKAS
For each integers 0 †a †kâ 2 and b â„ 2 we shall define vector bundles Ga,b over
Gkâ23kâ6 with fibre
Ga,b(C, L) = H0(â§aML â Lâb)
over every point (C, L) â Gkâ23kâ6. Note that Gi,2|Gkâ2
3kâ6= B, where B is the vector bundle
we introduced in Proposition 2.5. Of course, the question is how to extend this descrip-tion over the divisors â0
j . First we will extend Ga,b over Ïâ1(M0gâȘâ0
0âȘâ01) and we start
by constructing the vector bundles G0,b:
Proposition 3.9. For each b â„ 2 there exists a vector bundle G0,b over Ïâ1(M0g âȘâ0
1 âȘâ00) â
Gkâ23kâ6 of rank k(3b â 2) â 6b + 3 whose fibres admit the following description:
âą For (C, L) â Gkâ23kâ6, we have that G0,b
(C, L) = H0(C, Lâb).
âą For t = (C âȘy E, L) â Ïâ1(â01), where L is the linear series gkâ2
3kâ6 on C determining
a limit gkâ23kâ6 on C âȘy E, we have that
G0,b(t) = H0(C, Lâb(â2y)) + C · ub â H0(C, Lâb),
where u â H0(C, L) is any section such that ordy(u) = 0.
âą For t = (C/y ⌠q, L) â Ïâ1(â00), where q, y â C and L is a gkâ2
3kâ6 on C, we have that
G0,b(t) = H0(C, Lâb(ây â q)) â C · ub â H0(C, Lâb),
where u â H0(C, L) is a section such that ordy(u) = ordq(u) = 0.
Proof. Recalling that Ï : Grd â Mg denotes the variety of limit gr
dâs over all treelike
curves of genus g, the morphism Gkâ23kâ6 â G
k(3bâ2)â6b+2b(3kâ6) given by (C, L) 7â (C, Lâb) can
be extended to a morphism Îœb : Gkâ23kâ6 â G
k(3bâ2)â6b+2b(3kâ6) over MgâȘâ0
0âȘâ01 as follows. We
fix a point t =(C âȘy E, l = (lC , lE)
)a limit gkâ2
3kâ6, where lC = (LC , H0(LC)) â Gkâ23kâ6(C)
is such that H0(LC) = H0(LC(â2y))âC · u, for a certain u â H0(LC) with ordy(u) = 0,
while lE =(OE((3k â 6)y), (2k â 6)p + |VE |
)(any limit gkâ2
3kâ6 on C âȘy E is of this form).
Then we set Îœb(t) := (lbC , lbE), where lbC =(Lâb
C , H0(LâbC (â2y)) â C · ub â H0(Lâb
C ))
and
lbE =(OE(b(3k â 6)p), (2k â 3)p + |
(k(3bâ 2)â 6b + 3
)p|
). In other words, (lbC , lbE) is the
b-th power of the original limit linear series gkâ23kâ6.
For a point z = (C/y ⌠q, LC) â â00 with LC being a gkâ2
3kâ6 such that H0(LC) =
H0(LC(ây â q))âC · u, we define Îœb(z) :=(Lâb
C , H0(LâbC (ây â q))âC · ub â H0(Lâb
C )).
The fact that Îœb can be constructed algebraically follows easily from the equations of the
scheme Gkâ23kâ6 described in [EH1], pg. 358. The variety G
k(3bâ2)â6b+2b(3kâ6) carries a tauto-
logical vector bundle T = TC with fibre over each point corresponding to a curve fromM0
g âȘ â00 being the space of global sections of the linear series, while the fibre over a
point corresponding to a curve from â01 is the space of sections of the aspect of the limit
linear series corresponding to the curve of genus g â 1. We define G0,b := Îœâb (T ). The
description of the fibres of the vector bundle G0,b is now immediate. €
Having defined the vector bundles G0,b we now define inductively all vector bun-dles Ga,b: first we define G1,b as the kernel of the multiplication map G0,1âG0,b â G0,b+1,
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 19
that is, by the exact sequence
0 ââ G1,b ââ G0,1 â G0,b ââ G0,b+1 ââ 0.
Having defined Gl,b for all l †a â 1, we define the vector bundle Ga,b by the exactsequence
(4) 0 ââ Ga,b ââ â§aG0,1 â G0,bda,bââ Gaâ1,b+1 ââ 0.
We now prove the right-exactness of the sequence (4) which will ensure the correctnessof the definition of Ga,b:
Proposition 3.10. For integers b â„ 2 and 1 †a †i, the Koszul multiplication map da,b :â§aG0,1 âG0,b â Gaâ1,b+1 is well-defined and surjective. In particular one can define the vectorbundle Ga,b := Ker(da,b) over Ïâ1(M0
g âȘ â00 âȘ â0
1) and the sequence (4) makes sense.
Proof. We do induction on a and start with the case a = 1. We only check the surjectivityof the map G0,1âG0,b â G0,b+1 over the locus Ïâ1(â0
1), the other cases, namely Ïâ1(M0g)
and Ïâ1(â00) being quite similar. It is enough to show that if C is a sufficiently general
curve of genus 2k â 3, y â C is a point and L â W kâ23kâ6(C) is a linear system with a
cusp at y, then the map H0(L) â H0(Lâb(â2y)) â H0(Lâ(b+1)(â2y)) is onto. This isequivalent to H1(ML â Lâb(â2y)) = 0 which follows because of Proposition 2.1.
We now treat the general case 1 †a †i and we want to show that the Koszulmap da,b : â§aH0(L) â G0,b(L) â Gaâ1,b+1(L) is surjective for each b â„ 2 and for each(C, L) as above. For simplicity we have denoted Ga,b(L) = Ga,b(t), where t â Ïâ1(â0
1)is the point (C âȘy E, l), E is an arbitrary elliptic tail and l is a limit linear series havingL as its C-aspect. We can then write H0(L)/H0(L(â2y)) = C · u, where u â H0(L) isuniquely determined up to translation by an element from H0(L(â2y)). We first claimthat the restriction of the Koszul differential
d0a,b : â§aH0(L) â H0(Lâb(â2y)) â H0
(â§aâ1ML â Lâ(b+1)(â2y)
)
is surjective. This is so because Coker(d0a,b) = H1(â§aML â Lâb(â2y)) = 0 (use again
that ML is a stable bundle). But then since
H0(â§aâ1ML â Lâ(b+1)(â2y)
)â Gaâ1,b+1(L)
is a linear subspace of codimension(kâ2aâ1
), to prove that da,b itself is surjective it suffices
to notice that the image of the injective map â§aâ1H0(L(â2y)) â â§aâ1H0(L)âG0,b+1(L)given by
f1 ⧠. . . ⧠faâ1 7â f1 ⧠. . . ⧠faâ1 â ub+1 âaâ1â
l=1
(â1)lâ1u ⧠f1 ⧠. . . ⧠fl ⧠. . . ⧠faâ1 â flub
is entirely contained in Im(da,b) and is clearly disjoint from H0(â§aâ1MLâLâ(b+1)(â2y)
).
