TOWARDS THE AMPLE CONE OF Mg,n §0. Introduction and statement of results The moduli ...

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JOURNALOFTHE
AMERICANMATHEMATICALSOCIETY
Volume15,Number2,Pages273{294
S0894-0347(01)00384-8
ArticleelectronicallypublishedonDecember20,2001
TOWARDS THE AMPLE CONE OF M
g;n
ANGELA GIBNEY, SEAN KEEL, AND IAN MORRISON
To BillFulton on his sixtiethbirthday
x0. Introduction and statement of results
Themodulispaceofstablecurvesisamongthemoststudiedobjectsinalgebraic
geometry. Nonetheless,itsbirationalgeometryremainslargelyamysteryandmost
Mori theoretic problems in the area are entirely open. Here we consider one of the
most basic:
(0.1) Question (Mumford). What are the ample divisors on M ?
g;n
There is a strati(cid:12)cation of M by topological type where the codimension k
g;n
strata are the irreducible components of the locus parameterizing pointed curves
withatleastksingularpoints. Anampledivisormustintersectanyonedimensional
stratum positively. It is thus natural to consider the following conjecture.
(0.2) Conjecture: F (M ). A divisor on M is amplei(cid:11) it has positive inter-
1 g;n g;n
section withall 1-dimensional strata. In other words, anye(cid:11)ective curvein M is
g;n
numerically equivalent to an e(cid:11)ective combination of 1-strata. Said di(cid:11)erently still,
every extremal ray of the Mori cone of e(cid:11)ective curves NE (M ) is generated by
1 g;n
a one dimensional stratum.
F (M ) was previously conjectured by Fulton, whence our choice of notation.
1 0;n
Special cases of F (M ) are known (see [KeelMcKernan96], x7) but the general
1 0;n
case remains open. The central observation of this work is that the case of higher
genusisnoharder. Indeed,NE (M )isnaturallyaquotientofNE (M )(cid:2)
1 g;n 1 0;2g+n
R(cid:21)0 and Conjecture (0.2) is true for all g i(cid:11) it is true for g = 0; see (0.3){(0.5).
Further, the second formulation of Conjecture (0.2) holds except possibly for very
degeneratefamilies in M ; see (0.6). Moreover,we are able to give strong results
g;n
on the contractions of M (i.e. morphisms with geometrically connected (cid:12)bers
g;n
from it to other projective varieties). For example we show that for g (cid:21)2 the only
(cid:12)brations of M are compositions of a tautological (cid:12)bration, given by dropping
g;n
ReceivedbytheeditorsSeptember 5,2000.
2000 Mathematics Subject Classi(cid:12)cation. Primary14H10,14E99.
Key words and phrases. Amplecone,Moricone,modulispace, stablecurve.
During this research, the (cid:12)rst two authors received partial support from a Big XII faculty
researchgrant,andagrantfromtheTexasHigherEducationCoordinatingBoard.
The (cid:12)rst author also received partial support from the Clay Mathematics Institute, and the
secondfromtheNSF.
Thethirdauthor’sresearchwaspartiallysupportedbyaFordhamUniversityFacultyFellow-
shipandbygrantsfromtheCentredeRecercaMatem(cid:19)aticaforastayinBarcelonaandfromthe
ConsiglioNazionalediRicercheforstaysinPisaandGenova.
(cid:13)c2001 American Mathematical Society
273
274 ANGELA GIBNEY, SEAN KEEL, AND IAN MORRISON
points, with a birational morphism; see (0.9){(0.11). Here are precise statements
of our results:
Weworkoveranalgebraicallyclosed(cid:12)eldofanycharacteristicotherthan2(this
last assumption is used only at the end of the proof of (3.5)).
By the locus of (cid:13)ag curves, we shall mean the image F of the morphism
g;n
f : M =S ! M induced by gluing on g copies of the pointed rational
0;g+n g g;n
elliptic curve at g points (which the symmetric group S (cid:26) S permutes). The
g g+n
map f is the normalization of F .
g;n
(0.3) Theorem. g (cid:21) 1. A divisor D 2 Pic(M ) is nef i(cid:11) D has non-negative
g;n
intersection with all the one dimensional strata and Dj is nef. In particular,
Fg;n
F (M =S ) implies F (M ).
