Divisors class groups of singular surfaces

0.23 MB·English

by  Robin Hartshorne

DIVISORS CLASS GROUPS OF SINGULAR SURFACES
ROBINHARTSHORNEANDCLAUDIAPOLINI
ABSTRACT. Wecompute divisors classgroups of singularsurfaces. Most notablyweproduce an
3
exactsequencethatrelatestheCartierdivisorsandalmost Cartierdivisorsofasurfacetothethose
1 of itsnormalization. Thisgeneralizes Hartshorne’stheoremforthecubicruledsurface in P3. We
0
applytheseresultstolimitthepossiblecurvesthatcanbeset-theoreticcompleteintersectioninP3in
2
characteristiczero.
n
a
J

5 1. INTRODUCTION
1
On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire
]
C theory of curves and surfaces is dependent on this study of codimension one subvarieties and the
A
linearandalgebraic familiesinwhichtheymove.
.
h Thistheoryhasbeengeneralized intwodirections: theWeil divisors onanormalvariety, taking
t
a codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme,
m
based on locally principal codimension one subschemes. Most of the literature both in algebraic
[
geometryandcommutativealgebrauptonowhasbeenlimitedtothesekindsofdivisors.
1
v More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe
2
[9] introduced the notion of an almost Cartier divisor, which is locally principal off a subset of
2
2 codimensiontwo. Atheoryofgeneralizeddivisorswasproposedoncurvesin[14],andextendedto
3
any dimension in [15]. Thelatter paper gave acomplete description of the generalized divisors on
.
1
theruledcubicsurfacein P3.
0
3 In this paper we extend that analysis to an arbitrary integral surface X, explaining the group
1
: APicX of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group
v
i of the normalization S of X and certain local data at the singular points of X. We apply these
X
results to give limitations on the possible curves that can be set-theoretic compete intersections in
r
a
P3 incharacteristic zero
Insection 2 weexplain ourbasic set-up, comparing divisors on avariety X toits normalization
S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors
at a singular point of X in terms of the Cartier divisors along the curve of singularities and its
inverse image in the normalization. In Section 4 we derive some global exact sequences for the
groups PicX, APicX, and PicS, which generalize the results of [15, 6] to arbitrary surfaces
§
Theseresultsareparticularly transparent forsurfaceswithordinarysingularities, meaningadouble
curvewithafinitenumberofpinchpointsandtriplepoints.
AMS2010MathematicsSubjectClassification.Primary14C20,13A30;Secondary14M10,14J05.
ThesecondauthorwaspartiallysupportedbytheNSAandtheNSF.
1
2 R.HARTSHORNEANDC.POLINI
Insection 5wegathersomeresults oncurvesthatweneedinourcalculations onsurfaces. Then
in Section 6 wegive a number of examples of surfaces and compute their groups of almost Picard
divisors.
InSection7weapplytheseresultstolimitthepossibledegreeandgenusofcurvesinP3thatcan
beset-theoretic completeintersections onsurfaceswithordinarysingularities incharacteristic zero,
extendingearlierworkofJaffeandBoratynski. Weillustratetheseresultswiththedeterminationof
allset-theoretic completeintersections onanumberofparticularsurfacesin P3.
Our main results assume that the ground field k is of characteristic zero, so that a) we can use
theexponential sequence incomparing theadditiveandmultiplicative structures, andb)sothatthe
additivegroupofthefieldisatorsion-free abeliangroup.
The first author would like to thank the Department of Mathematics at the University of Notre
Dameforhospitality duringthepreparation ofthispaper.
2. DIVISORS AND FINITE MORPHISMS
AlltheringstreatedinthispaperareNoetherian, essentially offinitetypeoverafieldk whichis
algebraically closed. In our application we will often compare divisors on integral surface X with
itsnormalization S. Butsomeofourpreliminary results arevalidmoregenerally sowefixasetof
assumptions.
Assumptions2.1. Letπ : S X beadominant finite morphism ofreduced schemes. LetΓand
−→
LbecodimensiononesubschemesinS andX respectively suchthatπ restrictstoamorphismofΓ
toL. Assumethat S and X both satisfy G (i.e. Gorenstein incodimension 1) andS (i.e. Serre’s
1 2
condition S ) so that the theory of generalized divisors developed in [15] can be applied. Further
2
assumethattheschemesΓandLhavenoembeddedassociated primes,hencetheysatisfyS .