€
To compute the intersection numbers of the divisors Zg,i with the test curves C0
and C1 we need to understand the restriction of the vector bundle Gi,2 to X and Y .
20 G. FARKAS
Proposition 3.11. Let C be a general curve of genus 2k â 3 and k = 3i + 6 with i â„ 0. If C1
is the test curve in â1 obtained by attaching to C a fixed elliptic tail at a varying point of C andX = Ïâ1(C1), then we have the following formula:
c1(G2,i|X) =
(3i + 4
i
)(cx x + cη η + cγ γ + cΞ Ξ),
where
cη = â27i + 40, cx = â27i4 + 153i3 + 331i2 + 323i + 120
(3i + 4)(i + 1)
cÎł =5i + 8
3i + 4, cΞ = â
27i3 + 101i2 + 124i + 48
(3i + 3)(3i + 4).
Proof. For j â„ 2 we define the jet bundle J1(Mâj) := (uâ)
(vâ(OCĂCkâ2
(Mâj)) âO2â
),
where u, v : CĂCĂCkâ2 â CĂCkâ2 are the projections. We have two exact sequencesof vector bundles on X :
0 ââ uâ
(vâ(Mâj) âO(â2â)
)|X
ââ G0,j|X ââ Mâj|X ââ 0
and0 ââ uâ
(vâ(Mâj) âO(â2â)
)|X
ââ Ïâ2(Fj)|X ââ J1(M
âj)|X ââ 0,
from which we can write that
c1(G0,j |X) = c1(Ïâ2(Fj)|X) â c1
((Ïâ
1KC âMâj)|X)
=
= âj2Ξ â (3j + 4)(k â 2) η + jÎł â j((3j â 2)(k â 2) â 1) x.
From the exact sequences defining Gi,2 we then obtain that
c1(Gi,2|X) =iâ
l=0
(â1)lc1(â§iâlG0,1|X â G0,l+2|X).
Note that Riemann-Roch gives that rk(G0,l+2) = (3kâ6)l+4kâ9, while c1(â§iâlG0,1|X) =(
kâ2iâlâ1
)(âΞ â (k â 2)x). To obtain a closed formula for c1(Gi,2|X) we now specialize to
the case k = 3i + 6 and we write that
c1(Gi,2|X) =iâ
l=0
(â1)l[(
(9i + 12)l + (12i + 15))( 3i + 4
i â l â 1
)(âΞ â (3i + 4)x)+
+
(3i + 5
i â l
)(â(l + 2)2Ξ + (l + 2)Îł â (3i + 4)(3l + 10)η â (l + 2)((3l + 4)(3i + 4)â 1)x
)].
A long but elementary calculation leads to the stated formula.
€
A similar calculation yields the first Chern class of Gi,2|Y :
Proposition 3.12. Let (C, q) be a general pointed curve of genus 2k â 3 and C0 the test curvein â0 obtained by identifying the fixed point q with a varying point y on C. If Y = Ïâ1(C0),then we have the formula
c1(Gi,2|Y ) =
(3i + 4
i
)(dx x + η + dΞ Ξ),
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 21
where
dx = â27i4 + 153i3 + 331i2 + 323i + 120
(3i + 4)(i + 1), dΞ = â
27i3 + 101i2 + 124i + 48
(3i + 3)(3i + 4).
Proof. We define Îq := C Ă q Ă Ckâ2 and denote by Îč : C Ă Ckâ2 â Îq the inclusionmap. Then we have the exact sequences on Y
0 ââ (uâ)(vâ(Mâj) âO(ââ â Îq)
)|Y
ââ G0,j|Y ââ Mâj|Y ââ 0,
and
0 ââ Îčâ(vâ(Mâj) âO(ââ)
)|Y
ââ (uâ)(vâ(Mâj) âOâ+Îq
)|Y
ââ Ïâ2(Fj)|Y ââ 0.
Using that c1(M) = (3k â 6)η â Îł â x, it is now straightforward to check that
c1
(Îčâ(vâ(Mâj) âO(ââ))
)= âjx â η,
hence c1(G0,j|Y ) = c1(Ïâ2(Fj))|Y + jx + η = âj2Ξ + η â j((3j â 2)(k â 2)â 1)x. But then
as in the previous proposition we have that
c1(Gi,2|Y ) =
iâ
l=0
(â1)lc1(â§iâlG0,1|Y â G0,l+2|Y ),
which after some calculations yields the stated formula in the special case k = 3i+6. €
In what follows we extend the vector bundles Ga,b over the whole Gkâ23kâ6. Roughly
speaking, the fibre of Ga,b over a point corresponding to a singular curve C âȘy D will
be the space H0(C âȘD,â§aML â Lâb), where L will be a line bundle on C âȘD obtained
by twisting appropriately a limit gkâ23kâ6 on C âȘ D. The twisting is chosen such that,
the vector bundles Ga,b sit in the exact sequences (4). In the next statement we use thenotation introduced in Proposition 3.5:
Theorem 3.13. For integers 0 †a †i, b ℠2, or (a, b) = (0, 1), there exists a vector bundle
Ga,b over Gkâ23kâ6 having the following properties:
(1) If (C, L) â Gkâ23kâ6, then Ga,b(C, L) = H0(â§aML â Lâb), that is, Ga,b |Gkâ2
3kâ6= Ga,b.
If t = (C âȘy D, LC , LD) â Ïâ1(â0j ) â Gkâ2
3kâ6, with 2 †g(D) = j †k â 1, g(C) =
2k â 2 â j, LC â W kâ23kâ4âj(C) and LD â W kâ2
kâ2+j(D), then there are two situations:
(2) If 2 †j †2i + 2, then Ga,b(t) = H0(C âȘ D,â§aMLCâȘDâ Lâb
CâȘD), where LCâȘD =(LC(â(j + 2)y), LD(â(k â 2 â j)y)
)â Pic3kâ6â2j(C) Ă Pic2j(D) is a globally generated
line bundle on C âȘ D.
(3) If 2i+3 †j †3i+5 and c := [(j +2)/2], then Ga,b(t) = H0(C âȘD,â§aMLCâȘDâLâb
CâȘD),
where LCâȘD =(LC(âcy)), LD(â(k â c)y)
)â Pic3kâ4âjâc(C) Ă Picj+câ2(D) is a globally
generated line bundle on C âȘ D.
(4) There are exact sequences over Gkâ23kâ6: 0 ââ Ga,b ââ â§aG0,1 â G0,b ââ Gaâ1,b+1 ââ 0.
