The universal theta divisor over the moduli space of curves

0.23 MB·English

by  Gavril Farkas

THE UNIVERSAL THETA DIVISOR OVER THE MODULI SPACE OF CURVES
GAVRILFARKASANDALESSANDROVERRA
2
1
0 ABSTRACT. BycomputingtheclassoftheuniversalantiramificationlocusoftheGauss
2 map, we obtain a complete birational classification of the universal theta divisor Th
g
overthemodulispaceofcurves.
n
u The universal theta divisor over the moduli space A of principally polarized
g
J abelian varieties of dimension g is the divisor Θg inside the universal abelian variety
0 X → A ,characterizedbytwoproperties: (i)Θ = Θ,foreveryppav[A,Θ] ∈ A ,
1 g g g|[A,Θ] g
and (ii) the restriction s∗(Θ ) along the zero section s : A → X is trivial on A . The
g g g g
] study of the geometry of Θ mirrors that of A itself. Thus it is known that Θ is
G g g g
unirational forg ≤ 4; thecase g ≤ 3isclassical, for g = 4, werefer to[Ve1]. Whenever
A
A is of general type (that is, in the range g ≥ 7, cf. [Fr], [Mum], [T]), one can use
. g
h Viehweg’sadditivity theorem[Vi] for the fibre space Θ → A , to conclude that Θ is
g g g
at ofgeneraltypeaswell. OurfirstresultconcernsthebirationaltypeofΘ5:
m
Theorem0.1. TheuniversalthetadivisorΘ isuniruled.
[ 5

2 BymakinguseofthegenericallyfinitePrymmapP : R6 → A5,weshowthatthe
v pull-backoftheuniversalthetadivisoroverthePrymmodulispaceR ,thatis,
6
4
8 Θp := Θ × R
5 5 A5 6
1
0 isuniruled. WesketchtheideaoftheproofandrefertoSection4fordetails.
9. Wefix ageneralelement[C,η] ∈ R6, inducing an e´tale double cover f : C˜ → C.
0 The canonical curve C ⊂ P5 can be viewed as a quadratic section of a smooth quintic
0 delPezzosurfaceS ⊂ P5. AgeneralelementLonthethetadivisorΞ(C,η)ofthePrym
1
: variety P(C,η) is a line bundle L on C˜ such that Nmf(L) = KC and h0(C˜,L) = 2. We
v
associate to this data a rank 3 quadric Q ∈ PSym2H0(K ) everywheretangent to the
i L C
X canonicalcurveC ⊂ P5 alongaPrymcanonicaldivisor,thatis,
r
a C ·Q = 2d ,
L L
where d ∈ |K ⊗ η|. The pencil in | − 2K | generated by the curves C and S · Q
L C S L
inducesarationalcurveintheuniversalPrymthetadivisorΘp.
5
Nextweobtainacompletebirationalclassificationoftheuniversalthetadivisor
Th := M × Θ
g g Ag g
over the moduli space of curves. If [C] ∈ M is a smooth curve, the Abel-Jacobi map
g
C → Picg−1(C) provides a resolution of singularities of the theta divisor Θ of the
g−1 C
Jacobian of C. Thus one may regard the degree g − 1 universal symmetric product
C := M /S as a birational model of Th (having only finite quotient sin-
g,g−1 g,g−1 g−1 g
gularities), and ask for the place of Th in the classification of varieties. We provide a
g
completeanswertothisquestion. Forsmallgenus,Th enjoysrationalityproperties:
g
1
2 G.FARKASANDA.VERRA
Theorem0.2. Th isunirationalforg ≤ 9anduniruledforg ≤ 11.
g
The first part of the theorem, is a consequence of Mukai’s work [M1], [M2] on
representing canonical curves with general moduli as linear sections of certain homo-
geneous varieties. When g ≤ 9, there exists a Fano variety V ⊂ PNg of dimension
g
n := N −g+2andindexn −2, suchthatgeneral1-dimensionalcompleteintersec-
g g g
tionsofV arecanonicalcurves[C] ∈M havinggeneralmoduli. Thecorrespondence
g g
Σ := { (x ,...,x ),Λ) ∈ Vg−1×G(g,N +1) :x ∈ Λ, fori = 1,...,g−1
1 g−1 g g i
(cid:0) (cid:9)
mapsdominantlyontoTh via themap (x ,...,x ),Λ 7→ [V ∩Λ,x +···+x ].
g 1 g−1 g 1 g−1
Since Σ is a Grassmann bundle over th(cid:0)e rational variety(cid:1)Vg−1, it follows that Th is
g g
unirational in the range g ≤ 9. The cases g = 10,11 are settledby theobservation that
inthisrangethespaceM isuniruled,see[FP],[FV2].
g,g−1
For the remaining genera, we achieve a complete classification. This is the main
resultofthepaper:
Theorem0.3. TheuniversalthetadivisorTh isavarietyofgeneraltypeforg ≥ 12.
g
We also have a birational classification theorem for the universal degree n sym-
metricproductC := M /S forall1 ≤ n ≤ g−2,andrefertoSection3fordetails.
g,n g,n n
Our results are complete in degree g − 2 and less precise as n decreases. Similarly to
Theorem0.3,thenatureofC changeswheng = 12:
g,g−2
Theorem0.4. Theuniversaldegreeg−2symmetricproductC isuniruledforg < 12and
g,g−2
avarietyofgeneraltypeforg ≥ 12.