1
Now we recall the notion of generalized divisors from [15]. If X is a scheme satisfying G
1
and S , we denote by K the sheaf of total quotient rings of the structure sheaf . A general-
2 X X
O
ized divisor on X is a fractional ideal K , i.e. a coherent sub- -module of K , that is
X X X
I ⊂ O
nondegenerate, namely for each generic point η X, = K , and such that is a reflexive
η X,η
∈ I I
-module.
X
O
Wesay isprincipalifitisgeneratedbyasinglenon-zero-divisor f inK . Wesay isCartier
X
I I
ifitislocally principal everywhere. Wesay isalmost Cartier ifitislocally principal offsubsets
I
ofcodimension atleast2. WedenotebyCartX andbyACartX thegroupsofCartierdivisorsand
almost Cartier divisors, respectively, and dividing these by the subgroup of principal divisors we
obtain the divisors class groups PicX and APicX, respectively. The divisor is effective if it is
I
contained in . In that case it defines acodimension one subscheme Y X without embedded
X
O ⊂
components. Conversely, foranysuch Y,itssheafofideals isaneffectivedivisor.
Y
I
Werecallsomeproperties ofthesegroups.
Proposition 2.2. Adoptassumptions 2.1. Thefollowing hold:
DIVISORSCLASSGROUPSOFSINGULARSURFACES 3
(a) Thereisanaturalmapπ⋆ :PicX PicS
−→
(b) Thereisanaturalmapπ⋆ :APicX APicS
−→
(c) Thereisanexactsequence
0 PicX APicX APic(Spec )
X,x
→ −→ −→ O
x∈X
M
wherethesumistakenoverallpointsx X ofcodimension atleasttwo.
∈
Proof. For(a)and(b)see[15,2.18]. ThemaponPicmakessense foranymorphism ofschemes.
For APic, we need only to observe that since π is a dominant finite morphism, if Z X has
⊂
codimension two,then alsoπ−1(Z) S hascodimension two. Thesequence in(c)isduetoJaffe
⊂
forsurfaces(see[15,2.15]),butholdsinanydimension (sameproof).
Proposition2.3. Adoptassumptions2.1. FurtherassumethatX andS areaffineandS issmooth.
Thenthereisanaturalgrouphomomorphism
ϕ :APicX CartΓ/π∗CartL
−→
Proof. Givena divisor class d APicX, choose an effective divisor D dthat does not contain
∈ ∈
any irreducible component of Lin itssupport (this ispossible byLemma2.4below). Now restrict
thedivisorDtoX L,transportitviatheisomorphism πtoS Γ,andtakeitsclosureinS. Since
− −
S issmooth,thiswillbeaCartierdivisoronallofS,whichwecanintersectwithΓtogiveaCartier
divisoronΓ.
IfwechooseanothereffectivedivisorD′representingthesameclassd,thatalsodoesnotcontain
anycomponent ofLinitssupport, then D D′ isaprincipal divisor (f)forsomef K . Since
X
− ∈
π givesanisomorphism ofS ΓtoX Litfollows thatS andX arebirational, i.e. K = K .
X S
− −
Sotheequation D′ D = (f)persists on S,showingthattheambiguity ofourconstruction isthe
−
CartierdivisoronΓdefinedbytherestrictionoff. Notenowthatsince(f)= D D′,wecanwrite
−
D′ = f D where D′ and D are the ideals of D′ and D in X, and f KX. If λ is a generic
I I I I O ∈
point of L, then after localizing, the ideals D′,λ and D,λ are both the whole ring X,λ, since D
I I O
and D′ are effective divisors not containing any component of L in their support. Therefore f is a
unit in O . Thus f restricts to a non-zerodivisor in the total quotient ring K , whose stalk at λ
X,λ L
is isomorphic to O / . Thus the restriction of f defines a Cartier divisor on L whose image
X,λ L,λ
I
in Γ will be the same as the restriction of f from S to Γ. Hence our map ϕ is well-defined to the
quotient group CartΓ/π∗CartL.
Thefollowinglemmaistheaffineanalogueof[15,2.11].