22 G. FARKAS
Proof. For j â„ 2 the divisors â0j are mutually disjoint and we can carry out the construc-
tion of Ga,b over each open set Ïâ1(M0g âȘ â0
j ) â Gkâ23kâ6 and glue the resulting bundles
together. We denote by Ï : Cj â M0g âȘ â0
j the restriction of the universal curve and
by p : Y = Cj ĂM0gâȘâ0
jGkâ2
3kâ6 â Gkâ23kâ6. Then (Ïp)â1(â0
j ) = Dj + D2kâ2âj , where
Dj (resp. D2kâ2âj) denotes the divisor in Y corresponding to the marked point beingon the genus j (resp. 2k â 2 â j) component. If L = L0,3kâ6 is the Poincare bundleon Y chosen such that it parametrizes bundles having bidegree (0, 3k â 6) on curves[D âȘy C] â â0
j , then for b â„ 1 we set
G0,b := pâ(Lâb âOY(â2bj Dj)
), when 2 †j †2i + 2,
and G0,b := pâ(Lâb â OY(âb(j â 2 + c)Dj)
)when 2i + 3 †j †3i + 5. For each point
t = (R = C âȘy D, LC , LD) â Ïâ1(â0j ), we have that G0,b(t) = H0(R, Lâb
R ), where in the
case j †2i + 2 we have LR = (LC(â(j + 2)y), LD(â(k â 2 â j)y)) â Pic3kâ6(R) and
since H1(R, LâbR ) = 0 for b â„ 2, it follows that G0,b is a vector bundle for all b â„ 1. When
1 †a †i, the sheaves Ga,b are defined inductively using the sequences (4). For this tomake sense and in order to conclude that Ga,b are vector bundles, one has to show that
(5) H1(R,â§aMLRâ Lâb
R ) = 0,
for all b ℠2 and 0 †a †i. To achieve this we use the Mayer-Vietoris sequence on R:
0 â â§aMLRâLâb
R ââ(â§aMLR
âLâbR
)|Câ
(â§aMLR
âLâbR
)|D
ââ â§aMLRâLâb
R âC(y) â 0,
together with the exact sequences
0 ââ H0(LR|C(ây)) âOD ââ MLRâOD ââ MLR|D
ââ 0,
and
0 ââ H0(LR|D(ây)) âOC ââ MLRâOC ââ MLR|C
ââ 0.
We obtain that (5) holds if we can show that
H1(C,â§aMLR|Câ Lâb
R|C) = H1(C,â§aMLR|Câ Lâb
R|C(ây)) = 0
and
H1(D,â§aMLR|Dâ Lâb
R|D) = H1(D,â§aMLR|Dâ Lâb
R|D(ây)) = 0.
We only check this when j †2i + 2 the remaining case being similar. Since h0(LR|C) =
kâ1âj and h0(LR|D) = j +1, the inequalities g(C)+minh0(LR|C)â1, i †deg(LR|C)
and g(D) + minh0(LR|D)â 1, i †deg(LR|D) are satisfied precisely because j †2i + 2and we can invoke Lemma 2.1. €
Next, for 0 †a †i and b â„ 1 we define vector bundles Ha,b over Gkâ23kâ6 having
fibre Hi,j(C, L) = H0(â§iMPkâ2(j)) over each point (C, L) â Gkâ23kâ6 corresponding to an
embedding C|L|â Pkâ2. First we set H0,1 := G0,1 and H0,b := SymbH0,1 for b â„ 1. Having
already defined Haâ1,b for all b â„ 1, we define Ha,b via the exact sequence
(6) 0 ââ Ha,b ââ â§aH0,1 â SymbH0,1 ââ Haâ1,b+1 ââ 0.
The bundles Ha,b being defined entirely in terms of H0,1, the right exactness of thesequence (6) is obvious. There is a natural vector bundle morphism Ïa,b : Ha,b â Ga,b
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 23
over Gkâ23kâ6; when k = 3i + 6, a = i and b = 2, then rank(Hi,b) = rank(Gi,b) and the
degeneracy locus of Ïi,2 is the codimension 1 compactification over Gkâ23kâ6 of the locus
Ug,i defined in Proposition 2.5.
Next we compute the Chern classes of Ha,b along the curves X and Y :
Proposition 3.14. When k = 3i + 6, we have the following formulas for the first Chern classof Hi,2:
c1(Hi,2|X) = c1(Hi,2|Y ) = â(3i + 6)
(3i + 4
i
)((3i + 4)x + Ξ
).
Proof. Using repeatedly the sequence (6) we obtain the formula
c1(Hi,2|X) = c1(Hi,2|Y ) =iâ
l=0
(â1)lc1(â§iâlG0,1|X â Syml+2G0,1|X).
Since c1(G0,1) = âΞ â (3i + 4)x and c1(Symj(G0,1)) = j(3i+4+j)(3i+5)(3i+4)
(3i+j+3
j
)c1(G0,1), the
stated formula is obtained after a straightforward calculation. €
In the remainder of this section we will compute c1(Gi,2 â Hi,2) · Ïâ(Cj), whereCj â âj is the test curve associated to [C] â M2kâ2âj . We retain the notation intro-duced in Propositions 3.4 and 3.5. There are two definitions of the bundles Ga,b|Ïâ(â0
j )
depending whether j †2i+2 or j ℠2i+3 and we will explain in detail the calculationsonly in the first case, the second being similar. We start by describing the Chern number
of the restriction of G0,1 to the 1-cycleâ[j/2]
α=0 a(j, α) Xj,α on C Ă Pic3kâjâ4(C):
Proposition 3.15. Suppose C is a general curve of genus 2kâ 2â j where 2 †j †2i+2 andlet Cj â âj â M2kâ2 be the test curve associated to C. Then
c1(G0,1|P[j/2]
α=0 a(j,α)Xj,α) = âΞ â j
((3k â j â 4)η + Îł
)â j(j + 1)(2k â 3 â j)η + â”,
where â” = 6(j + 1)
(6i+10âj
3i+6
)(6i+8
j
)(3i+4
j
) .
Proof. For each 0 †α †j/2, we fix once and for all a linear series LD â W kâ2kâ2+j(D)
such that h0(LD(â(k + 1 â α)y)) = α and h0(LD(â(k + α â j)y)) = j â α (there are
a(j, α) such gkâ2kâ2+j âs and for each of them Xj,α Ă LD is a component of Ïâ(Cj)). Just
like in Proposition 3.8, we introduce the Poincare bundle L on Xj,α and we define thesheaf on Xj,α given by P := Ïâ
2(Ï2âL). Furthermore, for each a â„ 0 we consider theTaylor map Îœa : P â Ja(L) (the target being the a-th jet bundle of L), and we setFa := Im(Îœa) â Ja(L) and Pa+1 := Ker(Îœa).