1 0;g+n g 1 g;n
Using the results of [KeelMcKernan96], Theorem (0.3) has the following conse-
quence.
(0.4) Corollary. F (M ) holds in characteristic zero as long as g+n (cid:20) 7 and
1 g;n
when n=0 for g (cid:20)11.
For g (cid:20)4, F (M ) was obtained previously by Faber.
1 g
We call E (cid:26) M (g (cid:21)1) a family of elliptic tails if it is the image of the map
g;n
M !M obtained by attaching a (cid:12)xed n+1-pointed curve of genus g(cid:0)1 to
1;1 g;n
the moving pointed elliptic curve. Any two families of elliptic tails are numerically
equivalent. In (2.2), we will see that all the 1-strata except for E are numerically
equivalent to families of rational curves. Note that we abuse language and refer
to a curve as rational if all of its irreducible components are rational. Thus it is
immediate from (0.3) that the Mori cone is generated by E together with curves
in R (cid:26) M , the locus of rational curves. The locus R is itself (up to nor-
g;n g;n g;n
malization) a quotient of M : By deformation theory R is the closure of
0;2g+n g;n
the locus of irreducible g-nodal (necessarily rational) curves. The normalization
of such a curve is a smooth rational curve with 2g +n marked points. Thus, if
G(cid:26) S is the subgroup of permutations commuting with the product of g trans-
2g
positions (12)(34):::(2g(cid:0)12g), the normalization of R is naturally identi(cid:12)ed
g;n
with M =G. Thus (0.3) implies the following result:
0;2g+n
(0.5) Corollary. D 2 Pic(M ) is nef i(cid:11) Dj is nef. Equivalently, the
g;n Rg;n[E
natural map
NE (M =G[E)=NE (M =G)(cid:2)R(cid:21)0 !NE (M )
1 0;2g+n 1 0;2g+n 1 g;n
is surjective.
In fact we prove the following strengthening of (0.3):
(0.6) Theorem. Let g (cid:21) 1 and let N (cid:26) NE (M ) be the subcone generated by
1 g;n
the strata (1){(5) of (2.2) for g (cid:21) 3, by (1) and (3){(5) for g = 2, and by (1)
and (5) for g = 1. Then N is the subcone generated by curves C (cid:26) M whose
g;n
associated family of curves has no moving smooth rational components. Hence,
NE (M )=N +NE (M =S ):
1 g;n 1 0;g+n g
(See(1.2)forthemeaningofmovingcomponent.) Ananalogousresultforfamilies
whose general member has at most one singularity is given in [Moriwaki00, A].
TOWARDS THE AMPLE CONE OF Mg;n 275
Of course Theorem (0.6) implies (the second formulation of) Conjecture (0.2)
for any family C (cid:26) M with no moving smooth rational component. In fact,
g;n
F (M )holds,andsoConjecture(0.2)holdsaslongasthereisnomovingsmooth
1 0;7
rational component containing at least 8 distinguished (i.e. marked or singular)
points.
Theorem (0.3) also has a converse which we state as follows.
(0.7) Theorem. Every nef line bundle on M =S is the pullback of a nef
0;g+n g
line bundle on M and F (M =S ) is equivalent to F (M ). In particular,
g;n 1 0;g+n g 1 g;n
F (M ) is equivalent to F (M =S ) and F (M ) is equivalent to F (M ).
1 g 1 0;g g 1 1;n 1 0;n+1
(0.8) Theorem. The Mori cone of F is a face of the Mori cone of M : there
g;n g;n
is a nef divisor D such that
D?\NE (M )=NE (M =S ):
1 g;n 1 0;g+n g
Remark. An interesting problem is to determine whether or not F (M ) for g (cid:21)1
1 g
implies F (M ) for all n(cid:21)3. For example, if one maps M into M by gluing
1 0;n 0;n g
one pointed curve of very di(cid:11)erent genera, then the pullback on the Picard groups
will be surjective. However,it is not clear that this will be true for the nef cones.