Lemma2.4. LetX be an affine scheme satisfying G and S . Letd APicX be an equivalence
1 2
∈
classofalmostCartierdivisors. LetY ,...,Y beirreducible codimension onesubsetsofX. Then
1 r
thereexistsaneffective divisorD dthatcontains noneoftheY initssupport.
i
∈
4 R.HARTSHORNEANDC.POLINI
Proof. Theclass d corresponds to a reflexive coherent sheaf of -modules [15, 2.8], which is
X
L O
locally free at all points x X of codimension one because the divisors in d are almost Cartier.
∈
Since X is affine, the sheaf is generated by global sections. Thus for the generic point y of Y ,
i i
L
therewillbeasections Γ(X, )whoseimageinthestalk isnotcontainedinm . Those
i ∈ L Lyi yiLyi
sections s Γ(X, )nothaving thisproperty formapropersub-vector space V ofΓ(X, ). Now
i
∈ L L
ifwechooseasections Γ(X, )notcontainedinanyoftheV ,thecorresponding divisor D[15,
i
∈ L
2.9]willbeaneffectivedivisorintheclassd,notcontaining anyoftheY initssupport.
i
Remark 2.5. In Proposition 2.3 if S is not smooth the construction does not work because the
closureofDinS maynotbeCartier. Howeverifwedefine GtobethefollowingsubsetofAPicX,
namely
G = d APicX π∗(d) PicS
{ ∈ | ∈ }
then we can construct the map ϕ : G CartΓ/π∗CartL in the same way. The condition that
−→
the element π∗(d) of APicS lies in PicS is equivalent, by Proposition 2.2(c), to the vanishing of
itsimageinAPic(Spec )forallsingularpoints s S.
S,s
O ∈
Proposition2.6. Adoptassumptions2.1. Assumethatthemapinducedbyπfrom toπ ( )
L,X ∗ Γ,X
I I
isanisomorphism. Thenthemapofsheaves ofabelian groups γ : onX defined bythe
0
N −→ N
followingdiagram isanisomorphism:
∗ π ∗ 0
OX −→ ∗OS −→ N →
(1) α β γ
↓ ↓ ↓
∗ π ∗ 0
OL −→ ∗OΓ −→ N0 →
Proof. For every point x X, set (A,m ) to be the local ring and B to be the semi-local
A X,x
∈ O
ring S,π−1(x). As it is sufficient to check an isomorphism of sheaves on stalks, wecan restrict to
O
the local situation where X = SpecAandS = SpecB. LetA = A/I bethe local ring of L and
0
B = B/J bethe semi-local ring of Γ. Ourhypothesis says that the homomorphism from AtoB
0
inducesanisomorphism from I toJ. Nowweconsiderthediagramofabeliangroups:
A∗ B∗ N 0
−→ −→ →
a b c
↓ ↓ ↓
A∗ B∗ N 0
0 −→ 0 −→ 0 →
andweneedtoshowthattheinducedmapcisanisomorphism.
Since A A and B B are surjective maps of (semi)-local rings, the corresponding
0 0
−→ −→
mapsonunitsaandbaresurjective (seeLemma5.2). Thereforethethirdmap cissurjective.
To show c is injective, let a N go to 1 in N . Because the diagram is commutative a comes
0
∈
from an element b B∗ and b(b) = c B∗ whose image in N is 1. Hence c comes from an
∈ ∈ 0 0
element d A∗, which lifts to an e A∗. Let f be the image of e in B∗. Now b and f have the
∈ 0 ∈
sameimagecinB∗. RegardingthemaselementsoftheringB thismeansthattheirdifferenceisin
0
the ideal J. But J by hypothesis is isomorphic to I, hence there is anelement g I whose image
∈
DIVISORSCLASSGROUPSOFSINGULARSURFACES 5
givesb f inJ. Nowconsidertheelement h = g+e A. Sinceg I m ,theelementhisa
A
− ∈ ∈ ⊂
unitinA,i.e. itisanelement ofA∗. Furthermore itsimagein B∗ isb. Therefore theimageof bin
N,whichisa,isequalto1. Thusthemapcisanisomorphism. Sincethisholdsatallstalksx X,
∈
weconclude thatthemapγ : ofsheavesisanisomorphism.