For each pair (y, LC) â Xj,α we have an exact sequence
0 ââ H0(D, LR|D(ây)) ââ H0(C âȘy D, LR) ââ H0(C, LR|C) ââ 0,
where LR =(LC(â(j+2)y), LD(â(kâ2âj)y)
)is the line bundle on CâȘD whose global
sections give the fibre of G0,1 at each point in Gkâ23kâ6. This exact sequence globalizes to
24 G. FARKAS
an exact sequence of vector bundles on Xj,α
0 ââ H0(LD(â(k â 1 â j)y)) âOXj,α ââ G0,1|Xj,αââ Pj+2 ââ 0,
thus c1(G0,1|Xj,α) = c1(Pj+2). We now study how the bundles Fa relate to one another
via the exact sequences 0 ââ Ïâ1(K
âaC ) â L ââ Ja(L) ââ Jaâ1(L) ââ 0, linking
successive jet bundles. Recalling that for a generic point (y, LC) â Xj,α we have thataLC (y) = (0, . . . , α â 1, α + 1, . . . , j â α, j â α + 2, . . . , k) (cf. Proposition 3.4), it followsthat Fa = Ja(L) for a †α â 2, while Fαâ1 is obtained from Jαâ1(L) via an elementarytransformation along the divisor D1 â Xj,α consisting of the b(2k â 2â j, k â 2 + αâ j)
points (y, LC) â Xj,α where aLCαâ1(y) = α. More precisely, one has the exact sequence
0 ââ Ïâ1(K
â(αâ1)C ) â LâOXj,α(âD1) ââ Fαâ1 ââ Jαâ2(L) ââ 0.
Furthermore, we have exact sequences
0 ââ Ïâ1(K
â(α+1)C ) â L ââ Fα+1 ââ Fαâ1 ââ 0 and
0 ââ Ïâ1(K
âaC ) ââ Fa ââ Faâ1 ââ 0, for all α + 2 †a †j â α â 2.
If D2 â Xj,α is the divisor consisting of the b(2k â 2 â j, k â 1 + α â j) points (y, LC)
satisfying the condition aLCjâαâ1(y) = j â α + 1, then we also have the exact sequences
0 ââ Ïâ1(K
â(jâα)C ) â LâOXj,α(âD2) ââ Fjâα ââ Fjâαâ1 ââ 0,
0 ââ Ïâ1(K
â(jâα+2)C ) â L ââ Fjâα+2 ââ Fjâα ââ 0,
and finally
0 ââ Ïâ1(K
âaC ) â L ââ Fa ââ Faâ1 ââ 0, for j â α + 2 †a †k â 1.
We gather all the information contained in these sequences to obtain that
c1(Fj+1) = jc1(L) + j(j + 1)(g(C) â 1)η â deg(D1 + D2), and then
c1(G0,1|P
α a(j,α)Xj,α) = c1(P)âc1(Fj+1) = âΞâj((3kâjâ4)η+Îł)âj(j+1)(2kâ3âj)η+â”,
where â” :=
[j/2]â
α=0
a(j, α)(b(2k â 2 â j, k â 2 + α â j) + b(2k â 2 â j, k â 1 + α â j)
)=
= j!(6i + 10 â j)!
j+1â
α=0
(j â 2α â 1)(j â 2α)(j â 2α + 1)2
(3i + 5 â α)! (3i + 5 + α â j)! (j â α + 1)! α!.
This sum can be computed with Maple which finishes the proof. €
Next we compute the Chern class of G0,b with b â„ 2. The proof being similar tothat in Proposition 3.15 (in fact simpler), we decided to omit it.
Proposition 3.16. Let b ℠2 and 2 †j †2i + 2. Then for each 0 †α †j/2 we have that
c1(G0,b|Xj,α) = âb2Ξ â b2(j + 2)(Îł + (3k â 4 â j)η) â
(bj + 2b
2
)(2k â 3 â j)η.
We turn our attention to the remaining components of Ïâ(Cj). First we show that
YâČâČâČ
j,α does not appear in the computation of c1(Gi,2 âHi,2) · Ïâ(Cj):
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 25
Proposition 3.17. For 0 †α †j/2, a â„ 0 and b â„ 1, we have that c1(Ga,b) · YâČâČâČ
j,α =
c1(Ha,b) · YâČâČâČ
j,α = 0.
Proof. Clearly, it suffices to show that c1(G0,b) · YâČâČâČ
j,α = 0 for all b ℠1 and then use theexact sequences (4) and (6). We carry this out only for b = 1 the case b ℠2 being analo-
gous. Fix l = (LD, VD â H0(LD)) â YâČâČâČ
j,α. By duality KD âLâšD((k +2âα)y) is one of the
a(j, α) (that is, finitely many) linear systems g1jâα+1 on D with a (j â 2α + 1)-fold point
at y. The space of sections VD â H0(LD) is chosen such that H0(LD(â2y)) â VD,
hence YâČâČâČ
j,α can be identified with the disjoint union of a(j, α) copies of the projec-
tive line P(H0(LD)/H0(LD(â2y))
), one for each choice of LD. Then H0(LCâȘD|D) =
H0(LD(â(k â 1 â j)y)), for each l â YâČâČâČ
j,α (that is, G0,1(l) is independent of VD). Global-izing this we get that G
0,1|YâČâČâČj,α
is trivial. €
In a similar fashion we have the following:
Proposition 3.18. For 0 †α †j/2, a â„ 0 and b â„ 1, we have the equalities c1(Ga,b) · YâČ
j,α =
c1(Ha,b) · YâČ
j,α = 0.
Proposition 3.19. For 0 †α †j/2 and b ℠1, we have the equality
c1(G0,b) · YâČâČ
j,α = âb2(j â 2α)j!
α!(j â α)!.
Proof. We start by recalling that YâČâČ
j,α is the locus of line bundles LD â Pickâ2+j(D) such
that h0(LD(â(k + 1 + αâ j)y)) â„ j âαâ 1 and h0(LD(â(kâα + 1)y)) ℠α. Using [Fu],
Theorem 14.3, one finds that YâČâČ
j,α ⥠jâ2αα!(jâα)!Ξ
jâ1. Having fixed once and for all one of
the finitely many linear series LC â W kâ23kâjâ4(C) such that LC Ă Y
âČ
j,α â Gkâ23kâ6, there
is an identification
G0,1(LD) = H0(LD(â(k â 2 â j))y),
for each LD â YâČâČ
j,α. Since the vector bundle Pickâ2+j(D) 3 LD 7â H0(LD â OÎČy), isalgebraically trivial for each ÎČ â„ 0 (remember that y â D is a fixed point), we obtainthat c1(G0,1|Y
âČâČj,α
) = âΞ. Similarly, for b â„ 2 we get that c1(G0,b|YâČâČj,α
) = âb2Ξ from which
now the conclusion follows. €
Propositions 3.15, 3.16, 3.17, 3.18, 3.19 will enable us to compute the intersectionnumbers c1(Gi,2âHi,2)·Ï
â(Cj) needed to determine the coefficient of ÎŽj in the expansion
of [Z6i+10,i]. Repeatedly using (6) , one can write that
(7) c1(Hi,2) =iâ
l=0
(â1)lc1
(â§iâlG0,1 â Syml+2G0,1
)= (3i + 6)
(3i + 4
i
)c1(G0,1)
(note the similarity with Proposition 3.14). Using (4) we also have that
(8) c1(Gi,2) =
iâ
l=0
(â1)lc1
(â§iâlG0,1 â G0,l+2
),
26 G. FARKAS
and a simple calculation yields that
(9) c1(Gi,2 âHi,2) = â13i2 + 35i + 24
(3i + 4)(i + 1)
(3i + 4
i
)c1(G0,1) +
iâ
l=0
(â1)l
(3i + 5
i â l
)c1(G0,l+2).
