THE AMPLE CONE OF THE KONTSEVICH MODULI SPACE 1. Introduction Positive-dimensional ...

14 Pages·2012·0.2 MB·English

THE AMPLE CONE OF THE KONTSEVICH MODULI SPACE
IZZETCOSKUN,JOEHARRIS,ANDJASONSTARR
Abstract. We produceample,respectivelyNEF,eventually free,divisorsin
the Kontsevich space M0;n(Pr;d) of n-pointed, genus 0, stable maps to Pr,
givensuchdivisorsinM0;n+d. Weprovethisproducesallample,respectively
NEF,eventuallyfree,divisorsinM0;n(Pr;d). Asaconsequence,weconstruct
a contraction of the boundary [bkd==12c(cid:1)k;d(cid:0)k in M0;0(Pr;d) analogous to a
contraction ofthe boundary[bkn==32c(cid:1)~k;n(cid:0)k inM0;n (cid:12)rstconstructed byKeel
andMc Kernan.
1. Introduction
Positive-dimensionalfamiliesof varietiesspecialize {non-generalvarietiesinthe
family exhibit special properties. Given a parameter space, the subset parametriz-
ing varieties with a special property is typically closed. Which special properties
occurincodimension1,respectivelyforevery1-parameterfamilyofvarieties? More
precisely, when is the associated closed subset an e(cid:11)ective divisor, respectively an
ample divisor? These questions, among others, motivate the study of e(cid:11)ective and
ample divisors in parameter spaces of varieties.
TheparameterspacewestudyistheKontsevichmodulispaceofn-pointed,genus
0, stable maps to projective space, denoted M (Pr;d). Here we study the ample
0;n
cone, and more generally the NEF and eventually free cones. In a second article
[CHS], using signi(cid:12)cantly di(cid:11)erent methods and under additional hypotheses, we
study the e(cid:11)ective cone.
Our goal is to study families of curves in a general target X. Fortunately,
this largely reduces to the study for Pr: As the Kontsevich space is functorial
in the target, for every morphism X ! Pr, NEF and base-point-free divisors on
M (Pr;d)giveNEFandbase-point-freedivisorsonM (X;(cid:12))(thisfunctoriality
0;n 0;n
isoneofmanyadvantagesofthe KontsevichspaceovertheHilbertschemeandthe
Chow variety).
Here is our main result.
Theorem 1.1. Let r and d be positive integers, n a nonnegative integer such that
n+d(cid:21)3. There is an injective linear map,
v :Pic(M )Sd !Pic(M (Pr;d)) :
0;n+d Q 0;n Q
2000 Mathematics SubjectClassi(cid:12)cation. Primary:14D20,14E99,14H10.
During the preparation ofthis article the second author was partially supported by the NSF
grant DMS-0200659. The third author was partiallysupported by the NSF grant DMS-0353692
andaSloanResearchFellowship.
1
The NEF cone of M (Pr;d), respectively, the base-point-free cone, is the product
0;n
of the cone generated by H;T;L ;:::;L and the image under v of the NEF cone
1 n
of M ==S , respectively, the base-point-free cone.
0;n+d d
The action of S on M permutes the last d marked points. The map v
d 0;n+d
generatesNEFandbase-point-freedivisorsonthe KontsevichspacefromNEFand
base-point-freedivisorson M ==S . In particular,it generatescontractionsof
0;n+d d
the Kontsevich space from contractions of M ==S .
0;n+d d
Theorem 1.2. For every integer r (cid:21)1 and d(cid:21)2, there is a contraction,
cont:M (Pr;d)!Y;
0;0
restricting to an open immersion on the interior M (Pr;d) and whose restriction
0;0
to the boundary divisor (cid:1)k;d(cid:0)k (cid:24)= M0;1(Pr;k)(cid:2)Pr M0;1(Pr;d(cid:0)k) factors through
the projection to M (Pr;d(cid:0)k) for each 1 (cid:20) k (cid:20) bd=2c. The following divisor is
0;1
the pull-back of an ample divisor on Y,
bd=2c
D =T + k(k(cid:0)1)(cid:1) :
r;d k;d(cid:0)k
kX=2
SomeconnectionbetweentheampleconeoftheKontsevichspaceandtheample
cone of M is natural, and certainly not surprisingto experts. A similarconnec-
0;n
tion between the Fulton-MacPherson space and M was proved in [Che]. The
0;n
primaryimportanceofTheorem1.1istheprecise,simpledescriptionofv: withone
exception, it maps each boundary divisor of M to the corresponding bound-
0;n+d
ary divisor of the Kontsevich space. This is used to construct the contraction in
Theorem 1.2, which is analogous to the \democratic" contraction of the boundary
of M (cid:12)rst constructed in an unpublished note of Keel and Mc Kernan.