WenotethatinanumberofrespectsusingF toreducethegeneralconjecture
g;n
is more desirable than using R . Analogues of Theorems (0.7) and (0.8) fail for
g;n
R , indicating that F is the more natural locus from the Mori theoretic point
g;n g;n
of view (e.g. one is led to consider F by purely combinatorial consideration
g;n
of the Mori cone). Further the birational geometry of F is much simpler than
g;n
that of R . For example, when n = 0, the Picard group of M =S has rank
g;n 0;g g
roughly g=2 and is freely generated by the boundary divisors, while Pic(M =G)
0;2g
has rank roughly g2=2 and there is a relation among the boundary classes. Of
greater geometric signi(cid:12)cance is the contrast between the cones of e(cid:11)ective divi-
1
sors: NE (M =S ) is simplicial and is generated by the boundary divisors (see
0;g g
1
[KeelMcKernan96]), while the structure of NE (M =G) is unclear and it is def-
0;2g
initely not generated by boundary divisors (a counter-example is noted below).
Itisstraightforwardtoenumeratethepossibilitiesfortheonedimensionalstrata,
and then express (0.2) as a conjectural description of the ample cone as an inter-
section of explicit half spaces. This description is given for n=0 in [Faber96] and
for g =0 in [KeelMcKernan96]. We treat the general case in x2.
A fundamentalprobleminbirationalgeometryis to study morphismsofa given
varietytoothervarieties. Intheprojectivecategoryanysuchmorphismisgivenby
a semi-ample divisor | i.e. a divisor such that the linear system of some positive
multipleisbasepointfree. Asemi-ampledivisorisnecessarilynef. Thisimplication
is one of the main reasonsfor consideringnef divisors. Correspondingly,one of the
main reasons that (0.1) is interesting is its connection to:
Question. What are all the contractions (i.e. morphisms with geometrically con-
nected (cid:12)bers to projective varieties) of M ?
g;n
We have a number of results in this direction. Recall a de(cid:12)nition from [Keel99]:
For a nef divisor D on a variety X a subvariety Z (cid:26) X is called D-exceptional if
the D-degree of Z is zero,or equivalently, Dj is not big. The exceptionallocus of
Z
D, denoted E(D), is the union of the exceptional subvarieties. If D is semi-ample
and big, this is the exceptional locus for the associated birational map.
276 ANGELA GIBNEY, SEAN KEEL, AND IAN MORRISON
(0.9) Theorem. Let D 2Pic(M ) be a nef divisor, g (cid:21)2 (resp. g =1). Either
g;n
D is the pullback of a nef divisor on Mg;n(cid:0)1 via one of the tautological projections
(resp. is the tensor product of pullbacks of nef divisors on M and M via the
1;S 1;Sc
tautological projection for some subset S (cid:26)N) or D is big and E(D)(cid:26)@M .
g;n
(0.10) Corollary. Forg (cid:21)2any(cid:12)brationofM (toaprojectivevariety)factors
g;n
through a projection to some M (i<n), while M has no non-trivial (cid:12)brations.
g;i g
The above corollary and its proof are part of the (cid:12)rst author’s Ph.D. thesis
[Gibney00], written under the direction of the second author.
(0.11) Corollary. g (cid:21) 1. Let f : M ! X be a birational morphism to a
g;n
projective variety. Then the exceptional locus of f is contained in @M . In
g;n
particular X is again a compacti(cid:12)cation of M .
g;n
Note that our results give some support to the conjecture that the nef cone of
M behaves as if it were described by Mori’s cone theorem (i.e. like the cone
g;n
of a log Fano variety) with dual cone generated by (cid:12)nitely many rational curves
and having every face contractible. This is surprising, since for example M is
g;n
usually(e.g. forg (cid:21)24)ofgeneraltype, andconjecturally(by Lang,see [CHM97])
no rational curve meets the interior of M for g (cid:21) 2 and n su(cid:14)ciently big. Of
g;n
course,ourmainresultsreduceConjecture(0.2)togenus0,wherefrominitialMori
theoretic considerations it is much more plausible.
We note that prior to this work very few nef (but not ample) line bundles on
M were known. Myriad examples can be constructed using (0.3) as we indicate
g;n
in x6.