4. THE SLOPE OF Z6i+10,i
In this section we finish the calculation of the class of the virtual divisor Zg,i. Asbefore, we fix i â„ 0, k = 3i + 6 and g = 6i + 10, hence Ï : Hi,2 â Gi,2 is a morphism
between two vector bundles of the same rank defined over G3i+49i+12.
Theorem 4.1. We fix i â„ 0 and denote by Ï : G3i+49i+12 â M6i+10 the natural projection map.
Then we have the following formula for the class of the pushforward to M6i+10 of the virtualdegeneracy locus of the morphism Ï : Hi,2 â Gi,2:
Ïâ(c1(Gi,2 âHi,,2)) =1
3i + 5
(6i + 7
3i + 4, 2i + 1, i + 2
)(aλ â b0 ÎŽ0 â · · · â b3i+5 ÎŽ3i+5
),
where
a =(4i + 7)(6i2 + 19i + 12)
(2i + 3)(i + 2), b0 =
12i2 + 31i + 18
3(2i + 3), b1 =
12i2 + 33i + 20
i + 2and
bj =jf(i, j)
6(i + 1)(i + 2)(2i + 3)(6i + 9 â j), for 2 †j †2i + 2,
withf(i, j) = 864 i5 â (240j â 5256)i4 + (16j2 â 998j + 11830)i3+
+(31j2 â 1254j + 11585)i2 + (â5j2 â 286j + 3981)i â 24j2 + 240j â 216,
and
bj =2(13i2 + 35i + 24)(3i + 5)
(i + 1)(i + 2)(6i+10
j
)(
3i + 4
(j â 1)/2
)2
+g(i, j)
12(i + 1)(i + 2)(2i + 3)(6i + 9 â j)
for an odd j such that 2i + 3 †j †3i + 5, where
g(i, j) = 1728ji5 â (576j2 â 10512j + 540)i4 + (48j3 â 2592j2 + 24120j â 3234)i3+
(140j3â3926j2+25176jâ7278)i2+(109j3â2107j2+10875jâ7263)i+12j3â156j2+972jâ2700
and
bj =(13i2 + 35i + 24)((3i + 5)(2j + 1) â j2)
2(i + 1)(i + 2)(3i + 5)(6i+10
j
)(
3i + 5
j/2
)2
+h(i, j)
12(i + 1)(i + 2)(2i + 3)(6i + 9 â j),
for an even j such that 2i + 3 †j †3i + 5, where
h(i, j) = 1728ji5 â (576j2 â 10512j + 1020)i4 + (48j3 â 2592j2 + 24280j â 6034)i3+
(140j3â3942j2+25830jâ13400)i2+(109j3â2145j2+11774jâ13230)i+12j3â180j2+1392jâ4896.
In particular, bj ℠b0 for all 1 †j †3i + 5 and
s(Ïâ(c1(Gi,2 âHi,2))) =a
b0=
3(4i + 7)(6i2 + 19i + 12)
(12i2 + 31i + 18)(i + 2).
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 27
Proof. Since codim(Mg â Mg,Mg) â„ 2, it makes no difference whether we compute
the class Ïâ(Gi,2 âHi,2) on Mg or on Mg and we can write
Ïâ(Gi,2 âHi,2) = Aλ â B0 ÎŽ0 â B1 ÎŽ1 â · · · â B3i+5 ÎŽ3i+5 â Pic(Mg),
where λ, Ύ0, . . . , Ύ3i+5 are the generators of Pic(Mg). We start with the following:
Claim: One has the relation A â 12B0 + B1 = 0.
We pick a general curve [C, q] â M6i+9,1 and at the fixed point q we attach to
C a Lefschetz pencil of plane cubics. If we denote by R â Mg the resulting curve,then R · λ = 1, R · ÎŽ0 = 12, R · ÎŽ1 = â1 and R · ÎŽj = 0 for j â„ 2. The relationA â 12B0 + B1 = 0 follows once we show that Ïâ(R) · c1(Gi,2 â Hi,2) = 0. To achievethis we check that G0,b|Ïâ(R) is trivial and then use (4) and (6). We take [C âȘq E] to be an
arbitrary curve from R, where E is an elliptic curve. Using that limit g3i+49i+12 on CâȘqE are
in 1 : 1 correspondence with linear series L â W 3i+49i+12(C) having a cusp at q (this being
a statement that holds independent of E) and that G0,b|Ïâ(â01) consists on each fibre of
sections of the genus g â 1 aspect of the limit g3i+49i+12, the claim now follows.
Now we determine A, B0 and B1 explicitly. We fix a general pointed curve (C, q) âM6i+9,1 and construct the test curves C1 â â1 and C0 â â0. Using the notationfrom Proposition 3.3, we get that Ïâ(C0) · c1(Gi,2 â Hi,2) = c1(Gi,2|Y ) â c1(Hi,2|Y ) and
Ïâ(C1) · c1(Gi,2 âHi,2) = c1(Gi,2|X) â c1(Hi,2|X) (the other component T of Ïâ(C1) doesnot appear because G0,b|T is trivial for all b â„ 1). On the other hand
C0 · Ïâ(c1(Gi,2 âHi,2)) = (12i + 18)B0 â B1 and C1 · Ïâ(c1(Gi,2 âHi,2)) = (12i + 16)B1,
while we already know that A â 12B0 + B1 = 0. Using Propositions 3.11, 3.12, 3.14, theexpressions for [X] and [Y ] as well as the well-known formula Ξjxkâ2âj = g!/(g â j)!on Ckâ2, we get a linear system of 3 equations in A, B0 and B1 which leads to the statedformulas for the first three coefficients.
To compute Bj for 2 †j †3i + 5, we fix general curves [C] â M6i+10âj and[D, y] â Mj,1 and consider the test curve Cj â âj . Then on one hand we have that2(6i + 9 â j)Bj = Ïâ(Cj) · c1(Gi,2 âHi,2), on the other hand
Ïâ(Cj)·c1(Gi,2âHi,2) = c1(Gi,2âHi,2)·
[j/2]â
α=0
((a(j, α)Xj,α+b(6i+10âj, 3i+5+αâj)Y
âČâČ
j,α
).
Using (9) together with Proposition 3.19, we get that
c1(Gi,2 âHi,2) ·
[j/2]â
α=0
b(6i + 10 â j, 3i + 5 + α â j)YâČâČ
j,α =
=(8i2 + 19i + 12)j!(6i + 10 â j)!(3i + 2)!
i!(2i + 4)!
( jâ
α=0
(j â 2α â 1)(j â 2α)2(j â 2α + 1)
α! (j â α)! (3i + 5 â α)! (3i + 5 + α â j)!