TheproofofTheorem0.3reliesonthecalculationoftheuniversalantiramification
divisorclassoftheGaussmap. ForacurveC ofgenusg,letγ : C 99K Pg−1 ∨ bethe
g−1
Gaussgivenbyγ(D) := hDiforD ∈ C −C1 . ThebranchdivisorBr(γ(cid:0)) ⊂ ((cid:1)Pg−1)∨is
g−1 g−1
thedualofthecanonicalcurveC ⊂ Pg−1. TheclosureinC oftheramificationdivisor
g−1
Ram(γ) is the locus of divisors D ∈ C such that supp(D)∩supp(K (−D)) 6= ∅. 1
g−1 C
Theantiramification divisorAntram(γ),definedbythefollowingequalityofdivisors
γ∗(Br(γ)) = Antram(γ)+2·Ram(γ),
splitsintothelocusofnon-reduceddivisors∆ := {2p+D :p ∈ C, D ∈ C }andthe
C g−3
locus of divisors D ∈ C such that K (−D) has non-reduced support. Globalizing
g−1 C
thisconstructionoverM ,weareleadtoconsidertheuniversalantiramificationdivisor
g
Antram := [C,x ,...,x ] ∈ M :∃p ∈ C withH0 K (−x −···−x −2p) 6= 0 .
g 1 g−1 g,g−1 C 1 g−1
(cid:8) (cid:0) (cid:1) (cid:9)
WehavethefollowingformulafortheclassofAntram :
g
Theorem0.5. TheclosureinM oftheantiramification locusislinearlyequivalentto,
g,g−1
g−1
[Antram ]= −4(g−7)λ+4(g−2) ψ −2δ −(12g−22)δ −
g i irr 0:2
Xi=1
g i−1
− 2i3−5i2−3i+4g−4i2s+14si−6gs−s+2s2g−3s2+2 δ ∈Pic(M ).
i:s g,g−1
Xi=0Xs=0(cid:16) (cid:17)
1
ThedescriptionoftheramificationdivisoroftheGaussmapgivenin[ACGH]p.247iserroneous.
THEUNIVERSALTHETADIVISOROVERTHEMODULISPACEOFCURVES 3
By construction,Antram is S -invariant, thusit descendstoan effective divi-
g g−1
sor G on C which, as it turns out, spans an extremal ray of the cone Eff(C ).
g g,g−1 g,g−1
Theuniversalthetadivisorisequippedwiththeinvolutionτ :C 99K C ,
g,g−1 g,g−1
τ [C,x +···+x ] := [C,y +···+y ],
1 g−1 1 g−1
(cid:0) (cid:1)
where O (y + ··· + y + x + ··· + x ) = K . Then G is the pull-back of the
C 1 g−1 1 g−1 C g
boundarydivisor∆ ⊂ C underthismap. Sincetheextremalityof∆ iseasyto
0:2 g,g−1 0:2
establish,thefollowingresultcomesnaturally:
e e
Theorem 0.6. The effective divisor G is covered byirreducible curves Γ ⊂ C such that
g g g,g−1
Γ ·G < 0. InparticularG ∈ Eff(C )isanon-movableextremaleffectivedivisor.
g g g g,g−1
The curves Γ have a simple modular construction. One fixes a general linear
g
series A ∈ W2 (C), in particular A is complete and has only ordinary ramification
g+1
points. The generalpoint of Γ correspondsto an element [C,D] ∈ C , where D ∈
g g,g−1
C is an effective divisor such that H0 C,A ⊗ O (−2p − D) 6= 0, for some point
g−1 C
p ∈ C,thatis,Distheresidualdivisorcut(cid:0)outbyatangentlineto(cid:1)thedegreeg+1plane
modelofC givenbyA. WerefertoSection2fordetails.
The proofs of Theorems 0.3 and 0.4 rely on two ingredients. First, we use our
result[FV2],statingthatforg ≥ 4,thesingularitiesofC imposenoadjointconditions,
g,n
that is, pluricanonical forms defined on the smooth locus of C extend to a smooth
g,n
modelof thesymmetricproduct. Precisely, if ǫ : C → C denotesany resolutionof
g,n g,n
singularities,thenforanyℓ ≥ 0,thereisagroupisomorphism
e
ǫ∗ : H0 (Cg,n)reg,KC⊗g,ℓn →∼= H0 Cg,n,KCeg⊗,nℓ .
(cid:0) (cid:1) (cid:0) (cid:1)
e
In particular, Th is of general type when the canonical class K ∈ Pic(C ) is
g Cg,g−1 g,g−1
big. This makes the problem of understanding the effective cone of C of some
g,g−1
importance. Ifπ :M → C isthequotientmap,theHurwitzformulaimplies
g,g−1 g,g−1
(1) π∗(K ) ≡ K −δ ∈ Pic(M ).