0;n
Recently we were informed of di(cid:11)erent constructions of the contraction of The-
orem 1.2 in [Par] and by Anca Musta(cid:24)ta(cid:20) and Andrei Musta(cid:24)ta(cid:20). One advantage of
our proof is that it uses only the existence of the map v, which is itself a formal
consequence of the de(cid:12)nition of the Kontsevich space. The proof of Theorem 1.2
alsogivesanew,veryshortconstructionofKeel-Mc Kernan’scontractionofM .
0;n
Acknowledgments: We would like to thank B. Hassett and A. J. de Jong for
suggesting helpful references and for useful discussions.
2. Statement of results
TheKontsevichmoduli spaceM (Pr;d) compacti(cid:12)esthe schemeparameteriz-
0;n
ing smooth, rational curves of degree d in Pr. Precisely, it is the smooth, proper,
Deligne-Mumford stack parameterizing families of data (C;(p ;:::;p );f) of,
1 n
(i) a proper, connected, at-worst-nodal,genus 0 curve C,
(ii) an ordered sequence p ;:::;p of distinct, smooth points of C,
1 n
(iii) and a degree-d morphism f : C ! Pr satisfying the following stability
condition: every irreducible component of C mapped to a point under f
contains at least 3 special points, i.e., marked points p and nodes of C.
i
In [Pa] R. Pandharipande gives generators of the Kontsevich space:
(1) the class H of the divisor of maps whose images intersect a (cid:12)xed codimen-
sion two linear space in Pr (provided r >1 and d>0),
2
(2) the class Li of the pull-back ev(cid:3)i(OPr(1)), for 1 (cid:20) i (cid:20) n, associated to the
ith evaluation morphism, ev (C;(p ;:::;p );f):=f(p ),
i 1 n i
(3) and the classes (cid:1) of the boundary divisors consisting of maps
(A;dA);(B;dB)
with reducibledomains. HereAtB isanyorderedpartitionof themarked
points, andd and d arenon-negativeintegerssatisfyingd=d +d . If
A B A B
d =0 (respectively, if d =0), we demand #A(cid:21)2 (resp. #B (cid:21)2).
A B
The divisor classes H and L on M (Pr;d) are NEF and base-point-free. For
i 0;n
d (cid:21) 2, there is another NEF and base-point-free divisor class T, the tangency
divisor: Fixing a hyperplane (cid:5) (cid:26) Pr, T is the class of the divisor parametrizing
stable maps (C;p ;:::;p ;f) for which f(cid:0)1((cid:5)) is not simply d reduced, smooth
1 i
points of C. In terms of Pandharipande’sgenerators,the class of T equals,
bd=2c
d(cid:0)1 k(d(cid:0)k)
T = H+ 0 (cid:1) 1:
d d (A;k);(B;d(cid:0)k)
kX=0 AX;B
@ A
Finally, the map v from Theorem 1.1 is described in Section 3. Together, all
nonnegative-linearcombinationsofthesedivisorsgiveaconeinPic(M (Pr;d)) .
0;n Q
Weusethe methodof test families toprovethisisthe entireconeof NEFdivisors,
respectively, eventually free divisors. In other words, we (cid:12)nd morphisms from test
varieties to M (Pr;d) . Since every NEF divisor, resp. eventually free divisor,
0;n Q
pulls back to such a NEF divisor, resp. eventually free divisor, this constrains
the NEF and eventually free divisors among all divisors. By producing su(cid:14)ciently
many test families, we prove every NEF, resp. eventually free divisor, is in our
cone.
Hypothesis 2.1. For the rest of the paper assume that the triple (n;r;d) in
M (Pr;d) satis(cid:12)es r (cid:21)1; d(cid:21)1 and n+d(cid:21)3:
0;n
Pr
f
* p
C
L
L L L
Figure 1. The morphism (cid:11).
The morphism (cid:11). There is a 1-morphism (cid:11) : M (cid:2)Pr(cid:0)1 ! M (Pr;d)
0;n+d 0;n
de(cid:12)ned as follows. Fix a point p 2 Pr and a line L (cid:26) Pr containing p. To every
curveC inM attachacopyofLateachofthelastdmarkedpointsanddenote
0;n+d
the resulting curve by C0. Consider the morphism f : C0 ! Pr that contracts C
to p and maps the d rational tails isomorphically to L (see Figure 1). Since the
space of lines in Pr passing through the point p is parameterized by Pr(cid:0)1, there is
an induced 1-morphism (cid:11):M (cid:2)Pr(cid:0)1 !M (Pr;d):
0;n+d 0;n
Since(cid:11)isinvariantforthe actionofS permutingthe lastdmarkedpoints,the
d
pull-back map determines a homomorphism
(cid:11)(cid:3) =((cid:11)(cid:3);(cid:11)(cid:3)):Pic(M (Pr;d))!Pic(M )Sd (cid:2)Pic(Pr(cid:0)1):
1 2 0;n 0;n+d
3
We will denote the two projections of (cid:11)(cid:3) by (cid:11)(cid:3) and (cid:11)(cid:3).