),
which can be computed by Maple, whereas c1(Gi,2 â Hi,2) · (â[j/2]
α=0 a(j, α)Xj,α) can becomputed using (9) and Propositions 3.15 and 3.16 (note in particular that ┠has to bemultiplied by the coeffcient of c1(G0,1) in (9)). The rest now follows and checking thatbj ℠b0 for j ℠1 is elementary. €
28 G. FARKAS
Remark 4.2. If we retain the notation of Theorem 4.1, we see that if we fix j â„ 1 and
let i vary, we have that limiââbj
b0= 6j. This is the same asymptotical estimate as in the
case of the classical Brill-Noether divisors wherebj
b0= 6j(gâj)
g+3 for all j â„ 1 (cf. [EH3]).
Theorem 4.3. Assume that vector bundle morphism Ï : Hi,2 â Gi,2 is non-degenerate at a
general point from Ïâ1(â01). Then Z6i+10,i is a divisor on M6i+10 and
6 < s(Z6i+10,i) =3(4i + 7)(6i2 + 19i + 12)
(12i2 + 31i + 18)(i + 2)< 6 +
12
g + 1.
Remark 4.4. Since Ïâ1(â01) is irreducible, to verify the assumption made in Theorem
4.3 it suffices to show that if C â P3i+4 is a general 1-cuspidal curve with pa(C) = 6i+10and deg(C) = 9i + 12, then C satisfies property(Ni). We checked this for g = 16, 22 (cf.Remarks 2.9 and 2.11), while the case g = 10 was treated in [FP]. The assumption inTheorem 4.3 implies that Ï is non-degenerate over a general point from Ïâ1(â0
0) andalso over a general point from Ïâ1(M0
g), which is equivalent to Zg,i being a divisor on
M0g. Therefore the assumption is slightly stronger than Conjecture 2.6.
Proof of Theorem 4.3. We denote by Z(Ï) the degeneracy locus of Ï, thus Ïâ(Z(Ï)) is
a divisor on M6i+10,i such that Ïâ(Z(Ï)) = Zg,i +â3i+5
j=0 djâj , for certain coefficients
dj â„ 0. Using the Remark above we obtain that d0 = d1 = 0, hence s(Z6i+10,i) =s(Ïâ(c1(Gi,2 âHi,2))) and the rest follows from Theorem 4.1. €
Remark 4.5. In the simplest case i = 0, we can compare the formula for [Z6i+10,i] withour findings in [FP]. Theorem 4.1 gives the formula
Ïâ(c1(G0,2 âH0,2)) ⥠42(7λ â ÎŽ0 â 5ÎŽ1 â4
3ÎŽ2 â 7ÎŽ3 â
44
7ÎŽ4 â
685
42ÎŽ5),
whereas Theorem 1.6 from [FP] says that if K10 is the divisor on M10 of curves lying ona K3 surface then
K10 = 7λ â ÎŽ0 â 5ÎŽ1 â 9ÎŽ2 â 12ÎŽ3 â 14ÎŽ4 â b5ÎŽ5,
where b5 â„ 6. Since we have also established the set-theoretic equality Z10,0 = K10 (cf.[FP], Theorem 1.7), it follows that there is a scheme-theoretic equality Z10,0 = 42K10.Here 42 is the number of pencils g1
6 on a general curve of genus 10, and its appearancein the formula of [Z10,0] has a clear geometric meaning: if a Brill-Noether general curve[C] â M10 fails (N0) for one linear system g4
12 = KC(âg16), then it fails (N0) for all 42
linear systems g412 it possesses, that is, the map Ï : U10,0 â K10,0 is 42 : 1. Moreover, the
vector bundle morphism H0,2 â G0,2 is degenerate along each of the boundary divisorsâ2, . . . ,â5. This situation presumably extends to higher genera as well hence the mainthrust of Theorem 4.1 is that it computes s(Z6i+10,i).
To finish the proof of Theorems 1.1 and 1.2 we specialize to the cases g = 16 and22 and note that the assumption made in Theorem 4.3 is satisfied in these situations (cf.Theorems 2.7 and 2.10).
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 29
5. AN EFFECTIVE DIVISOR ON M14,1
In this section we describe how to construct effective divisors on Mg,n usingsyzygy type conditions for pointed curves. We treat only one example. We denote
by Z114,0 the closure in M14,1 of the locus Z1
14,0 of smooth pointed curves [C, p] â M14,1
for which there exists a linear series L â W 618(C) such that the map
”L,p : Sym2H0(C, L(âp)) â H0(C, Lâ2(â2p))
is not an isomorphism. Just like the in the case of the loci Zg,i on Mg, the locus Z114,0
can be naturally viewed as the pushforward of the degeneracy locus of a morphismbetween two vector bundles of the same rank 21 over G6
18 ĂM14 M14,1.
We are going to show that Z114,0 is a divisor and in order to compute its class we
need some preparations. Recall that for g â„ 3, the group Pic(Mg,1) is freely generatedby λ, the tautological class Ï and the boundary classes ÎŽi = [âi] with 0 †i †g â 1,where for i â„ 1, the generic point of âi is a union of two smooth curves of genus i andg â i meeting at a point, the marked point lying on the genus i component. We denote
by Ï : Mg,1 â Mg the natural forgetful map. We can then write the class of Z114,0 on
M14,1 as
Z114,0 ⥠a λ + c Ï â b0 ÎŽ0 â b1 ÎŽ1 â · · · â b13 ÎŽ13.
As before, we will determine the relevant coefficients in the expression of [Z114,0]
by intersecting our divisor with various test curves.
Proposition 5.1. Let C be a general curve of genus 14 and C = Ïâ1([C]) â M14,1, the
test curve obtained by letting the marked point vary along C. Then C · Z114,0 = 12012, hence
c = C · Z114,0/(2g â 2) = 462.
Proof. Let us fix a linear series L â W 618(C). We count the number of points p â C, for
which the multiplication map ”L,p not injective. If p1 : C Ă C â C and p2 : C Ă C â Care the two projections, we define the vector bundles
E := (p2)â(pâ1(L) âOCĂC(ââ)) and F := (p2)â(p
â1(L
â2) âOCĂC(â2â)).
There is a natural multiplication map ”L : Sym2(E) â F , and the cardinality of its
degeneracy locus is just c1(F) â c1(Sym2(E)). A simple calculation shows that c1(E) =
â4x â Ξ = â18, hence c1(Sym2(E)) = â126, while c1(F) = â98. We obtain that ”L,p isnot an isomorphism for precisely 28 points p â C. Since C has 429 linear series g6
18 (cf.