The above results indicate that (regular) contractions give only a narrow view
of the geometry of M . For example, (0.10) gives that the only (cid:12)brations are
g;n
(induced from) the tautological ones and, by (0.11), birational morphisms only
a(cid:11)ect the boundary. On the other hand the birational geometry should be very
rich. For example, as is seen in (6.4), all but one of the elementary (i.e. relative
Picard number one) extremal contractions of M are small (i.e. isomorphisms in
g
codimension one), so there should be a wealth of (cid:13)ips. Also one expects in many
casesto(cid:12)ndinterestingrational(cid:12)brations. Forexample,wheng+1hasmorethan
onefactorization,theBrill-Noetherdivisorshouldgiveapositivedimensionallinear
system on M which is conjecturally (see [HarrisMorrison98, 6.63]) not big.
g
Oneconsequenceof(0.9)andthe factthatE((cid:21))=@M isthat@M isintrinsic.
g g
(0.12) Corollary. Any automorphism of M must preserve the boundary.
g
Inview ofourresults,F (M )is obviouslyofcompellinginterest. Itisnatural
1 0;n
to wonder:
(0.13) Question. If a divisor on M has non-negative intersection with all one
0;n
dimensionalstrata,doesitfollowthatthedivisorislinearlyequivalenttoane(cid:11)ective
combination of boundary divisors?
Apositiveanswerto(0.13)wouldimply(0.2). Theanalogouspropertydoeshold
onM (see(3.3)), butthe questioninthatcaseisvastlysimpler asthe boundary
1;n
divisors are linearly independent.
It is not true that every e(cid:11)ective divisor on M is an e(cid:11)ective sum of bound-
0;n
ary divisors. In other words, F1(M ) is false. In particular, the second author
0;n
TOWARDS THE AMPLE CONE OF Mg;n 277
has shown (in support of the slope conjecture, [HarrisMorrison98, 6.63]) that the
pullback of the Brill-Noether divisor from M to M is a counter-example to
g 0;2g
F1(M =G) for any g (cid:21)3 with g+1 composite. In fact, he has shownthat for
0;2g+n
g = 3 the unique non-boundary component of this pullback gives a new vertex of
1
NE (M =G).
0;6
Question (0.13) can be formulated as an elementary combinatorial question of
whether one explicit polyhedral cone is contained in another. As this restatement
mightbeofinteresttosomeonewithoutknowledgeofM ,wegivethisformulation:
g
(0.14) Combinatorial Formulation of (0.13). LetVbetheQ-vectorspacethat
is spanned by symbols (cid:14) for each subset T (cid:26)f1;2;:::;ng subject to the relations
T
(1) (cid:14) =(cid:14) for all T;
T Tc
(2) (cid:14) =0 for jTj(cid:20)1; and
T
(3) for each 4-element subset fi;j;k;lg(cid:26)f1;2;:::;ng
X X
(cid:14) = (cid:14) :
T T
i;j2T;k;l2Tc i;k2T;j;l2Tc
P
Let N (cid:26)V be the set of elements b (cid:14) satisfying
T T
bI[J +bI[K +bI[L (cid:21)bI +bJ +bK +bL;
for each partition of f1;2;:::;ng into 4 disjoint subsets I;J;K;L. Let E (cid:26) V be
the a(cid:14)ne hull of the (cid:14) .
T
Question: Is N (cid:26)E?
In theory (0.14) can be checked, for a given n, by computer. For n(cid:20)6 we have
done so, with considerable help from Maroung Zou, using the programPorta. Un-
fortunately the computational complexity is enormous, and beyond our machine’s
capabilities already for n = 7. These cases (n (cid:20) 6) were proved previously (by
hand) by Faber.
The remainder of the paper is organized as follows. In x1, we (cid:12)x notation for
variousboundarydivisorsandgluingmaps. Inx2,wegiveasmallsetofgenerators
fortheconespannedbyonedimensionalstrata(2.2)andinequalitieswhichcutout
thedualconeofdivisors(2.1)|wetermeachtheFabercone. Themainresultofx3
is(3.1)statingthataclassnefontheboundaryofM isnef. Inx4theresultsfrom
g;n
the previous section are used to deduce (0.6) and hence (0.3). We also study the
nefness and exceptional loci of certain divisors in order to deduce (0.7) and (0.8).