1 2
C
p
i
L
*
slide p along L
i
Figure 2. The morphism (cid:12) .
i
The morphisms (cid:12) . For each 1 (cid:20) i (cid:20) n, there is a 1-morphism (cid:12) : P1 !
i i
M (Pr;d) de(cid:12)ned as follows. Fix a degree-(d (cid:0) 1), (n (cid:0) 1)-pointed curve C
0;n
containingallexcepttheith markedpoint. AtageneralpointofC,attachalineL.
The resulting degree-d, reducible curve will be the domain of our map. The (cid:12)nal,
ith markedpointisinL. Varyingp inLgivesa1-morphism(cid:12) :P1 !M (Pr;d)
i i 0;n
(seeFigure2). Thisde(cid:12)nitionhastobeslightlymodi(cid:12)edinthecases(n;d)=(1;1)
or (2;1). When (n;d)=(1;1), we assume that the line L with the varying marked
point p constitutes the entire stable map. When (n;d) = (2;1), we assume that
i
the map has L as the only component. One marked point is allowed to vary on L
and the remaining marked point is held (cid:12)xed at a point p2L.
slide attachment point Pr
C C L
f
L
L
Figure 3. The morphism (cid:13).
The morphism (cid:13). If d (cid:21) 2, there is a 1-morphism (cid:13) : P1 ! M (Pr;d)
0;n
de(cid:12)ned as follows. Take two copies of a (cid:12)xed line L attached to each other at a
variable point. Fix a point p in the second copy of L. Let C be a smooth, degree-
(d(cid:0)2), genus 0, (n+1)-pointed stable map to Pr whose (n+1)-st point maps
to p. Attach this to the second copy of L at p. Altogether, this gives a degree-d,
n-pointed, genus 0 stable maps with three irreducible components. The n marked
pointsarethe (cid:12)rstnmarkedpointsof C. Theonlyvaryingaspectof thisfamily of
stablemapsistheattachmentpointofthetwocopiesofL. Varyingtheattachment
point in L(cid:24)=P1 givesa stablemap parameterizedbyP1, hence thereis an induced
1-morphism (cid:13) :P1 !M (Pr;d) (see Figure 3). When (n;d) =(1;2), we modify
0;n
the de(cid:12)nition by assuming that the map consists only of the two copies of the line
L and the marked point is held (cid:12)xed at the point p on the second copy of L.
Notation 2.2. If d(cid:21)2, denote by P the Abelian group
r;n;d
P :=Pic(M )Sd (cid:2)Pic(Pr(cid:0)1)(cid:2)Pic(P1)n(cid:2)Pic(P1):
r;n;d 0;n+d
Denote by u=u :Pic(M (Pr;d))!P the pull-back map
r;n;d 0;n r;n;d
u =((cid:11)(cid:3);((cid:12)(cid:3);:::;(cid:12)(cid:3));(cid:13)(cid:3)):
r;n;d 1 n
4
If d=1, denote by P the Abelian group
r;n;1
P :=Pic(M )Sd (cid:2)Pic(Pr(cid:0)1)(cid:2)Pic(P1)n
r;n;1 0;n+d
and denote by u=u :Pic(M (Pr;1))!P the pull-back map
r;n;1 0;n r;n;1
u =((cid:11)(cid:3);((cid:12) ;(cid:3);:::;(cid:12)(cid:3)))
r;n;1 1 n
Theorem 1.1 is equivalent to the following.
Theorem 2.3. The map u (cid:10)Q:Pic(M (Pr;d)) !P (cid:10)Q is an isomor-
r;n;d 0;n Q r;n;d
phism. The image under u (cid:10)Q of the ample cone, resp. NEF, eventually free
r;n;d
cone of M (Pr;d) equals the product of the ample cones, resp. NEF, eventually
0;n
free cones of Pic(M0;n+d)Sd, Pic(Pr(cid:0)1), and the factors Pic(P1).
This is equivalent to Theorem 1.1 because the linear map u is simply the
r;n;d
inverse of the product of the linear map v and the maps Q ! Pic(M (Pr;d))
0;n Q
associated to each generator H, T and L ;:::;L .
1 n
Notation 2.4. Denote by n the set f1;:::;ng. Denote by 4 = 4 the set of
n;d
4-tuples ((A;d );(B;d )) of an ordered partition AtB of n and an ordered pair
A B
of nonnegative integers (d ;d ) such that d +d = d and #A (cid:21) 2 (#B (cid:21) 2) if
A B A B
d = 0 (d = 0, respectively). Denote by 40 the subset of 4 of data such that
A B
#A+d (cid:21)2 and #B+d (cid:21)2.