Cg,g−1 Mg,g−1 0:2 g,g−1
Thesum g−1ψ ∈ Pic(M )Sg−1 ofcotangenttautologicalclassesdescendstoabig
i=1 i g,g−1
and nef cPlass on Cg,g−1 (cf. Proposition 1.2), thus in order to conclude that Thg is of
generaltype,itsufficestoexhibitaneffectivedivisorD∈ Eff(C ),suchthat
g,g−1
g−1
(2) π∗(K ) ∈Q ψ +φ∗Eff(M )+Q λ,π∗([D]),δ : i≥ 0,c ≥2 .
Cg,g−1 >0DXi=1 iE g ≥0D i:c E
Inthisformula,φ: M → M denotesthemorphismforgettingthemarkedpoints,
g,g−1 g
and refer to Section 1 for the standard notation for boundary divisor classes on M .
g,n
Comparing condition (2) against the formula for K given by (4), if one writes
Cg,g−1
π∗(D)≡ aλ−b δ +c g−1ψ − b δ ∈ Pic(M ),thefollowinginequality
irr irr i=1 i i,c i:c i:c g,g−1
P P
(3) 3c < b
0:2
is a necessary condition for the existence of a divisor D satisfying (2). It is straightfor-
wardtounravelthegeometricsignificanceofthecondition(3). If[C]∈ M isageneral
g
curve, there is a rational map u : C 99K C given by restriction. Denoting by
g−1 g,g−1
4 G.FARKASANDA.VERRA
x,θ ∈ N1(Cg−1)Q the standard generators of the Ne´ron-Severi group of the symmet-
ricproduct,theinequality (3)characterizes preciselythosedivisorsD ∈ Pic(C )for
g,g−1
which u∗([D]) lies in the fourth quarter of the (θ,x)-plane (see [K1] for details on the
effectiveconeofC ). ThedivisorD ⊂ C playingthisroleispreciselyAntram .
g−1 g,g−1 g
Weexplain brieflyhowTheorem0.5implies thestatementabouttheKodairadi-
[g/2]
mension of C . We choose an effective divisor D ≡ aλ− b δ ∈ Eff(M ) on
g,g−1 i=0 i i g
the moduli space ofcurves, with a,bi ≥ 0, having slopes = sP(D) := minaibi as small as
possible. Thennotethatthefollowinglinearcombination
g−1
1 3
π∗(K )− [Antram ]−(12g−25)φ∗(D)− ψ − (84g−185)−(12g−25)s λ
Cg,g−1 6g−11(cid:16)2 g Xi=1 i (cid:0) (cid:1) (cid:17)
is expressible as a positive combination of boundary divisors on M . Since, as al-
g,g−1
g−1
ready pointed out, the class ψ ∈ Pic(M ) descends to a big class on C ,
i=1 i g,g−1 g,g−1
oneobtainsthefollowing: P
Corollary0.7. Forallgsuchthattheslopeofthemodulispaceofcurvessatisfiestheinequality
84g−185
s(M ):= inf s(D)< ,
g D∈Eff(Mg) 12g−25
theuniversalthetadivisorTh isofgeneraltype.
g
The bound appearing in Corollary 0.7 holds precisely when g ≥ 12; for g such
that g + 1 is composite, the inequality s(M ) ≤ 6 + 12/(g + 1) is well-known, and
g
D can be chosen to be a Brill-Noether divisor Mr corresponding to curves with a gr
g,d d
when the Brill-Noether number ρ(g,r,d) = −1, cf. [EH1]. When g + 1 is prime and
g 6= 12, thenin practice g = 2k −2 ≥ 16, and D can be chosentobe theGieseker-Petri
GP1 consisting of curves C possessing a pencil A ∈ W1(C) such that the Petri map
g,k k
µ (C,A) : H0(C,A) ⊗ H0(C,K ⊗ A∨) → H0(C,K ) is not an isomorphism. When
0 C C
g = 12, one has to use the divisor constructed on M in [FV1]. Finally, when g ≤ 11
12
it is knownthats(M ) ≥ 6+12/(g +1)and inequality (0.7) isnotsatisfied. Infact, as
g
alreadypointedoutκ(Th ) = −∞inthisrange.
g
The proof of Theorem 0.4 proceeds along similar lines, and relies on finding an
explicitS -invariantextremalrayoftheconeofeffectivedivisorsonM . Arepre-
g−2 g,g−2
sentativeofthisrayischaracterizedbythegeometricconditionthatthemarkedpoints
appearinthesamefibreofapencilofdegreeg−1. Onecanconstructsuchdivisorson
allmodulispacesM with1≤ n ≤ g−2,cf. Section3.