The proof of (0.9) is contained in x5 following preparatory lemmas ((5.1){(5.3))
dealingwithdivisorsonCn=GwhereC isasmoothcurvewithautomorphismgroup
G. In x6, we collect some more ad hoc results. In particular, we answer a question
posed by Faber (6.2) and recoverthe classical ampleness result of Cornalba-Harris
(6.3). Finally, x7 contains a geometric reformulation of F (M ) and a review of
1 0;n
the evidence (to our minds considerable) for (0.2).
x1. Notation
For the most part we use standard notation for divisors, line bundles, and loci
on M . See e.g. [Faber96] and [Faber97] with the following possible exception.
g;n
By ! on M , g (cid:21) 2, we mean the pullback of the relative dualizing sheaf of the
i g;n
universal curve over M by the projection given by dropping all but the ith point.
g
278 ANGELA GIBNEY, SEAN KEEL, AND IAN MORRISON
Notethisisnot(forn(cid:21)3)therelativedualizingsheafforthemaptoMg;n(cid:0)1 given
by dropping the ith point (the symbol is used variously in the literature).
We note that the Q-Picard group of M is the same in all characteristics, by
g;n
[Moriwaki01].
We let N :=f1;2;:::;ng. The cardinality of a (cid:12)nite set S is denoted jSj.
For a divisor D on a variety X andfor Y a closedsubset of X, we say that D is
nef outside Y if D(cid:1)C (cid:21)0 for all irreducible curves C 6(cid:26)Y.
To obtain a symmetric description of boundary divisors on M , we use parti-
0;n
tionsT =[T;Tc]ofN into disjointsubsets, eachwith atleasttwoelements, rather
than subsets T. We write the corresponding boundary divisor as (cid:14)0;T, or (cid:14)T. For
a partition Q of a subset S (cid:26)N we write T >Q providedthe equivalence relation
induced by Q on S re(cid:12)nes that obtained by restricting T to S. We write Q=Tj
S
provided these two equivalence relations are the same.
We writeB forthe sumofboundarydivisorsonM =S thataretheimageof
k 0;n n
any (cid:14)T with jTj=k (thus Bk =Bn(cid:0)k). We will abuse notation byPusing the same
expression for its inverse image (with reduced structure) on M0;n, T;jTj=k(cid:14)T.
We make repeated use of the standard product decomposition for strata as a
(cid:12)nite image of products of various M . For precise details see [Keel99, pg. 274]
g;n
and [Faber97]. To describe this decomposition for boundary divisors we use the
notations
(cid:1)~i;S :=Mi;S[(cid:3)(cid:2)Mg(cid:0)i;Sc[(cid:3) (cid:16)(cid:1)i;S
and
(cid:1)~irr :=Mg(cid:0)1;N[p[q (cid:16)(cid:1)irr:
We refer to Mi;S[(cid:3) (and any analogous term for a higher codimension stratum) as
a factor of the stratum.
The key fact about these maps used here is the
(1.1) Lemma. The pullback to (cid:1)~ of any line bundle is numerically equivalent
i;S
to a tensor product of unique line bundles from the two factors. The given line
bundle is nef on (cid:1) i(cid:11) each of the line bundles on the factors is nef. Dually, let
i;S
C be any curve on the product, and let Ci;Cg(cid:0)i be its images on the two factors
(with multiplicity for the pushforward of cycles) which we also view as curves in
Mg;n by the usual device of gluing on a (cid:12)xed curve. Then, C and Ci +Cg(cid:0)i are
numerically equivalent.
Proof. The initial statement implies the other statements, and follows from the
explicit formulae of [Faber97].
Any curve E in M induces a decomposition of the curves it parameterizes
g;n
into a subcurve (cid:12)xed in the family E and a moving subcurve. Arguing inductively,
(1.1) yields
(1.2) Corollary. Every curve in M is numerically equivalent to an e(cid:11)ective
g;n
combination of curves whose moving subcurves are all generically irreducible.
x2. The Faber cone
In this section we consider the subcone of NE (M ) generatedby one dimen-
1 g;n
sional strata, and its dual. We refer to both as the Faber cone, appending of
curves, or of divisors if confusion is possible. We will call a divisor F-nef if it lies
TOWARDS THE AMPLE CONE OF Mg;n 279
in the Faber cone. Of course (0.2) is then the statement that F-nef implies nef, or
equivalently, the Mori and Faber cones of M are the same.
g;n
The next result describes the Faber cone of divisors as an intersection of half
spaces.