A B
Recall that the groupPic(M ) is generatedby boundary divisors (cid:1)~ , where
0;n A;B
AtB is an ordered partition of n with #A (cid:21)2 and #B (cid:21)2. Let (cid:1)~ denote
k;n(cid:0)k
the sum of the boundary divisors (cid:1)~ , where the sum runs over pairs
(A;B) A;B
(A;B) suchthat #A=k and#B =Pn(cid:0)k. The groupPic(M0;n+d)Sd is generated
by boundary divisors (cid:1)~ , where ((A;d );(B;d )) 2 (cid:1)0. The divisor
(A;dA);(B;dB) A B
(cid:1)~(A;dA);(B;dB)denotestheSd-invariantsumofboundarydivisors (A0;B0)(cid:1)~(A;A0);(B;B0),
where the sum runs over pairs (A0;B0) such that A0 tB0 is a paPrtition of the last
d points and #A0 =d and #B0 =d .
A B
To apply Theorem 2.3, we need to express the images of the standard gener-
ators of Pic(M0;n(Pr;d)) in terms of the standard generators for Pic(M0;n+d)Sd,
Pic(Pr(cid:0)1) and Pic(P1) factors.
Proposition 2.5. (i) Assume d(cid:21)2 so that (cid:13) is de(cid:12)ned. Then
(cid:13)(cid:3)T =O (2); (cid:13)(cid:3)H=0; (cid:13)(cid:3)L =0; for 1(cid:20)i(cid:20)n:
P1 i
The pull-back (cid:13)(cid:3)(cid:1) = 0 unless (#A;d ) or (#B;d ) is equal to (0;1)
(A;dA);(B;dB) A B
or (0;2). Moreover, if (n;d)6=(0;3),
(cid:13)(cid:3)(cid:1) =O (4) and (cid:13)(cid:3)(cid:1) =O ((cid:0)1):
(;;1);(n;d(cid:0)1) P1 (;;2);(n;d(cid:0)2) P1
If (n;d)=(0;3), then (cid:13)(cid:3)(cid:1) =O (3).
(;;1);(n;d(cid:0)1) P1
(ii) Assume n(cid:21)1 so that (cid:12) ;:::;(cid:12) are de(cid:12)ned. Then
1 n
(cid:12)(cid:3)H=0; (cid:12)(cid:3)L =O (1); (cid:12)(cid:3)L =0 if j 6=i; and (cid:12)(cid:3)T =0:
i i i P1 i j i
For every 1 (cid:20) i (cid:20) n, the pull-back (cid:12)(cid:3)(cid:1) equals O (1) if (n;d) 6= (1;2),
i (;;1);(n;d(cid:0)1) P1
and equals O (2) if (n;d) = (1;2). If (n;d) 6= (1;2), then (cid:12)(cid:3)(cid:1)
P1 i (fig;1);(figc;d(cid:0)1)
equals O ((cid:0)1). And (cid:12)(cid:3)(cid:1) equals 0 if neither (A;d ) nor (B;d ) equal
P1 i (A;dA);(B;dB) A B
(;;1) or (fig;1).
(iii) (cid:11)(cid:3)H=(0;OPr(cid:0)1(d)); (cid:11)(cid:3)Li =0; for 1(cid:20)i(cid:20)n; (cid:11)(cid:3)T =0:
5
Divisors in M (Pr;d) (cid:11)(cid:3) (cid:11)(cid:3) (cid:12)(cid:3) (cid:13)(cid:3)
0;n 1 2 i
T 0 0 0 O (2)
P1
H 0 OPr(cid:0)1(d) 0 0
L 0 0 O (1) 0
i P1
L 0 0 0 0
j6=i
(cid:1)(;;1);(n;d(cid:0)1) c OPr(cid:0)1((cid:0)d) OP1((cid:0)1) OP1(4)
(cid:1) (cid:1)~ 0 0 O ((cid:0)1)
(;;2);(n;d(cid:0)2) (;;2);(n;d(cid:0)2) P1
(cid:1) (cid:1)~ 0 O ((cid:0)1) 0
(fig;1);(figc;d(cid:0)1) (fig;1);(figc;d(cid:0)1) P1
(cid:1) (cid:1)~ 0 0 0
(A;dA);(B;dB) (A;dA);(B;dB)
all others
Figure 4. The pull-backs of the standard generators
If #A+d ;#B+d (cid:21) 2, then (cid:11)(cid:3)(cid:1) equals (cid:1)~ . The pull-
A B (A;dA);(B;dB) (A;dA);(B;dB)
back (cid:11)(cid:3)(cid:1)(;;1);(n;d(cid:0)1) equals (c;OPr(cid:0)1((cid:0)d)), where c is the class
(cid:0)1
c= d (d +#B)(d +#B(cid:0)1)(cid:1)~ :
(n+d(cid:0)1)(n+d(cid:0)2) A B B (A;dA);(B;dB)
((A;dAX);(B;dB))
2(cid:1)0
Proof. Items (i) and (ii) follow fromLemma 3.5 and Lemma 4.1. Item (iii), except
for the computation of c, is straightforward. The class c equals (cid:0) d . To
i=1 n+i
rewrite this as above, use [Pa, Lemma 2.2.1] (cf. also [dJS, LemmaP6.10]). (cid:3)
With the exceptions of (n;d) =(0;3), (1;2), and (1;3), Proposition 2.5 is sum-
marized by Figure 4. The phrase \all others" means, all pairs ((A;d );(B;d ))
A B
such that neither ((A;d );(B;d )) nor ((B;d );(A;d )) already occur in the ta-
A B B A
ble. The lines (cid:13)(cid:3) and (cid:1) apply only if d (cid:21) 2. The lines L , L and
(;;2);(n;d(cid:0)2) i j
(cid:1) only apply if n(cid:21)1.