0
N −→ N
Remark2.7. Inapplications wewilloften consider asituation where X isanintegral scheme and
S is its normalization. Then is a generalized divisor on X, whose inverse = a
S X
O I { ∈ O |
a is just the conductor of the integral extension. If wedefine L by this ideal, and Γ as
S X
O ⊂ O }
π∗(L),thenthemapinducedbyπ from toπ ( )isanisomorphism andthehypothesison
L,X ∗ Γ,X
I I
theidealsheavesissatisfied. Conversely,ifthemapfrom toπ ( )isanisomorphism,then
L,X ∗ Γ,X
I I
restrictingtoX L,wefind π ( )isanisomorphismthere,soS π−1(L) X L
X ∗ S
− O −→ O − −→ −
isanisomorphism.
Proposition 2.8. Ifπ : S X isafinitemorphism ofschemes,thenthenaturalmap
→
H1(X,π ( ∗)) H1(S, ∗)
∗ OS −→ OS
isanisomorphism.
Proof. First we will show that the first higher direct image sheaf R1π ( ∗) is zero. This sheaf
∗ OS
is the sheaf associated to the presheaf which to each open subset V in X associates the group
H1(π−1(V),OS∗|π−1V) [11, III, 8.1]. Hence the stalk of this sheaf at a point x ∈ X is the direct
limit
limH1(π−1(V),OS∗|π−1V).
x−∈→V
Anelementinthisdirectlimitisrepresentedbyapair(V,α)whereV isanopensetofX containing
xandα ∈ H1(π−1(V),OS∗|π−1V). ThisgroupisjustPic(π−1V),sotheelementαcorresponds to
aninvertible sheaf onπ−1V. WemayassumethatV isaffine,sinceaffineopensetsformabasis
L
forthetopology. Therefore, since π isfinite,theopensubsetπ−1V ofS isalsoaffine,andhence
L
isgeneratedbyglobalsections. Letz ,...,z π−1V bethefinitesetofpointsinπ−1(x). Wecan
1 r
∈
findasections H0(π−1V, )thatdoesnotvanishatanyofthez ,...,z . Sothezerosetofsis
1 r
∈ L
adivisorDwhosesupport doesnotcontainanyofthez . Sinceπ isfinite,itisapropermorphism,
i
so π(D) is closed in V and does not contain x. Let V′ = V π(D). Then π−1(V′) is free, and
− L|
since π−1(V′) π−1(V), theimage ofαinthe abovedirect limitiszero. Hence R1π ( ∗) = 0.
⊂ ∗ OS
Now the statement of the lemma follows from the exact sequence of terms of low degree of the
Lerayspectralsequence
0 H1(X,π ∗) H1(S, ∗) H0(X,R1π ( ∗)= 0
→ ∗OS −→ OS −→ ∗ OS
6 R.HARTSHORNEANDC.POLINI
Note 2.9. Since H1(S, ∗) computes PicS see [11, III, Ex. 4.5], this means that we can also
OS
computePicS asH1(X,π ∗).
∗OS
3. A LOCAL ISOMORPHISM FORAPic
In this section we prove a fundamental local isomorphism that allows us to compute the APic
groupofasurfacelocallyintermsofCartierdivisorsonthe curvesLandΓ. Wefirstobservethatif
Aisa local ring of dimension two satisfying G and S withspectrum X and punctured spectrum
1 2
X′ then APicX = PicX′. Indeed APicX = APicX′ (see [15, 1.12]), and X′ has no points of
codimension two,soAPicX′ = PicX′.
Theorem3.1. Adoptassumptions2.1. FurtherassumethatX isthespectrumofatwodimensional
localring,S issmooth,andthemapinducedbyπ from toπ ( )isanisomorphism. Then
L,X ∗ Γ,X
I I
themap
ϕ :APicX CartΓ/π∗CartL
−→
inProposition 2.3isanisomorphism.