g,n
Theorem0.8. TheclosureinsideM ofthelocus
g,g−2
g−2
F := {[C,x ,...,x ]∈ M : ∃A∈ W1 (C) with H0 C,A(− x ) 6= 0}
g,1 1 g−2 g,g−2 g−1 i
(cid:0) Xi=1 (cid:1)
isanon-movable,extremal effectivedivisoronM . Itsclassisgivenbytheformula:
g,g−2
g−2 g−2
1
F ≡ −(g−12)λ+(g−3) ψ −δ − s(g−4+sg−2s)δ −··· ∈Pic(M ).
g,1 i irr 0:s g,g−2
2
Xi=1 Xs=2
THEUNIVERSALTHETADIVISOROVERTHEMODULISPACEOFCURVES 5
Note that again, inequality (3) is satisfied, hence F can be used to prove that
g,1
K is big. Moreover, F descends to an extremal divisor F ∈ Eff(C ). In
Cg,g−2 g,1 g,1 g,g−2
fact,weshallshowthatFg,1 issweptbycurvesintersectingitsclaessnegatively.
1. CONES OFeDIVISORS ON UNIVERSAL SYMMETRIC PRODUCTS
The aim of this section is to establish certain facts about boundary divisors on
M andC ,see[AC]forastandardreference. Wefollowtheconventionsetin[FV2],
g,n g,n
thatis,ifMisaDeligne-Mumfordstack,wedenotebyMitscoarsemodulispace.
For an integer 0 ≤ i ≤ [g/2] and a subset T ⊂ {1,...,n}, we denote by ∆
i:T
theclosureinM ofthelocusofn-pointedcurves[C ∪C ,x ,...,x ],whereC and
g,n 1 2 1 n 1
C are smooth curves of genera i and g − i respectively meeting transversally in one
2
point, and the marked pointslying on C are precisely thoseindexed by T. We define
1
δi:T := [∆i:T]Q ∈Pic(Mg,n). For0 ≤ i≤ [g/2]and0 ≤ s≤ g,weset
∆i:s := δi:T, δi:s := [∆i:s]Q ∈ Pic(Mg,n).
X
#(T)=s
By convention, δ := ∅, for s < 2, and δ := δ . If φ : M → M is the
0:s i:s g−i:n−s g,n g
morphism forgetting the marked points, we set λ := φ∗(λ) and δ := φ∗(δ ), where
irr irr
δ := [∆ ] ∈ Pic(M ) denotes the class of the locus of irreducible nodal curves.
irr irr g
Furthermore, ψ ,...,ψ ∈ Pic(M ) are the cotangent classes corresponding to the
1 n g,n
markedpoints. ThecanonicalclassofM iscomputedviaKodaira-Spencertheory:
g,n
n
(4) K ≡ 13λ−2δ + ψ −2 δ −δ ∈ Pic(M ).
Mg,n irr i i:T 1:∅ g,n
Xi=1 T⊂{X1,...,n}
i≥0
LetC := M /S betheuniversalsymmetricproductandπ : M → C (respec-
g,n g,n n g,n g,n
tivelyϕ : C → M )theprojection(respectivelytheforgetfulmap), sothatφ = ϕ◦π.
g,n g
Wedenotebyλ,δ ,δ := [∆ ] ∈ Pic(C )thedivisorclassesonthesymmetricprod-
irr i:c i:c g,n
uct pulling-back to the same symbols on M . Clearly, π∗(λ) = λ, π∗(δ ) = δ ,
e e e e g,n irr irr
π∗(δ ) = δ ;inthecasei = 0,c = 2,thisreflectsthebranchingofthemapπ alongthe
i:c i:c e e
divisor∆ ⊂ C . Following[FV2],letLdenotethelinebundleonC ,havingfibre
e 0:2 g,n g,n
e L[C,x1+···+xn] := Tx∨1(C)⊗···⊗Tx∨n(C),
overapoint[C,x +···+x ] := π([C,x ,...,x ])∈ C . Wesetψ := c (L),andnote:
1 n 1 n g,n 1
n n n
e
(5) π∗(ψ) = ψ − δ = ψ − sδ ∈ Pic(M ).
i 0:T i 0:s g,n
Xi=1(cid:16) i∈T⊂X{1,...,n} (cid:17) Xi=1 Xs=2
e
Proposition 1.1. For g ≥ 3 and n ≥ 0, the morphism π∗ : Pic(C ) → Pic(M ) is
g,n Q g,n Q
injective. Furthermore,thereisanisomorphism ofgroupsPic(C ) →∼= N1(C ) .
g,n Q g,n Q
Proof. Thefirstassertionisanimmediateconsequenceoftheexistenceofthenormmor-
phism Nm : Pic(M ) → Pic(C ), such that Nm (π∗(L)) = L⊗deg(π), for every L ∈
π g,n g,n π
Pic(Cg,n). The second part comes from the isomorphism Pic(Mg,n)Q →∼= N1(Mg,n)Q,
coupledwiththecommutativityoftheobviousdiagramsrelatingthePicardandNe´ron-
SeverigroupsofM andC respectively. (cid:3)
g,n g,n
6 G.FARKASANDA.VERRA
One may thus identify Pic(Cg,n)Q ∼= Pic(Mg,n)SQn. The Riemann-Hurwitz for-
mulaappliedtothebranchedcoveringπ : M → C yields,
g,n g,n
n n
π∗(K )= K −δ ≡ 13λ+ ψ −2δ −3δ −2 δ −··· .