(fig;1);(figc;d(cid:0)1)
In [Kaw], Kawamata associated an e(cid:11)ective, NEF Q-Cartier divisor L on M
0;n
to every n-tuple of rational numbers, (d ;:::;d ) satisfying 0 < d (cid:20) 1 and d +
1 n i 1
(cid:1)(cid:1)(cid:1)+d =2. In an unpublished note Keel and Mc Kernan proved the following.
n
Theorem 2.6 (Keel-Mc Kernan). The Q-Cartier divisor L is eventually free.
In particular, when d =(cid:1)(cid:1)(cid:1)=d =2=n, the divisor class of L equals (1=n(n(cid:0)
1 n
1))D , where
n
bn=2c
D = k(k(cid:0)1)(cid:1)~ :
n k;n(cid:0)k
kX=2
6
This is the divisor class giving the \democratic" contraction of the boundary of
M , cf. [Has, x2.1.2]. One application of Theorem 2.3 is the construction of the
0;n
analogous contraction in Theorem 1.2 as well as a new, short construction of the
democratic contraction.
Theorem 2.7. For every integer n(cid:21)4, there is a contraction
cont:M !Y
0;n
restricting to an open immersion on the interior M and whose restriction to
0;n
the boundary divisor (cid:1) = M (cid:2)M factors through projection to
k;n(cid:0)k 0;k+1 0;n+1(cid:0)k
M for each 3 (cid:20) k (cid:20) bn=2c. The divisor D is the pull-back of an ample
0;n+1(cid:0)k n
divisor on Y.
It follows easily that for every rational number b satisfying
2 2 2
<b< resp. b= ;
(n(cid:0)1) bn=2c bn=2c
settingB =b(cont) ((cid:1)~ ),K +B isanampleQ-Cartierdivisor,and(Y;B)is
(cid:3) 2;n(cid:0)2 Y
Kawamata log terminal, resp. log canonical (for this one only needs the existence
of the contraction and the formula for L).
3. The splitting homomorphism
In this section we de(cid:12)ne a map v : Pic(M0;n+d)Sd ! Pic(M0;n(Pr;d))(cid:10)Q that
mapstheNEFdivisorsinPic(M0;n+d)Sd toNEFdivisorsinM0;n(Pr;d). Themap
v givesasplittingof the map(cid:11)(cid:3) de(cid:12)nedinthe introductionandisessentialforthe
1
proof of Theorem 2.3.
Let (cid:5)(cid:26)Pr be a hyperplane not containing the point p used to de(cid:12)ne the mor-
phisms (cid:11) and (cid:13). Assume that the degreed(cid:0)1 curveused to de(cid:12)ne the morphisms
(cid:12) is not tangent to (cid:5), and none of the marked points on this curve are contained
i
in (cid:5). Finally, assume that the degreed(cid:0)2 curveused to de(cid:12)ne the morphism (cid:13) is
not tangent to (cid:5) and none of the marked points are contained in (cid:5).
Denote by M (Pr;d) the open substack of M (Pr;d) parameterizing
0;n+d 0;n+d
stable maps with irreducible domain. Let
ev :M (Pr;d)!(Pr)d
n+1;:::;n+d 0;n+d
be the evaluation morphism associated to the last d marked point. Denote by
M (Pr;d) the inverse image of (cid:5)d and by M (Pr;d) the closure of
0;n+d (cid:5) 0;n+d (cid:5)
M (Pr;d) in M (Pr;d).