In order to give a symmetric description of the cone we write (cid:14)0;fig for (cid:0) i in
Pic(M ) (so (cid:14) is de(cid:12)ned whenever jIj(cid:21)1).
g;n 0:I
(2.1) Theorem. Consider the divisor
X
D =a(cid:21)(cid:0)b (cid:14) (cid:0) b (cid:14)
irr irr i;I i;I
[g=2](cid:21)i(cid:21)0
I(cid:26)N
jIj(cid:21)1fori=0
on M (with the convention that we omit for a given g;n any terms for which the
g;n
corresponding boundary divisor does not exist). Consider the inequalities
(1) a(cid:0)12birr+b1;; (cid:21)0,
(2) b (cid:21)0,
irr
(3) b (cid:21)0 for g(cid:0)2(cid:21)i(cid:21)0,
i;I
(4) 2b (cid:21)b for g(cid:0)1(cid:21)i(cid:21)1,
irr i;I
(5) bi;I +bj;J (cid:21)bi+j;I[J for i;j (cid:21)0, i+j (cid:20)g(cid:0)1, I\J =;,
(6) bi;I +bj;J +bk;K +bl;L (cid:21) bi+j;I[J +bi+k;I[K +bi+l;I[L for i;j;k;l (cid:21) 0,
i+j+k+l=g, I;J;K;L a partition of N,
where bi;I is de(cid:12)ned to be bg(cid:0)i;Ic for i>[g=2].
For g (cid:21)3, D has non-negative intersection with all 1-dimensional strata i(cid:11) each
of the above inequalities holds.
For g = 2, D has non-negative intersection will all 1-dimensional strata i(cid:11) (1)
and (3){(6) hold.
For g =1, D has non-negative intersection with all 1-dimensional strata i(cid:11) (1)
and (5){(6) hold.
For g =0, D has non-negative intersection with all 1-dimensional strata i(cid:11) (6)
holds.
Proof. (2.2) below lists the numerical possibilities for a stratum. Each inequality
abovecomesfromstandardintersectionformulae(seee.g. [Faber96],[Faber97]),by
intersecting with the corresponding curve of (2.2). Intersecting with (2.2.4) gives
theinequality2b (cid:21)b ,whichbyshiftingindicesandnotationgives(2.1.4).
irr i+1;I
Theorem(2.2)givesalistingofnumericalpossibilitiesforonedimensionalstrata,
giving explicit representatives for each numerical equivalence class. The parts in
Theorems (2.1) and (2.2) correspond: that is, for each family X listed in parts (2)
to (6) of (2.2), the inequality which expresses the condition that a divisor D given
as in (2.1) meet X non-negatively is given in the corresponding part of (2.1).
Weobtainthesejustasin[Faber96],byde(cid:12)ningamapM !M byattach-
0;4 g;n
ing a (cid:12)xed pointed curve in some prescribed way. For a subset I (cid:26) N, by a k+I
pointed curve we mean a k+jIj pointed curve, where jIj of the points are labeled
by the elements of I.
(2.2) Theorem. Let X (cid:26)M be a one dimensional stratum. Then X is either
g;n
(1) For g (cid:21)1, a family of elliptic tails; or, numerically equivalent to the image
of M !M de(cid:12)ned by one of the attaching procedures (2){(6) below.
0;4 g;n
(2) For g (cid:21)3. Attach a (cid:12)xed 4+N pointed curve of genus g(cid:0)3.
280 ANGELA GIBNEY, SEAN KEEL, AND IAN MORRISON
(3) For g (cid:21) 2, I (cid:26) N, g(cid:0)2 (cid:21) i (cid:21) 0, jIj+i > 0. Attach a (cid:12)xed 1+I-pointed
curve of genus i and a (cid:12)xed 3+Ic-pointed curve of genus g(cid:0)2(cid:0)i.
(4) For g (cid:21) 2 , I (cid:26) N, g (cid:0)2 (cid:21) i (cid:21) 0. Attach a (cid:12)xed 2+I-pointed curve of
genus i and a (cid:12)xed 2+Ic-pointed curve of genus g(cid:0)2(cid:0)i.
(5) For g (cid:21) 1, I \J = ;, I;J (cid:26) N, i+j (cid:20) g(cid:0)1;i;j (cid:21) 0, jIj+i;jJj+j > 0.