Cg,n Mg,n 0:2 i irr 0:2 0:s
Xi=1 Xs=3
Asexpected,thesumofcotangentclassesdescendstoabiglinebundleonC .
g,n
Proposition1.2. ThedivisorclassN := ψ+ n sδ ∈Eff(C )isbigandnef.
g,n s=2 0:s g,n
P
Proof. The class N is characterized by tehe propertey that π∗(N ) = n ψ . This is
g,n g,n i=1 i
a nef class on Mg,n, in particular, Ng,n is nef on Cg,n. To establish thatPNg,n is big, we
expressitasacombinationofeffectiveclassesandtheclassκ ∈ Pic(C ),where
1 g,n
n [g/2] e
π∗(κ )= κ = 12λ+ ψ −δ − δ ∈ Pic(M ).
1 1 i irr i:s g,n
Xi=1 Xi=0 Xs≥0
e
Since π∗(κ ) is ample on M , it follows that κ is ample as well. To finish the proof,
1 g,n 1
we exhibit a suitable effective class on M having negative λ-coefficient. For that
g,n
e e
purpose,wechooseW ⊂ C tobethelocusofeffectivedivisorshavingaWeierstrass
g,n g,n
pointin theirsupport. Fori = 1,...,n,wedenotebyσ : M → M themorphism
i g,n g,1
forgettingallbutthei-thpoint,andlet
g−1
g+1 g−i+1
W ≡ −λ+ ψ− δ ∈ Eff(M ),
i:1 g,1
(cid:18) 2 (cid:19) (cid:18) 2 (cid:19)
Xi=1
betheclassofthedivisorofWeierstrasspointsontheuniversalcurve. Thenonefinds
n n n
g+1 g+1
π∗(W )≡ σ∗(W) = −nλ+ ψ − sδ −··· ∈ Pic(M ),
g,n i (cid:18) 2 (cid:19) i (cid:18) 2 (cid:19) 0:s g,n
Xi=1 Xi=1 Xs=2
and W ≡ −gλ+ g+1 ψ − [g/2] b δ , where b > 0. One checks that N
g,n 2 i=1 s≥0 i:s i:s i:s g,n
can be written aes a Q(cid:0)-co(cid:1)mebinaPtion wPith positeive coefficients of the ample class κ1, the
effectiveclass[W ]andotherboundarydivisorclasses. Inparticular,N isbig. (cid:3)
g,n g,n
e
2. THE UNIVERSAL ANTIRAMIFICATION LOCUS OF THE GAUSS MAP
We begin the calculation of the divisor Antram , and for a start we consider its
g
restrictionAntram toM . RecallthatAntram isdefinedastheclosureofthelocus
g g,g−1 g
ofpointedcurves[C,x ,...,x ] ∈ M ,suchthatthereexistsaholomorphicform
1 g−1 g,g−1
onC vanishingatx ,...,x andhavinganunspecifieddoublezero.
1 g−1
Letu : M(1) → M betheuniversalcurveoverthestackof(g−1)-pointed
g,g−1 g,g−1
(1)
smoothcurvesandwedenoteby [C,x ,...,x ],p ∈ M ageneralpoint,where
1 g−1 g,g−1
[C,x ,...,x ] ∈ M and p(cid:0)∈ C is an arbitrar(cid:1)y point. For i = 1,...,g − 1, let
1 g−1 g,g−1
(1)
∆ ⊂ M be the diagonal divisor given by the equation p = x . Furthermore, for
ip g,g−1 i
i = 1,...,g −1weconsiderasbefore theprojectionsσ : M(1) → M (respectively
i g,g−1 g,1
σ : M(1) → M ), obtained by forgetting all marked points except x (respectively
p g,g−1 g,1 i
THEUNIVERSALTHETADIVISOROVERTHEMODULISPACEOFCURVES 7
p), and then set K := σ∗(ω ) ∈ Pic(M(1) ) and K := σ∗(ω ) ∈ Pic(M(1) ). We
i i φ g,g−1 p p φ g,g−1
considerthefollowingcartesiandiagramofstacks
X −−−q−→ M(1)
g,g−1
f
 
My −−−φ−→ My
g,1 g
inwhichallthemorphismsaresmoothandφ(hencealsoq)isproper. For1≤ i ≤ g−1
there are tautological sections r : M(1) → X as well as r : M(1) → X, and set
i g,g−1 p g,g−1
g−1
E := Im(r ),E := Im(r ). Thus{E } andE arerelativedivisorsoverq.
i i p p i i=1 p
(1) g−1
Forapoint [C,x ,...,x ],p ∈ M ,wedenoteD := x +2p ∈ C ,
1 g−1 g,g−1 i=1 i g+1
andhavethefollo(cid:0)wingexactsequen(cid:1)ce: P
H0 O (D)
0 → C → H0 O (D) α→D H1(O ) → H1 O (D) → 0.