Proof. Let x be the closed point of X. Set X′ = X x and S′ = S π−1(x) . As we
− { } − { }
noted above wecancalculate APicX asPicX′ whichisalso H1(X′, ∗ ). Weconsider sheaves
OX′
ofabelian groupsonX
(2) 0 ∗ π ( ∗) 0
→ OX −→ ∗ OS −→ N →
andsimilarlywithprimes
(3) 0→ OX∗′ −→ π∗(OS∗′)−→ N′ → 0
Computingcohomology onX along(2)weobtaintheexactsequence:
0 H0(X, ∗ ) H0(X,π ( ∗)) H0(X, ) PicX = 0
→ OX −→ ∗ OS −→ N −→
whereH1(X, ∗ ) = PicX = 0sinceX isalocalaffinescheme. Nowcomputingcohomologyon
OX
X′ along(3)weobtaintheexactsequence
0→ H0(X′,OX∗′)→ H0(X′,π∗(OS∗′)) → H0(X′,N′)→ APicX → H1(X′,π∗(OS∗′)) = 0
whereH1(X′, ∗ ) = APicX andH1(X′,π ( ∗ )) = 0because bytheProposition 2.8wehave
OX′ ∗ OS′
H1(X′,π ( ∗ )) = H1(S′, ∗ ). Now the latter is PicS′, which in turn is equal to APicS. But
∗ OS′ OS′
APicS = PicSbecauseSissmoothandfinallyPicS = 0becauseSisasemi-localaffinescheme.
SinceX andS bothsatisfy S anysection of or overX′ orS′ extends toallofX orS.
2 X S
O O
Thus H0(X, ∗ ) = H0(X′, ∗ ) and H0(X, ∗) = H0(X′, ∗ ). This allows us to combine
OX OX′ OS OS′
theabovetwosequences ofcohomology intoone:
(4) 0 H0(X, ) H0(X′, ′) APicX 0
→ N −→ N −→ −→
DIVISORSCLASSGROUPSOFSINGULARSURFACES 7
By Proposition 2.6 applied to both maps S X and S′ X′, we obtain H0(X, ) =
−→ −→ N
H0(X, ) and H0(X′, ′) = H0(X′, ′). Thus we turn (4) into the following short exact
N0 N N0
sequence
0 H0(X, ) H0(X′, ′) APicX 0
→ N0 −→ N0 −→ −→
Usingthisexactsequence wecanderivethefollowingdiagram:
0 0 0
↓ ↓ ↓
0 H0(X, ∗) H0(X,π ( ∗)) H0(X, ) 0
→ OL −→ ∗ OΓ −→ N0 →
↓ ↓ ↓
0 H0(X′, ∗ ) H0(X′,π ( ∗ )) H0(X′, ′) 0
→ OL′ −→ ∗ OΓ′ −→ N0 →
↓ ↓ ↓
0 CartL CartΓ APicX 0
→ −→ −→ →
↓ ↓ ↓
0 0 0
Thefirsttworowsinthediagram areobtained applying cohomology totheshortexactsequences
0 ∗ π ( ∗) 0
→ OL −→ ∗ OΓ −→ N0 →
and
0 → OL∗′ −→ π∗(OΓ∗′)−→ N0′ → 0
and observing that again H1(X, ∗) = PicL = 0 because L is a local affine scheme, and
OL
H1(X′, ∗ ) = PicL′ = 0 because L′ as a scheme is adisjoint union of generic points. The ver-
OL′
tical columns arise from thefact that L and Γare (semi)-local curves, sothat when weremove the
closed points weobtain the local rings of the generic points, namely the total quotient rings of
L
O
and ,andtheCartierdivisorsarenothing elsethanthequotients oftheunitsinthetotalquotient
Γ
O
rings divided by the units of the (semi)-local rings, i.e. CartL = K∗/ ∗ and CartΓ = K∗/ ∗.
L OL Γ OΓ
Now the Snake Lemma yields the diagram. The last row of the above diagram implies the desired
statement, namely
APicX = CartΓ/π∗CartL.
∼
Remark3.2. IfS isnotsmooth, wecanreplace APicX withthegroup GdefinedinRemark2.5,
inwhichcasetheproofofTheorem3.1showsthatϕ: G CartΓ/π∗CartLisanisomorphism,
−→
because the cokernel of H0( ) H0( ′)is just the kernel of APicX APicS, and in our
N0 −→ N0 →
localcase,fortheimageanelementofAPicX tovanishinAPicSisthesameassayingitislocally
free,henceitisinPicS,whichiszerobecause S isasemilocalring.
Proposition 3.3. IfAisalocalringofdimension onesatisfying S andAisitscompletion, then
1
CartA = CartA
b
b
8 R.HARTSHORNEANDC.POLINI
Proof. This follows for instance from the proof of [15, 2.14] where it is shown that a a gives
7→
a one-to-one correspondence between ideals of finite colength of A and A, under which principal
idealscorresponds toprincipal ideals. b
b
Thefollowingproposition showsthatthelocalcalculation ofAPic dependsonlyontheanalytic
isomorphism classofthesingularity whenthenormalization issmooth.