Attach a (cid:12)xed 1+I-pointed curve of genus i, a (cid:12)xed 1+J-pointed curve of
genus j, and a (cid:12)xed 2+(I [J)c-pointed curve of genus g(cid:0)1(cid:0)i(cid:0)j.
(6) For g (cid:21) 0, [I;J;K;L] a partition of N into disjoint subsets, i;j;k;l (cid:21) 0,
i+j+k+l=g,andi+jIj;j+jJj;k+jKj;l+jLj>0. Attach1+I,1+J,1+K,
and 1+L pointed curves of genus i;j;k;l respectively.
Figure (2.3)showsschematic sketchesof eachof these 5 families numberedas in
(2.2). The generic (cid:12)ber is shown on the left of each sketch and the 3 special (cid:12)bers
(with, up to dual graph isomorphism, any multiplicities) are shown on the right.
The bolder curves are the component(s) of the M piece. Boxes give the type
0;4
(i.e. genus and marked point set) of each (cid:12)xed component and of each node not of
irreducible type.
Inanyofthefamilies,ifthecurvesketchedisnotstable,wetakethestabilization.
Thus e.g. in (6) for n = 4, g = 0, the map M ! M is the identity, while in
0;4 0;4
(3) for g = 2;n= 0, i = j = 0, the image of a generic point of M is the moduli
0;4
point of an irreducible rational curve with two ordinary nodes.
Figure (2.3).
TOWARDS THE AMPLE CONE OF Mg;n 281
Strata of type (6) play distinctly di(cid:11)erent roles, both geometrically and combi-
natorially, from those of type (1){(5). Those of type (6) come from the (cid:13)ag locus,
theyarethe onlystrataingenus0,andthere isaface ofNE (M )thatcontains
1 0;n
exactly these strata;see (4.9). The cone generatedby the strata (1){(5) has a nice
geometric meaning given in (0.6). Our proofs make no direct use of strata of type
(6); in particular we will only directly use the inequalities (2.1.1){(2.1.5).
Strata of type (2){(6) are all (numerically equivalent to) curves lying in R .
g;n
Those of type (6) are distinguished geometrically by the fact that the curve corre-
sponding to a generalpoint has only disconnecting nodes, and algebraicallyby the
fact that the corresponding inequality in (2.1) has more than 3 (in fact 7) terms.
Proof of (2.2). The proof is analogous to that of the case of n = 0 treated in
[Faber96]. Here are details.
LetX (cid:26)M beaonedimensionalstratum: X isa(cid:12)nite imageofaproductof
g;n
modulispaces(oneforeachirreduciblecomponentofthe curveC correspondingto
itsgeneralpoint). Allbut oneofthese spacesarezerodimensional,thus M , and
0;3
thereisadistinguishedonedimensionalfactorwhichiseitherM orM . Inthe
1;1 0;4
M case, X is a family of elliptic tails and gives (2.1.1) exactly as in [Faber96].
1;1
Consider the M case. Let C be the stable pointed curve corresponding to a
0;4
general point of X and let E (cid:26) C be the moving irreducible component (which is
rational). Let h : E~ ! E be the normalization and let M (cid:26) E~ be the union of
the following disjoint subsets: h(cid:0)1(Z\E) for each connected component Z of the
closureofCnE,h(cid:0)1(p)foreachsingularpointpofE,andh(cid:0)1(p )foreachlabeled
i
point p of C on E. These subsets partition M into equivalence classes.
i
We assign to each equivalence class Z a triple (a(Z);S(Z);h(Z)): if Z =
h(cid:0)1(Z\E),thenhisthearithmeticgenusofZ,S (cid:26)f1;2;:::;ngisthecollectionof
labeledpointsofC lyingonZ,andaisthecardinalityofZ\E. t(h(cid:0)1(p))=(2;;;0)
for a singular point p of E, and t(h(cid:0)1(p )) = (1;fig;0) for a labeled point p of
i i
C on E. The numerical class of X is determined by the collection of triples Z by
tPhe standard intersection productSformulae in, for example, [Faber97]. Note that
PZ(h(Z)+a(Z)(cid:0)1) = g, that ZS(Z) is a partition of f1;2;:::;ng, and that
a(Z)=4.