H(cid:0)0(O ) (cid:1) D C C
C (cid:0) (cid:1) (cid:0) (cid:1)
Inparticular,themorphismsα globalizetoamorphismofvectorbundlesoverM(1)
D g,g−1
g−1
α : A := q O E +2E /O → R1q O .
∗ X i p X ∗ X
(cid:16) (cid:0)Xi=1 (cid:1) (cid:17)
ThesubvarietyZ := [C,x ,...,x ],p ∈ M(1) :H0 K (−2p− g−1x ) 6= 0 is
1 g−1 g,g−1 C i=1 i
thenon-surjectivitylo(cid:8)c(cid:0)usofαandAntram(cid:1)g := u∗(Z) ⊂ M(cid:0)g,g−1. ThecPlassofZ(cid:1)iseq(cid:9)ual
to
g−1
[Z]= c A∨− R1q O ∨ = c −qO ( E +2E ) ∈ A2(M(1) ),
2 ∗ X 2 ! X i p g,g−1
(cid:16) (cid:0) (cid:1) (cid:17) (cid:16) Xi=1 (cid:17)
wherethelasttermcanbecomputedbyGrothendieck-Riemann-Roch:
g−1 ( g−1E +2E )k c (ω ) c2(ω )
ch qO E +2E = q i=1 i p · 1− 1 q + 1 q +··· ,
! X i p ∗
P k! 2 12
(cid:16) (cid:0)Xi=1 (cid:1)(cid:17) h(cid:16)Xk≥0 (cid:17) (cid:16) (cid:17)i
andweare interestedin evaluating thetermsofdegree1and2in thisexpression. The
resultofapplyingGRRtothemorphismq,canbesummarizedasfollows:
Lemma2.1. OnehasthefollowingrelationsinA∗(M(1) ):
g,g−1
(i)
g−1 g−1 g−1
ch q O ( E +2E ) =λ− K −3K +2 ∆ .
1 ∗ X i p i p ip
(cid:16) (cid:0) Xi=1 (cid:1)(cid:17) Xi=1 Xi=1
(ii)
g−1 g−1 g−1
5 1
ch q O ( E +2E ) = K2+ K2−2 (K +K )·∆ .
2 ∗ X i p 2 p 2 i i p ip
(cid:16) (cid:0) Xi=1 (cid:1)(cid:17) Xi=1 Xi=1
Proof. Weapplysystematicallythepush-pullformulaandthefollowingidentities:
E2 = −E ·q∗(K ), E2 = −E ·q∗(K ), E ·c (ω ) =E ·q∗(K ), E ·c (ω )= E ·q∗(K ),
i i i p p p i 1 q i i p 1 q p p
E ·E = 0 fori 6= j, E ·E = E ·q∗(∆ ), and q (c2(ω )) = 12λ.
i j i p i ip ∗ 1 q
8 G.FARKASANDA.VERRA
(cid:3)
Proposition2.2. Onehas[Antram ] =−4(g−7)λ+(4g−8) g−1ψ ∈ Pic(M ).
g i=1 i g,g−1
P
Proof. We apply the results of Lemma 2.1, as well as the formulas from [HM] p. 55, in
order to estimate the push-forward under u of the degree 2 monomials in tautological
g−1
classes. SettingF := q O ( E +2E ) ,weobtainthat
∗ X i=1 i p
(cid:0) P (cid:1)
g−1 g−1
u ch2(F) = −(8g−116)λ+(8g−24) ψ , and u ch (F) = 30λ−4 ψ ,
∗ 1 i ∗ 2 i
(cid:0) (cid:1) Xi=1 (cid:0) (cid:1) Xi=1
hence [Antram ] = u ch2(F)−2ch (F) /2, and the claimed formula follows at once.
g ∗ 1 2
(cid:3)
(cid:0) (cid:1)
WeproceednowtowardsprovingTheorem0.5andexpandtheclass[Antram ]in
g
thestandardbasisofthePicardgroup,thatis,
g−1 g i−1
[Antram ] = aλ+c ψ −b δ − b δ .
g i irr irr i:s i:s
Xi=1 Xi=0Xs=0
Wehavejustcomputeda = −4(g−7)andc = 4(g −2). Theremainingcoefficientsare
determined by intersecting Antram with curves lying in the boundary of M and
g g,g−1
understandinghowAntram degenerates. Webeginwiththecoefficientb :
g 0:2
Proposition2.3. Onehas the relation (4g −6)c−(g −2)b = (4g −2)(g −2). Itfollows
0:2
thatb = 12g−22.