0;n+d (cid:5) 0;n+d
M (Pr;d) is S -invariant under the action of S on M (Pr;d) per-
0;n+d (cid:5) d d 0;n+d
muting the last d marked points. Denote by
(cid:25) :M (Pr;d)!M (Pr;d)
0;n+d 0;n
the forgetful 1-morphism that forgets the last d marked points and stabilizes the
resulting family of prestable maps. This is S -invariant. Denote by
d
(cid:26):M (Pr;d)!M
0;n+d 0;n+d
the1-morphismthatstabilizestheuniversalfamilyofmarkedprestablecurvesover
M (Pr;d). This is S -equivariant.
0;n+d d
7
Lemma 3.1. The 1-morphism (cid:25) :M (Pr;d) !M (Pr;d) is (cid:19)etale. Denot-
0;n+d (cid:5) 0;n
ing the image by O , the morphism (cid:25) :M (Pr;d) !O is an S -torsor.
(cid:5) 0;n+d (cid:5) (cid:5) d
Proof. Let (C;(p ;:::;p ;q ;:::;q );f) be a stable map in M (Pr;d) . Then
1 n 1 d 0;n+d (cid:5)
(C;(p ;:::;p );f) satis(cid:12)es,
1 n
(i) C is irreducible,
(ii) f(cid:0)1((cid:5)) is a reduced Cartier divisor, and
(iii) none of the marked points p is contained in f(cid:0)1((cid:5)).
i
Conversely, for every stable map (C;(p ;:::;p );f) satisfying (i){(iii), for every
1 n
labeling of f(cid:0)1((cid:5)) as q ;:::;q , (C;(p ;:::;p ;q ;:::;q );f) is a stable map in
1 d 1 n 1 d
M (Pr;d) . Thus O is the open substack of stable maps satisfying (i){(iii)
0;n+d (cid:5) (cid:5)
andM (Pr;d) istheS -torsoroverO parameterizinglabelingsofthe(cid:12)bers
0;n+d (cid:5) d (cid:5)
of f(cid:0)1((cid:5)). (cid:3)
Denote by q : M ! M =S the geometric quotient. The composition
0;n+d 0;n+d d
q(cid:14)(cid:26):M (Pr;d) !M =S isS -equivariant. BecauseM (Pr;d) is
0;n+d (cid:5) 0;n+d d d 0;n+d (cid:5)
an S -torsor over O , there is a unique 1-morphism (cid:30)0 : O ! M =S such
d (cid:5) (cid:5) (cid:5) 0;n+d d
that (cid:30)0(cid:14)(cid:25) =q(cid:14)(cid:26).
De(cid:12)nition 3.2. De(cid:12)ne U to be the maximal open substack of M (Pr;d) over
(cid:5) 0;n
which (cid:30)0 extends to a 1-morphism, denoted
(cid:5)
(cid:30) :U !M =S :
(cid:5) (cid:5) 0;n+d d
De(cid:12)neI tobethenormalizationoftheclosureinM (Pr;d)(cid:2)M =S ofthe
(cid:5) 0;n 0;n+d d
image of the graph of (cid:30)0 , i.e., I is the normalization of the image of ((cid:25);q (cid:14)(cid:26)).
(cid:5) (cid:5)
De(cid:12)ne I to be the normalization of the image of ((cid:25);(cid:26)) in M (Pr;d)(cid:2)M .
(cid:5) 0;n 0;n+d
Finally, de(cid:12)ne U to be the inverse image of U in I .
e (cid:5) (cid:5) (cid:5)
e e
There is a pull-back map of S -invariant invertible sheaves,
d
(cid:26)(cid:3) :Pic(M )Sd !Pic(I )Sd;
0;n+d (cid:5)
whichfurtherrestrictstoPic(U(cid:5))Sd. After(cid:19)etalebasee-changefromU(cid:5) toascheme,
the morphism U ! U is the geometric quotient of U by the action of S .
(cid:5) (cid:5) e (cid:5) d
Therefore the peull-back map Pic(U(cid:5)) ! Pic(U(cid:5))Sd is ane isomorphism after ten-
soringwith Q; in fact, both the kerneland cokernelareannihilated by d!. Because
e
M =S is a proper scheme and because M (Pr;d) is separated and normal,
0;n+d d 0;n
bythevaluativecriterionofpropernessthecomplementofU hascodimension(cid:21)2.
(cid:5)
The smoothness of M (Pr;d) and [Ha, Prop. 6.5(c)] imply that the restriction
0;n
map Pic(M (Pr;d))!Pic(U ) is an isomorphism.
0;n (cid:5)
De(cid:12)nition3.3. De(cid:12)nev :Pic(M0;n+d)Sd !Pic(M0;n(Pr;d))(cid:10)Qtobetheunique
homomorphism commuting with (cid:26)(cid:3) via the isomorphisms above.
The map v is independent of the choice of (cid:5), hence it sends NEF divisors to
NEF divisors.