Z
Now (2){(6) are obtained by enumerating the possibilities for the collection of
a(Z) which are f4g, f3;1g, f2;2g, f2;1;1g and f1;1;1;1g. These correspond to
(the obvious generalizations to pointed curves of) the families (B), (C), (D), (E),
and(F) of [Faber96] andyield (2){(6)by considering the possibilities forh(Z) and
S(Z) subject to the above constraints.
Throughout the paper we will separate the and (cid:14) classes (because although
they have similar combinatorial properties, they are very di(cid:11)erent geometrically,
e.g. is nef, while the boundary divisors have very negative normal bundles).
i
Thus, we will write a divisor as
Xn X X
(2.4) c +a(cid:21)(cid:0)b (cid:14) (cid:0) b (cid:14) (cid:0) b (cid:14)
i i irr irr 0;S 0;S i;S i;S
i=1 S(cid:26)N [g=2](cid:21)j(cid:21)1;S(cid:26)N
n(cid:0)2(cid:21)jSj(cid:21)2
where b0;fig has become ci. As examples, in this notation (2.1.5) with i = j =
0;I =fig becomes
ci+b0;J (cid:21)b0;J[fig
282 ANGELA GIBNEY, SEAN KEEL, AND IAN MORRISON
which in turn specializes to
ci+cj (cid:21)b0;fi;jg
in case J =fjg while (2.1.3) for (i;I)=(0;fig) becomes c (cid:21)0.
i
Forg (cid:21)3,theexpressionaboveisunique(i.e. thevarioustautologicalclassesare
linearly independent), but for smaller genera there are relations which are listed in
[ArbarelloCornalba98]. For g = 1, the boundary divisors are linearly independent
and span the Picard group so we can assume above that a and all c are zero and,
i
if we do so, the resulting expression is unique.
x3. Nef on the boundary implies nef
In this section, we prove:
(3.1) Proposition. If g (cid:21) 2, or g = 1;n (cid:21) 2, a divisor D 2 Pic(M ) is nef i(cid:11)
g;n
its restriction to @M is nef.
g;n
(3.2) Lemma. g (cid:21) 2. Let D 2 Pic(M ) be a divisor expressed as in (2.1). For
g;n
[g=2](cid:21)i(cid:21)1 let
b =maxb
i i;S
S(cid:26)N
and de(cid:12)ne A2Pic(M ) by
g
X
A=a(cid:21)(cid:0)b (cid:14) (cid:0) b (cid:14) :
irr irr i i
[g=2](cid:21)i(cid:21)1
Ifthecoe(cid:14)cients of D satisfy one ofthe inequalities (2.1.1){(2.1.5)for (cid:12)xedindices
i;j and all subsets of N, then the coe(cid:14)cients of A satisfy the analogous inequality.
In the example of (2.1.5), bi +bj (cid:21) bi+j if bi;I +bj;J (cid:21) bi+j;I[J for all i and j
between 0 and g(cid:0)1 and for all subsets I;J (cid:26)N with I\J =;.
Proof. This is clear since the b are de(cid:12)ned by taking a maximum. For example,
i
suppose bi+j =bi+j;T. Then, bi+bj (cid:21)bi;;+bj;T (cid:21)bi+j;T =bi+j.
Since all the strata are contained in @M , (3.1) will follow immediately from:
g;n
(3.3) Lemma. Let D 2 Pic(M ). If g (cid:21) 3 [resp. g = 2;g = 1] and D has
g;n
non-negative intersection with all strata in (2.2) of types (1){(5)[resp. (1){(2) and
(4){(5); (1) and (5)], then D is nef outside of @M . Furthermore if g = 1, then
g;n
D is numerically equivalent to an e(cid:11)ective sum of boundary divisors.
Proof. We express D as in (2.4). We consider (cid:12)rst the case of g = 1. We can as
remarkedabove assume a=c =0 for all i. Then it is immediate from (2.1.1) and
i
(2.1.5) (using the translations after (2.4)) that all the coe(cid:14)cients are non-positive,
so D is linearly equivalent to an e(cid:11)ective sum of boundary divisors.
Now assume g (cid:21)2. De(cid:12)ne b and A as in (3.2).
i
We use the relation
X
=! + (cid:14)
i i 0;S
i2S