0:2
Proof. Wefixageneralpointedcurve[C,x ,...,x ]∈ M andconsiderthefamily
1 g−2 g,g−2
C := [C,x ,...,x ,x ] : x ∈ C ⊂ M .
xg−1 1 g−2 g−1 g−1 g,g−1
(cid:8) (cid:9)
The curve C is the fibre over [C,x ,...,x ] of the morphism M → M
xg−1 1 g−2 g,g−1 g,g−2
forgetting the point labeled by x . Note that C ·ψ = 1 for i = 1,...,g −2 and
g−1 xg−1 i
C ·ψ = 3g−4 = 2g−2+(g−2). ObviouslyC ·δ = g−2andthepointsin
xg−1 g−1 xi 0:2
theintersectioncorrespondtothe case whenx collides with oneofthefixedpoints
g−1
x ,...,x . The intersection of C with the remaining generators of Pic(M ) is
1 g−2 xi g,g−1
equal to zero. We set A := K ⊗ O (−x − ··· − x ) ∈ W1(C). By the generality
C C 1 g−2 g
assumption, h0(C,A) = 2, and all ramification points of A are simple. Pointed curves
intheintersectionC ·Antram correspondtopointsx ∈ C,suchthatthereexists
xg−1 g g−1
a(ramification)pointp ∈ C withH0 C,A⊗O (−2p−x ) 6= 0. ThepencilAcarries
C g−1
4g − 2 ramification points. For each(cid:0)of them there are g − (cid:1)2 possibilities of choosing
x ∈ C inthesamefibreastheramification point,hencetheconclusionfollows. (cid:3)
g−1
Nextwedeterminethecoefficientb . Firstwenotethattherelation
irr
(6) a−12b +b = 0
irr 1:0
holds. Indeed,thedivisorAntram isdisjointfromthecurvein∆ ⊂M ,obtained
g 1:0 g,g−1
from a fixed pointed curve [C,x ,...,x ,q] ∈ M , by attaching at the point q a
1 g−1 g−1,g
pencilofplanecubicsalongasectionofthepencilinducedbyoneofthe9basepoints.
Proposition2.4. Onehastherelation b = 2.
irr
THEUNIVERSALTHETADIVISOROVERTHEMODULISPACEOFCURVES 9
Proof. Wefixageneralcurve[C,q,x ,...,x ]∈ M ,andwedefinethefamily
1 g−1 g−1,g
C := [C/t ∼ q,x ,...,x ]: t ∈ C}⊂ ∆ ⊂ M .
irr 1 g−1 irr g,g−1
(cid:8)
ThenC ·ψ = 1fori = 1,...,g−1,C ·δ = −(deg(K )+2) = −2g+2,andfinally
irr i irr irr C
C ·δ = 1. AllotherintersectionnumberswithgeneratorsofPic(M )equalzero.
irr 1:0 g,g−1
g−1
WefixaneffectivedivisorD ∈C ofdegreee ≥ g(forinstanceD = q+ x ).
e i=1 i
Foreachpairofpoints(t,p) ∈ C ×C,thereisanexactsequenceonC P
g−1 g−1
0 → H0 C,K (q+t−2p− x ) → H0 C,K (D+q+t−2p− x ) β→t,p
C i C i
(cid:0) Xi=1 (cid:1) (cid:0) Xi=1 (cid:1)
g−1 g−1
H0 D,K (D+q+t−2p− x ) → H1 C,K (q+t−2p− x ) → 0.
C i C i
(cid:0) Xi=1 (cid:1) (cid:0) Xi=1 (cid:1)
The intersection C · Antram corresponds to the locus of pairs (t,p) ∈ C × C such
irr g
that the map β is not injective. On the triple product of C, we consider two of the
t,p
projections f : C × C × C → C × C and p : C × C × C → C given by f(x,t,p) =
1
(t,p) and p (x,t,p) = x, then set A := K (q − g−1x ) ∈ Picg−2(C). We denote by
1 C i=1 i
∆12,∆13 ⊂ C × C × C the corresponding diagPonals, and finally, introduce the line
bundleonC ×C ×C
F := p∗(A)⊗O (∆ −2∆ ).
1 C×C×C 12 13
ApplyingthePorteousformula,onecanwrite
ch2(fF)+2ch (fF)
C ·Antram = c (R1f F −R0f F) = 1 ! 2 ! ∈ A2(C ×C).
irr g 2 ∗ ∗
2
Weevaluatech (fF)usingGRRappliedtothemorphismf,thatis,
i !
p∗(A)+∆ −2∆ a 1
ch(fF) = f 1 12 13 · 1− p∗(K ) .
! ∗ (cid:0) a! (cid:1) 2 1 C
h(cid:16)Xa≥0 (cid:17) (cid:16) (cid:17)i
DenotingbyF ,F ∈ H2(C ×C)theclassofthefibres,aftercalculations onefindsthat
1 2
ch (f F) = −(g−2)F −4(g−2)F −2∆ ∈ H2(C ×C,Q),
1 ∗ 1 2 C
ch (f∗F) = −2(g−2) ∈ H4(C ×C,Q),
2
thatis,c (R1f F −R0f F) = 4(g−2)(g−1). Coupledwith(6),thisyieldsb = 2. (cid:3)
2 ∗ ∗ irr
Weareleftwiththetaskofdeterminingthecoefficientofδ . Thisrequiressolving
i:s
anumberofenumerativegeometryproblemsinthespiritofdeJonquie`res’formula. We
fix integers 0 ≤ i ≤ g and s ≤ i−1 as well as general pointed curves [C,x ,...,x ] ∈
1 s
M and [D,q,x ,...,x ] ∈ M , then construct a pencil of stable curves of
i,s s+1 g−1 g−i,g−s
genusg, by identifying the fixed point q ∈ D with a variable point, also denotedby q,
onthecomponentC:
C := [C ∪ D,x ,...,x ,x ,...,x ]: q ∈ C ⊂ ∆ ⊂ M .