Proposition 3.4. Let A be a reduced two dimensional local ring satisfying G and S whose nor-
1 2
malization Aisregular. Then
APic(SpecA) = APic(SpecA).
e
Proof. We let S be the normalization of X = SpecA, take Lb to be the conductor and Γ to be
π−1(L). ThenbyTheorem3.1(cf. Remark2.7)wecancomputeAPicX = CartΓ/π∗CartL. On
theotherhand,wehaveshown(Proposition3.3)thattheCartierdivisorsofaone-dimensional local
ringarethesameasthoseofitscompletion. Soapplying Theorem 3.1alsotoSpecAweprovethe
assertion.
b
Thenextexampleshowsthatwecannotdroptheassumption onAbeingregular.
Example 3.5. The divisor class groups of normal local rings andetheir behavior under comple-
tion have been studied by Mumford, Samuel, Scheja, Brieskorn, and others. In particular, if
A = C[x,y,z]/(xr + ys + zt) localized at the maximal ideal (x,y,z), where r < s < t are
pairwise relatively prime, then A is a unique factorization domain, so APic(SpecA) is 0. How-
ever,thecompletion AofAisnotauniquefactorization domain,soAPic(SpecA)isnot0,except
intheuniquecase(r,s,t) = (2,3,5) [24].
e e
4. GLOBAL EXACT SEQUENCES
InthissectionwecomparePic andAPic ofanysurfaceX toitsnormalization. Thisgeneralizes
[15, 6.3] which dealt with the case of a ruled cubic surface. In particular our result applies to a
surface withordinary singularities whosenormalization issmooth, thusproviding ananswertothe
hopeexpressed in[15,6.3.1].
Theorem4.1. Adoptassumptions2.1. FurtherassumethatX isasurfaceeitheraffineorprojective
and the map induced by π from to π ( ) is an isomorphism. Then there is an exact
L,X ∗ Γ,X
I I
sequence:
(a) PicX PicS PicΓ/π∗PicL.
−→ −→
Furthermore, ifS issmooth,thenthereisalsoanexactsequence
(b) APicX PicS CartΓ/π∗CartL PicΓ/π∗PicL 0.
−→ ⊕ −→ →
DIVISORSCLASSGROUPSOFSINGULARSURFACES 9
Proof. For(a)weusethenaturalmapfromProposition2.2(a)ofPicX toPicS andtherestriction
map of PicS to PicΓ/π∗PicL. The composition is clearly zero, since a divisor class originating
onX willlandinπ∗PicL. Toshowexactness inthemiddle, werecalltheresultofProposition 2.6
whichshowsthatthesheaves , inthefollowingdiagram areisomorphic:
0
N N
0 ∗ π ∗ 0
→ OX −→ ∗OS −→ N →
α↓ β↓ γ↓∼=
0 ∗ π ∗ 0
→ OL −→ ∗OΓ −→ N0 →
Takingcohomology onX andusingProposition 2.8,weobtainadiagram ofexactsequences
0 H0( ∗ ) H0(π ∗) H0( ) PicX PicS H1( )
→ OX → ∗OS → N → → → N
∼= ∼=
↓ ↓ ↓ ↓ ↓ ↓
0 H0( ∗) H0(π ∗) H0( ) PicL PicΓ H1( )
→ OL → ∗OΓ → N0 → → → N0
From this sequence, we see that if an element of PicS becomes zero in PicΓ/π∗PicL, then it is
zeroinH1( ) = H1( )hencecomesfromanelementofPicX.
0
N N
For(b)wefirstdefinethemapsinvolved inthesequence. WeusethenaturalmapfromProposi-
tion2.2(b)ofAPicX toAPicS,whichisequaltoPicS sinceS issmooth, togetherwiththemap
ϕofProposition 2.3 applied locally. Notethat CartΓ/π∗CartLissimply thedirect sum of allits
contributionsateachoneofitspoints,sinceLandΓarecurves. Thesecondmapof(b)iscomposed
ofthemapPicS PicΓ/π∗PicLof(a)andthenaturalmapsofCartierdivisorsto Pic.
→
The composition of the two maps is zero, because if we start with something in APicX, then
according totheconstruction ofProposition 2.3,itstwoimagesinPicΓ/π∗PicLwillbethesame.