[ACGH]), we obtain c = (C · Z114,0)/26 = 429 · 28/26 = 462. €
For more relations among the coefficients of [Z114,0] we define the map Îœ : M1,2 â Mg,1
obtained by attaching to each 2-pointed elliptic curve [E, q, p] a fixed general 1-pointedcurve [C, q] â Mgâ1 (the point of attachment being q). One has the pullback formulas
Îœâ(λ) = λ, Îœâ(Ï) = Ïp, Îœâ(ÎŽ0) = ÎŽ0, Îœâ(ÎŽ1) = âÏq and Îœâ(ÎŽgâ1) = ÎŽqp,
where Ïp and Ïq are the tautological classes corresponding to the marked points p andq, while ÎŽqp is the boundary component of curves having a rational tail containing both
30 G. FARKAS
q and p. On M1,2 these classes are not independent and we have the relations Ïq = Ïp,λ = Ïp â ÎŽqp and ÎŽ0 = 12(Ïp â ÎŽqp) (see e.g. [AC2], Proposition 1.9).
Proposition 5.2. If Îœ : M1,2 â M14,1 is as above, then Îœâ(Z114,0) = â . It follows that we
have the relations
a â 12b0 + b13 = 0 and c + b1 = b13.
Proof. We assume that [X = C âȘq E, p â E] â Z114,0. Then there exists a limit linear
series g618 on X determined by its aspects LC â W 6
18(C) and LE â W 618(E), together
with compatible elements
ÏC â Ker”LC: Sym2H0(LC) â H0(Lâ2
C ), ÏE â Ker”LE: Sym2H0(LE) â H0(Lâ2
E ),
satisfying the inequality
ordq(ÏC) + ordq(ÏE) â„ deg(LC) + deg(LE) = 36
and such that ÏE â Sym2H0(LE(âp)) (see [FP], Section 4, for how to study multiplica-tion maps in the context of limit linear series). Because (C, q) â M13,1 is general, thevanishing sequence of LC at q equals (0, 2, 3, 4, 5, 6, 7), the vanishing sequence of LE
at q is (11, 12, 13, 14, 15, 16, 18) and finally, the vanishing sequence of LE at p is either(0, 1, 2, 3, 4, 5, 6) or (0, 1, 2, 3, 4, 5, 7), depending on whether pâq â Pic0(E) is a 7-torsionclass or not.
We claim that ordq(ÏE) †29(= 13 + 16 = 14 + 15). Indeed, otherwise ordq(ÏE) â„30(= 14+16 = 15+15), and since ÏE â Sym2H0(LE(âp)), after subtracting base points
ÏE becomes a 6= 0 element in the kernel of the map Sym2H0(N) â H0(Nâ2), whereN = LE(âpâ14q) â Pic3(E). This is obviously impossible. Therefore ordq(ÏE) †29, soby compatibility, ordq(ÏC) â„ 7(= 2+5 = 3+4). We now show that when (C, p) â M13,1
is chosen generically, there can be no such element ÏC which leads to a contradiction.
Claim: Suppose Ï0, Ï2, Ï3, Ï4, Ï5, Ï6, Ï7 is a basis of H0(LC) adapted to the point q in
the sense that ordq(Ïi) = i. If W (q, LC) â Sym2H0(LC) denotes the 17-dimensionalsubspace spanned by the elements Ïi · Ïj with 4 †i †j †7, Ï3 · Ïj for j â„ 4 and Ï2 · Ïj
for j â„ 5, then the restriction of the multiplication map W (q, LC) â H0(L2C(â7q)) is an
isomorphism. (Note that W (q, LC) does not depend on the chosen basis Ïi).
The proof of this claim is similar to the proof of Theorem 5.1 in [FP]. It is enoughto construct a single 1-cuspidal curve X â P6 with pa(X) = 14 and deg(X) = 18,such that if Îœ : C â X is the normalization of X and q â C is the inverse imageof the cusp, then X does not lie on any quadric contained in W
(q, Îœâ(OX(1))
). We
construct the following cuspidal curve: define Î â P6 to be the image of the map tf7â
[1, t2, t3, t4, t5, t6, t7], then choose a general hyperplane H â P6 which intersects Î indistinct points p1, . . . , p7. Take D â H to be a general smooth curve of genus g(D) = 7and deg(D) = 11 which passes through p1, . . . , p7. Then X := Î âȘ D is a curve ofarithmetic genus 14 and degree 21 having a cusp at the point q = f(0) â Î. The quadricsin W
(q, Îœâ(OX(1))
)can of course be written down explicitly and to show that D can be
chosen such that it is not contained in any quadric from W(q, Îœâ(OX(1))
)amounts to a
simple counting argument. €
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 31
Remark 5.3. The claim we have just proved also shows that Z114,0 is a divisor on M14,1.
Alternatively, this can be proved in the same way as Theorem 2.7.
We now determine the coefficient b1:
Proposition 5.4. Consider general curves [E, p, q] â M1,2 and [C] â M13, and denote by
C2 â M14,1 the test curve consisting of points [C âȘq E, p â E]qâC. Then C2 · Z114,0 =
133848, therefore b1 = C2 · Z114,0/24 = 5577.
Proof. Throughout the proof we will use the notations introduced in Proposition 3.11.
From the proof of Proposition 5.2 it is clear that the intersection number C2 ·Z114,0 equals
the number of pairs (q, D) â X â C Ă C6 such that the map
W (q, KC(âD)) â H0(Kâ2
C (â2D â 7q))
is not an isomorphism. To compute this number we construct the rank 17 vector bun-dle W over X having fibre W (q, D) = W (q, KC(âD)) over each point (q, D) â X . The
curve C being general, the vanishing sequence aKC(âD)(q) will be generically equalto (0, 2, 3, 4, 5, 6, 7), while at a finite number of points (q, D) â X we will have that
aKC(âD)(q) = (0, 2, 3, 4, 5, 6, 8). In order to compute c1(W ) we note that W has a sub-bundle W1 â W which fits into two exact sequences:
0 ââ Sym2uâ
(vâ(M) âO(â4â)
)|X
ââ W1 â uâ
(vâ(M) âO(â4â)
)|XâP ââ 0,
where P = uâ
(vâ(M) â I3
â/I4â
)|X
, and
0 ââ W1 ââ W ââ uâ
(vâ(M) âO(â5â)
)|Xâuâ
(vâ(M) â I2
â/I3â
)|X
ââ 0.
Next, we consider the multiplication map W â uâ
(vâ(Mâ2) â O(â7â)
)|X
whose de-
generation locus we want to compute. The intersection number C2 · Z114,0 is equal to
(c1
(uâ(v
â(Mâ2) âO(â7â)))â c1(W )
)· [X] = (3Ξ â 7x â Îł + 18η) · [X] = 133848,
as it turns out after a short calculation. Here we have used that c1(W ) can be computedfrom the two exact sequences involving W and W1, while
c1
(uâ(v
â(Mâ2) âO(â7â)))
= c1(F2) â c1(J6(Mâ2)) = â4Ξ â 34x + 14Îł â 756η.