i:s q 1 s s+1 g−1 i:s g,g−1
(cid:8) (cid:9)
Wesummarizethenon-zerointersectionnumbersofC withgeneratorsofPic(M ):
i:s g,g−1
C ·ψ =··· = C ·ψ = 1, C ·δ = i, C ·δ = 2i−2+s.
i:s 1 i:s s i:s i:s−1 i:s i:s
10 G.FARKASANDA.VERRA
Theorem2.5. Wefixintegers0 ≤ i≤ gand0 ≤ s ≤ i−1. Then,thefollowingformulaholds:
b =2i3 −5i2−3i+4g−4i2s+14si−6gs−s+2s2g−3s2+2.
i:s
Intheproofanessentialroleisplayedbythefollowingcalculation:
Proposition 2.6. Let i,s be integers such that 0 ≤ s ≤ i−1, and [C,x ,...,x ] ∈ M a
1 s i,s
generalpointedcurve. Thenumberofpairs(q,p) ∈ C ×C suchthat
H0 C,K ⊗O (−x −···−x −(i−s−1)q−2p) 6= 0,
C C 1 s
(cid:0) (cid:1)
isequalto a(i,s) := 2(i−s−1)(2i3 −5i2+i+2−2i2s+3is).
Remark 2.7. By specializing, one recovers well-known formulas in enumerative ge-
ometry. For instance, a(3,0) = 56 is twice the number of bitangents of a smooth
plane quartic, whereas a(4,0) = 324 equals the number of canonical divisors of type
3q+2p+x ∈ |K |,where[C] ∈ M . ThismatchesdeJonquie`res’formula,cf. [ACGH]
C 4
p.359.
Proof of Theorem 2.5. We fix a general point [C ∪ D,x ,...,x ] ∈ C · Antram ,
q 1 g−1 i:s g
corresponding to a point q ∈ C. We shall show that q is not one of the marked points
x ,...,x on C, then give a geometric characterization of such points and count their
1 s
number. Let
ω ∈ H0 D,K ⊗O (2iq) and ω ∈H0 C,K ⊗O (2g−2i)q
D D D C C C
(cid:0) (cid:1) (cid:0) (cid:1)
be the aspects of the section of the limit canonical series on C ∪ D, which vanishes
q
doubly at an unspecifiedpoint p ∈ C ∪D as well as along the divisor x +···+x .
1 g−1
Theconditionord (ω )+ord (ω ) ≥ 2g−2,comesfromthedefinitionofalimitlinear
q C q D
series. Wedistinguishtwocasesdependingonthepositionofthepointp. Ifp ∈ Dthen,
div(ω ) ≥ x +···+x , div(ω )≥ x +···+x +2p.
C 1 s D s+1 g−1
Since the points q,x ,...,x ∈ D are general, we find that ord (ω ) ≤ i+s−2.
s+1 g−1 q D
Moreover,K ⊗O ((i−s+2)q−x −···−x ) ∈ W1 (D)isapencil,andp ∈ D
D D s+1 g−1 g−i+1
isoneofits(simple)ramificationpoints. TheHurwitzformulagives4(g−i)choicesfor
suchp ∈ D.
Bycompatibility,ord (ω ) ≥ 2g−i−s. Aparametercountimpliesthatequality
q C
musthold. TheconditionH0 C,K ⊗O (−x −···−x −(i−s)q 6= 0,isequivalentto
C C 1 s
askingthatq ∈ C bearamific(cid:0)ation pointofK ⊗O (− s x )(cid:1)∈ Wi−s−1 (C). Since
C C j=1 j 2i−2−s
the pointsx1,...,xs ∈ C are chosento be general, all ramPification pointsof this linear
series are simple and occur away from the marked points. From Plu¨cker’s formula,
thenumberoframification pointsequals(i−s)(i2 −1−is). Multiplyingthiswiththe
numberofchoicesforp ∈ D,weobtainatotalcontributionof4(g−i)(i−s)(i2−is−1)
to the intersectionC ·Antram , stemming from thecase when p ∈ D. The proofthat
i:s g
each ofthesepointsofintersectionistobecountedwithmultiplicity 1isstandardand
proceedsalongthelinesof[EH2]Lemma3.4.
We assume now that p ∈ C. Keeping the notation from above, it follows that
ord (ω ) = i+s−1andord (ω )= 2g−i−s−1,therefore
q D q C
s
06= σ ∈ H0 C,K ⊗O (− x −(i−s−1)q−2p) .
C C C j
(cid:0) Xj=1 (cid:1)