Thesecondmapofthesequence (b)isclearlysurjective.
Toshowexactnessinthemiddleof(b),supposethataclassd PicS andadivisorDinCartΓ
∈
have the same image in PicΓ/π∗PicL. First, we can modify D by an element of π∗CartL so
that d and D will have the same image in PicΓ. Next, by adding some effective divisors linearly
equivalent to mH, where H = 0in the affine case and H is a hyperplane section in the projective
case,wecanreducetothecasewheredandDareeffective. Considertheexactsequenceofsheaves
0 (d) (d) (d) 0
Γ S Γ
→ I −→ O −→ O →
IfXisaffine,thenH0(X, (d)) H0(X, (d))issurjective,sothesections H0(X, (d))
S Γ Γ
O → O ∈ O
definingthedivisorD willlifttoasection s′ H0(X, (d)). Thissectiondefinesacurve inS
S
∈ O C
in the divisor class d, not containing any component of Γ in its support, whose intersection with Γ
is D. We can transport restricted to S Γ to X L, and take its closure in X. This will be an
C − −
elementofAPicX givingrisetothedandD westartedwith.
IfX isprojective, weuseahyperplane section H. SinceH comesfrom PicX,itissufficientto
provetheresultford+mH andD+mH. Nowform 0thecokernelofthemapH0(X, (d+
S
≫ O
mH)) H0(X, (d+mH))landsinH1(X, (d+mH)),whichiszerobySerre’svanishing
Γ Γ
→ O I
theorem. Thentheproofproceeds asintheaffinecase.
Proposition 4.2. Withthehypotheses ofTheorem4.1thefollowingconditions areequivalent:
(i) themapPicX PicS isinjective
−→
10 R.HARTSHORNEANDC.POLINI
(ii) themapPicL PicΓisinjective and
→
coker(H0( ∗ ) π H0( ∗)) = coker(H0( ∗) π H0( ∗)).
OX → ∗ OS OL → ∗ OΓ
FurthermoreifS issmooth,conditions (i)and(ii)arealsoequivalent to
(iii) ThefirstmapofTheorem4.1(b)isinjective
Inaddition, withoutassumingS smooth,ifconditions(i)and(ii)holdandthemapPicL PicΓ
→
isanisomorphism thenPicX PicS isalsoanisomorphism.
→
Proof. Fromthediagramofexactsequences intheproofofTheorem4.1(a),statement(i)isequiv-
alenttotheexactnessofthesequence:
(5) 0 H0( ∗ ) H0(π ∗) H0( ) 0.
→ OX → ∗OS → N →
SinceH0(X, ) = H0( ),theexactness of(5)impliestheexactness of
N ∼ N0
0 H0( ∗) H0(π ∗) H0( ) 0.
→ OL → ∗OΓ → N0 →
Looking again at the diagram of exact sequences in the proof of Theorem 4.1(a), this implies (ii).
On the other hand, (ii) clearly implies (i). Now if S is smooth, because of the local isomorphism
ofTheorem3.1,anyelementinthekernelofthefirstmapofTheorem4.1(b)iszeroinallthelocal
groups APic(Spec ), hence by Jaffe’s sequence (see Proposition 2.2(c)) is already in PicX.
X,x
O
Thus(iii)isalsoequivalent to(i)and(ii).
The last statement follows again from the diagram of exact sequences in the proof of Theorem
4.1(a).
Remark 4.3. If X is integral and projective in Proposition 4.2 then so is S, thus H0(X, ∗ ) =
OX
H0(X,π ∗) = k∗, where k is the ground field. Therefore coker(H0( ∗ ) π H0( ∗)) = 0
∗OS OX → ∗ OS
andtheequalityofthecokernels in(ii)holdsifandonlyif
H0( ∗) = H0(π O∗).
OL ∗ Γ
Remark4.4. IfS isnotsmooth, then asinRemarks 2.5and 3.2wecanobtain thesameresults as
inTheorem4.1,withAPicX replacedbyG.
The next theorem shows, that at least over the complex number C, the map PicX PicS is
→
alwaysinjective.
Theorem4.5. LetX beanintegral surface inP3 overk = C. IfS isthenormalization ofX,then
thenaturalmap
PicX PicS
−→
isinjective.