€
Since b1 = 5577 we now obtain that b13 = b1+c = 6039. To compute the coefficientb0 (and thus the λ-coefficient a) we use our last test curve:
Proposition 5.5. Let (C, q, p) â M13,2 be a general 2-pointed curve. We denote by C3 â
M14,1 the family consisting of curves [C/y ⌠q, p]yâC. Then C3 ·Z114,0 = c+26b0âb13 =
24453. It follows that b0 = 1155 and a = 7821.
Proof. We retain the notations introduced in Proposition 3.12. We construct a vector
bundle map Sym2(E) â F over the curve Y â CĂC6, where E = uâ
(vâ(M)âO(âÎp)
)
is the bundle with fibre E(y, D) = H0(KC(âD â p)
), while F is the bundle with fibre
F(y, D) = H0(Kâ2
C (â2D â 2p â y â q))â C · t2 â H0
(Kâ2
C (â2D â 2p)),
32 G. FARKAS
where H0(KC(âD â p)
)/H0
(KC(âD â p â y â q)
)= C · t, for every (y, D) â Y .
It is easy to show that c1(E) = â5xâΞ, hence c1(Sym2(E)) = â35xâ7Ξ. The classc1(F) can be computed from the exact sequence
0 ââ uâ
(vâ(Mâ2)âO(â2Îp â Îq ââ)
)|Y
ââ F ââ(Mâ2 âO(â2Îp)
)|Y
(âA) ââ 0,
where A = Y â© Ïâ11 (p) is the effective divisor on the curve Y consisting of all points
(p, D) such that h0(p + q + D) â„ 2. Using the formula for [Y ] (cf. Proposition 3.12),we get that |A| = 429. We also compute that c1
(uâ(v
â(Mâ2) â O(â2Îp â Îq â â)))
=
â4Ξ â 40x â 33η + 2Îł and of course c1(Mâ2 â O(â2Îp)) = 2(17η â Îł â x). Therefore
we can write that
C3 · Z114,0 =
(c1(F) â c1(Sym2(E))
)· [Y ] â |A| = (â7x + 3Ξ + η) · [Y ] â |A| = 24453.
€
We have thus far determined the coefficients a, c, b0, b1, b13 in the expansion of
[Z114,0]. This is already enough to conclude that [Z
114,0] lies outside the cone of Pic(M14,1)
spanned by pullbacks of effective divisors from M14, Brill-Noether divisors on M14,1
and boundary divisors (see the discussion after Theorem 1.4 for the relevance of thisresult). To get a bound on the remaining coefficients bj for 2 †j †12, we use a variant
of Theorem 1.1 from [FP]. The boundary divisor âj â M14,1 with 3 †j †12, j 6= 4 isfilled-up by pencils Rj obtained by attaching to a fixed 2-pointed curve [B, p, q] â Mj,2
a variable 1-pointed curve [C, q] â M14âj,1 moving in a Lefschetz pencils of curves ofgenus 14 â j sitting on a fixed K3 surface. Deformations of Rj cover âj for j 6= 2, 4,
hence we have that Rj · Z114,0 ℠0. Since one also has the relations (see [FP], Lemma 2.4)
Rj · λ = 15 â j, Rj · ÎŽj = â1, Rj · ÎŽ0 = 6(17 â j), Rj · Ï = 0 and Rj · ÎŽi = 0 for i 6= 0, j,
we immediately get the estimate bj â„ 15 + 27j for all 3 †j †12, j 6= 4. To obtain thebounds b2 â„ 325 and b4 â„ 271 we use similar pencils filling up â2 and â4 respectively.We skip these details. This completes the proof of Theorem 1.4.
6. THE KODAIRA DIMENSION OF Mg,n
In this last section we use the effective divisors Z16,1 and Z22,2 to improve Loganâs
results about which moduli spaces Mg,n are of general type. For a general reference
about Pic(Mg,n) we refer to [AC2] and [Log]. For each 1 †i †n we denote by Ïi :
Mg,n â Mg,1 the morphism forgetting all marked points except the one labelled by i
and we also consider the map Ï : Mg,n â Mg which forgets all marked points. We
recall that the canonical class of Mg,n is given by the formula
KMg,n= 13λ â 2ÎŽ0 +
nâ
i=1
Ïi â 2â
iâ„0, S
ÎŽi:S ââ
S
ÎŽ1:S .
Here Ïi is the tautological class corresponding to the i-th marked point, while ÎŽi:S withi â„ 0 and â 6= S â 1, . . . , n denotes the class of the boundary divisor with genericpoint being a union of two curves of genus i and g â i such that the marked points onthe genus i component are precisely those labelled by S.
SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg 33
Proof of Theorem 1.3. We start with the case g = 22 and we show that M22,2 is of general
type. On M22,2 we consider the averaged pullback of the Weierstrass divisor
W12 := Ïâ1(W) + Ïâ
2(W) ⥠â2 λ +g(g + 1)
2(Ï1 + Ï2) â
â
iâ„0,S 6=â
ci:S ÎŽi:S ,
where ci:S â„ 0, and significantly, the coefficient of ÎŽ0 is 0. By Theorem 1.2, we have an-other effective class, namely [Ïâ(Z22,2)] = c(1665
256 λâÎŽ0â· · · ), where c > 0. One can easily
check that KM22,2can be written as a positive combination of [W12], [Ï
â(Z22,2)], Ï1 + Ï2
and some other boundary classes. Since Ï1 + Ï2 is big and nef, it follows that M22,2 isof general type.
When g = 21 we use the maps Ïi,j : M21,5 â M22 for 1 †i < j †5, whereÏi,j identifies the marked points labelled by i and j and forgets those labelled by i, jc.The Q-divisor class
â
i<j
Ïâi,j(Z22,2) ⥠c(
1665
256λ â ÎŽ0 +
2
5
5â
i=1
Ïi â · · · ), where c > 0,
is obviously effective on M21,5. Since the Q-class âλ + 11â5
i=1 Ïi â 0 · ÎŽ0 â · · · , is alsoeffective (cf. [Log], Theorem 5.4 - we have retained only the coefficients that play arole in our argument), once again we see that KM21,5
can be written as the sum of an
effective divisor and a positive multiple ofâ5
i=1 Ïi.
Finally we settle the case g = 16: we adapt Theorem 5.4 from [Log] to conclude
that the Q-class âλ + 239 (
â9i=1 Ïi) â 0 · ÎŽ0 â · · · , is effective on M16,9 (precisely, this
is the class of the S9-orbit of the closure in M16,9 of the effective divisor D on M16,9
consisting of points [C, p1, . . . , p9] such that h0(C, 2p1 + · · · + 2p7 + p8 + p9) â„ 2 and wehave explicitly indicated that the coefficient of ÎŽ0 is 0 and retained only the coefficientsthat are significant for this calculation). The class [Ïâ(Z16,1)] = c(407
61 λâÎŽ0â· · · ), wherec > 0, is also effective and one writes KM16,9
as a positive sum of these two effective
classes, boundary classes and the big and nef classâ9
i=1 Ïi. €
Remark 6.1. Theorem 6.3 from [Log] claims that M22,4 is of general type but the nu-merical argument used in the proof seems to be incorrect.
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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TEXAS, AUSTIN, TX 78712
E-mail address: [email protected]