S
Lemma 3.4. For every base-point-free invertible sheaf L in Pic(M0;n+d) d, v(L)
is base-point-free. In particular, for every ample invertible sheaf L, v(L) is NEF.
Thus, by Kleiman’s criterion, for every NEF invertible sheaf L, v(L) is NEF.
8
Proof. For every [(C;(p ;:::;p );f)] in M (Pr;d), there exists a hyperplane (cid:5)
1 n 0;n
satisfying the conditions above and such that f(cid:0)1((cid:5)) is a reduced Cartier divisor
containing none of p ;:::;p . By Lemma 3.1, (C;(p ;:::;p );f) is contained in
1 n 1 n
U . Since L isbase-point-free,thereexistsa divisorD in the linearsystemjLj not
(cid:5)
containing(cid:30) [(C;(p ;:::;p );f)]. Bytheproofof[Ha,Prop. 6.5(c)],theclosureof
(cid:5) 1 n
(cid:30)(cid:0)1(D) is in the linear system jv(L)j; and it does not contain [(C;(p ;:::;p );f)].
(cid:5) 1 n
(cid:3)
Lemma 3.5. (i) The images of (cid:11), (cid:12) and (cid:13) are contained in U .
i (cid:5)
(ii) The morphisms (cid:30) (cid:14)(cid:12) and(cid:30) (cid:14)(cid:13) areconstantmorphisms. Therefore (cid:12)(cid:3)(cid:14)v
(cid:5) i (cid:5) i
and (cid:13)(cid:3)(cid:14)v are the zero homomorphism.
(iii) The composition of (cid:11) with (cid:30) equals q(cid:14)pr . Therefore
(cid:5) M
0;n+d
(cid:11)(cid:3)(cid:14)v :Pic(M )Sd !Pic(M )Sd (cid:2)Pic(Pr(cid:0)1)
0;n+d 0;n+d
is the homomorphism whose projection on the (cid:12)rst factor is the identity,
and whose projection on the second factor is 0.
Proof. (i): The image of (cid:11) is contained in O . Denote by q the intersection point
(cid:5)
of L and (cid:5).
The image (cid:12) (L(cid:0)fqg) is contained in O . The stable map (cid:12) (q) sends the ith
i (cid:5) i
markedpointinto(cid:5). Uptolabelingthedpointsofthe inverseimageof(cid:5), thereis
onlyone(n+d)-pointedstablemapinM (Pr;d) thatstabilizestothisstable
0;n+d (cid:5)
map. Itisobtainedfrom(cid:12) (q)byremovingthe ith markedpointfromL,attaching
i
a contracted component C0 to L at q, containing the ith marked point and exactly
one of the last d marked points, and labeling the d(cid:0)1 points in C \(cid:5) with the
remaining d(cid:0)1 marked points.
Similarly, (cid:13)(L(cid:0)fqg)is containedin O . The stablemap (cid:13)(q) has twocopies of
(cid:5)
L attached to each other at q. This appears to be a problem, because the inverse
image of (cid:13)(q) in M (Pr;d) is 1-dimensional, isomorphic to M . The stable
0;n+d (cid:5) 0;4
maps have a contracted component C0 such that both copies of L are attached to
C0 and 2 of the d new marked points are attached to C0. The remaining d(cid:0)2
marked points are the points of C \(cid:5). However, the map (cid:26) that stabilizes the
resulting prestable (n+d)-marked curveis constant on this M . Indeed, the (cid:12)rst
0;4
copy of L has no marked points and is attached to C0 at one point. So the (cid:12)rst
stepin stabilizationwill pruneL reducingthe numberof special pointson C0 from
4 to 3.
(ii): In the family de(cid:12)ning (cid:12) , only the ith marked point on L varies. After
i
adding the d new marked points, L is a 3-pointed prestable curve; marked by the
node p, the ith marked point, and the point q. For every base the only family of
genus0, 3-pointed, stable curvesis the constant family. So upon stabilization, this
family of genus 0, 3-pointed, stable curves becomes the constant family.
Inthefamilyde(cid:12)ning(cid:13),onlythe attachmentpointofthe twocopiesofLvaries.
The (cid:12)rst copy of L gives a family of 2-pointed, prestable curves; marked by q and
the attachment point of the two copies of L. This is unstable. Upon stabilization,
the(cid:12)rstcopyofLisprunedandthemarkedpointqonthe(cid:12)rstcopyisreplacedbya
markedpointonthesecondcopyattheoriginalattachmentpoint. Nowthesecond
copy of L gives a family of 3-pointed, prestable curves; marked by the attachment
pointpofthesecondandthirdirreduciblecomponents,theattachmentpointofthe
(cid:12)rst and second components, and q. For the same reason as in the last paragraph,
this becomes a constant family.
9
(iii): Eachstable map in (cid:11)(M (cid:2)Pr(cid:0)1) is obtained froma genus0, (n+d)-
0;n+d
pointed,stablecurve(C ;(p ;:::;p ;q ;:::;q ))andalineLinPr containingpby
0 1 n 1 d
attaching for each 1(cid:20)i(cid:20)n, a copy C of L to C where p in C is identi(cid:12)ed with
i 0 i
q in C . The map to Pr contracts C to p, and sends each curve C to L via the
i 0 0
identity morphism. Denoting by r the intersection point of L and (cid:5), the inverse
image of (cid:5) consists of the d points r ;:::;r , where r is the copy of r in C .
1 d i i
The component C is a 2-pointed, prestable curve: marked by the attachment
i
point p of C and by r . This is unstable. So, upon stabilization, C is pruned
i i i
and the marked point r is replaced by a marking on C at the point of attach-
i 0
ment of C and C , namely q . Therefore, up to relabeling of the last d marked
0 i i
points, the result is the genus 0, (n+d)-pointed, stable curve we started with,
(C ;(p ;:::;p ;q ;:::;q )). (cid:3)
0 1 n 1 d
4. More divisors
In the previous section we constructed a map (see De(cid:12)nition 3.3)
v :Pic(M )Sd !Pic(M (Pr;d))(cid:10)Q:
0;n+d 0;n
In this section we prove that the image of v, the divisor classes H, T and the
tautological divisors L , generate Pic(M (Pr;d))(cid:10)Q.
i 0;n
The divisor class H , [Pa, Prop. 1] is the class of stable maps whose image
(cid:3)
intersects a (cid:12)xed codimension 2 linear space (cid:3) of Pr. This is de(cid:12)ned to be the
empty divisor if r =1. For convenience, assume (cid:3) is contained in (cid:5) and does not
intersect L or the curves C used to de(cid:12)ne (cid:12) and (cid:13). If n (cid:21) 1, the divisors L ,
i i;(cid:5)
i=1;:::;n,[Pa,Prop. 1]arethepull-backbyev oftheCartierdivisor(cid:5). Ifd(cid:21)1,
i
the last divisoris T , [Pa, x2.3];the divisorof stable maps (C;(p ;:::;p );f) such
(cid:5) 1 n
thatf(cid:0)1((cid:5))isnotareduced,(cid:12)nitesetofdegreed. Thisisde(cid:12)nedtobetheempty
divisorifd=1. In[Pa]PandharipandeprovesthatH ,L andT areirreducible
(cid:3) i;(cid:5) (cid:5)
Cartier divisors (when they are nonempty).
Lemma 4.1. (i) The Cartier divisors T , L and H are NEF.
(cid:5) i;(cid:5) (cid:3)
(ii) The pull-backs (cid:11)(cid:3)(T ) and (cid:11)(cid:3)(L ) are zero. The pull-back (cid:11)(cid:3)(H ) equals
(cid:5) i;(cid:5) (cid:3)
(0;OPr(cid:0)1(d)) in Pic(M0;n+d)Sd (cid:2)Pic(Pr(cid:0)1); if r =1, then OPr(cid:0)1(1) is the
trivial invertible sheaf.
(iii) Assume n (cid:21) 1 so that (cid:12) is de(cid:12)ned for 1 (cid:20) i (cid:20) n. The pull-backs (cid:12)(cid:3)(T )
i i (cid:5)
and (cid:12)(cid:3)(H ) are zero. For 1 (cid:20) j (cid:20) n di(cid:11)erent from i, (cid:12)(cid:3)(L ) is zero.
i (cid:5) i j;(cid:5)
Finally, (cid:12)(cid:3)(L ) is O (1).
i i;(cid:5) P1
(iv) Assume d (cid:21) 2 so that (cid:13) is de(cid:12)ned. The pull-backs (cid:13)(cid:3)(H ) and (cid:13)(cid:3)(L )
(cid:3) i;(cid:5)
are zero, and (cid:13)(cid:3)(T ) is O (2) in Pic(P1).
(cid:5) P1
Proof. (i): By an argument similar to the one in Lemma 3.4, these divisors are
base-point-free(whenevertheyarenon-empty). ThedivisorH isbigifr(cid:21)2,and
(cid:3)
T is big if d(cid:21)2. The divisors L are not big.
(cid:5) i
(ii): Bythe proof of Lemma 3.5, the imageof (cid:11) isin O , which isdisjoint from
(cid:5)
T . Also,ev (cid:14)(cid:11)istheconstantmorphismwith imagep,sothe inverseimageofL
(cid:5) i i
is empty. Finally, the pull-back of H equals the pull-back under the diagonal (cid:1)
(cid:5)
of the Cartier divisor d pr(cid:0)1((cid:3)) in (Pr(cid:0)1), where (cid:3) is considered as a divisor
j=1 j
in Pr(cid:0)1 via projectionPfrom p.
10