Both in order to understand the structure of the Jacobian, and as an aid in its construction, we shall need to study the symmetric powers ofC.
For any varietyV, the symmetric groupSr onr letters acts on the product ofr copies VrofV by permuting the factors, and we want to define therth symmetric powerV.r/ of V to be the quotientSrnVr. The next proposition demonstrates the existence ofV.r/ and lists its main properties.
A morphism'WVr !T is said to besymmetricif' D' for all inSr.
PROPOSITION3.1. Let V be a variety over k. Then there exists a variety V.r/ and a symmetric morphismWVr !V.r/having the following properties:
(a) as a topological space,.V.r/; /is the quotient ofVrbySrI
(b) for any open affine subsetU ofV,U.r/is an open affine subset ofV.r/and .U.r/;OV.r//D .Ur;OVr/Sr
(set of elements fixed by the action ofSr/.
The pair .V.r/; / has the following universal property: every symmetric k-morphism 'WVr !T factors uniquely through.
The mapis finite, surjective, and separable.
PROOF. IfV is affine, sayV DSpecm.A/, defineV.r/ to beSpecm..A˝k:::˝kA/Sr/.
In the general case, writeV as a unionS
Ui of open affines, and constructV by patching together theUi.r/. See Mumford 1970, II,7, p66, and III,11, p112, for the details. 2 The pair.V.r/; /is uniquely determined up to a unique isomorphism by its universal property. It is called therth symmetric powerofV.
PROPOSITION3.2. The symmetric powerC.r/of a nonsingular curve is nonsingular.
PROOF. We may assume that k is algebraically closed. The most likely candidate for a singular point onC.r/ is the imageQof a fixed point.P; :::; P /ofSronCr, whereP is a closed point of C. The completionOOP of the local ring atP is isomorphic tokŒŒX , and so
O
O.P;:::;P / kŒŒX ˝O:::˝OkŒŒX kŒŒX1; :::; Xr.
It follows thatOOQ kŒŒX1; :::; XrSr whereSr acts by permuting the variables. The fun-damental theorem on symmetric functions says that, over any ring, a symmetric polynomial can be expressed as a polynomial in the elementary symmetric functions 1; :::; r. This implies that
kŒŒX1; :::; XrSr DkŒŒ1; :::; r, which is regular, and soQis nonsingular.
For a general pointQ D .P; P; :::; P0; :::/with P occurringr0 times,P0 occurring r00times, and so on,
O
OQ kŒŒX1; :::; Xr0Sr0˝OkŒŒX1; :::; Xr00Sr00˝O ...,
which the same argument shows to be regular. 2
3. THE SYMMETRIC POWERS OF A CURVE 95 REMARK3.3. The reader may find it surprising that the fixed points of the action ofSron Crdo not force singularities onC.r/. The following remarks may help clarify the situation.
LetG be a finite group acting effectively on a nonsingular varietyV, and supppose that the quotient varietyW DGnV exists. ThenV !W is ramified exactly at the fixed points of the action. A purity theorem Grothendieck 1971, X, 3.1, saysW can be nonsingular only if the ramification locus is empty or has pure codimension1inV. As the ramification locus ofVr overV.r/ has pure codimensiondim.V /, this implies thatV.r/ can be nonsingular only ifV is a curve.
Let K be field containing k. If K is algebraically closed, then (3.1a) shows that C.r/.K/ D SrnC.K/r, and so a point of C.r/ with coordinates in K is an unordered r-tuple ofK-rational points. This is the same thing as an effective divisor of degreer on CK. WhenK is perfect, the divisors onCK can be identified with those onCK fixed un-der the action ofGal.Kal=K/. Since the same is true of the points onC.r/, we see again that C.r/.K/ can be identified with the set of effective divisors of degreer on C. In the remainder of this section we shall show thatC.r/.T /has a similar interpretation for any k-scheme. (Since this is mainly needed for the construction ofJ, the reader more interested in the properties ofJ can pass to the section 5.)
LetX be an algebraic space overk. Recall Hartshorne 1977, II 6 p145, that a Cartier divisorDis effective if it can be represented by a family.Ui; gi/iwith thegiin .Ui;OX/.
Let I.D/be the subsheaf of OX such that I.D/jUi is generated by gi. Then I.D/ D L. D/, and there is an exact sequence
0!I.D/!OX !OD !0
whereOD is the structure sheaf of the closed algebraic subspace ofT associated withD.
The closed subspaces arising from effective Cartier divisors are precisely those whose sheaf of ideals can be locally generated by a single element that is not a zero-divisor. We shall often identifyDwith its associated closed subscheme.
For example, letT DA1 D Specm kŒY , and letD be the Cartier divisor associated with the Weil divisornP, whereP is the origin. ThenDis represented by.Yn;A1/, and the associated algebraic subspace isSpecm.kŒY =.Yn//.
DEFINITION 3.4. LetWX !T be a morphism ofk-schemes. Arelative effective Cartier divisoronX=T is a Cartier divisor onX that is flat overT when regarded as an subspace ofX.
Loosely speaking, the flatness condition means that the divisor has no vertical compo-nents, that is, no components contained in a fibre. WhenT is affine, sayT DSpecm.R/, an algebraic subspaceDofX is a relative effective Cartier divisor if and only if there exists an open affine coveringX DS
Ui andgi 2 .Ui;OX/DRi such that (a) D\Ui DSpecm.Ri=giRi/,
(b) gi is not a zero-divisor, and (c) Ri=giRi is flat overR, for alli.
Henceforth all divisors will be Cartier divisors.
LEMMA3.5. IfD1andD2are relative effective divisors onX=T, then so also is their sum D1CD2.
PROOF. It suffices to prove this in the case thatT is affine, sayT DSpecm.R/. We have to check that if conditions (b) and (c) above hold for gi andg0i, then they also hold for gig0i. Condition (b) is obvious, and the flatness ofRi=gig0iRi overR follows from the exact sequence
0!Ri=giRi gi0
!Ri=gigi0Ri !Ri=g0iRi !0;
which exhibits it as an extension of flat modules. 2
REMARK3.6. Let Dbe a relative effective divisor onX=T. On tensoring the inclusion I.D/ ,! OX with L.D/we obtain an inclusion OX ,! L.D/and hence a canonical global sectionsD ofL.D/. For example, in the case thatT is affine andD is represented as in the above example,L.D/jUi isgi 1Ri andsDjUi is the identity element inRi.
The mapD 7! .L.D/; sD/defines a one-to-one correspondence between relative ef-fective divisors onX=T and isomorphism classes of pairs.L; s/whereLis an invertible sheaf onX ands2 .X;L/is such that
0!OX !s L!L=sOX !0 is exact andL=sOX is flat overT.
Observe that, in the case thatX is flat overT,L=sOX is flat overT if and only if, for allt inT,sdoes not become a zero divisor inL˝OXt. (Use that anR-moduleM is flat ifTorR1.M; N /D0for all finitely generated modulesN, and that any such moduleN has a composition series whose quotients are the quotient ofRby a prime ideal; therefore the criterion has only to be checked withN equal to such a module.)
PROPOSITION3.7. Consider the Cartesian square
X X0
?
? y
?
? y
X T0
IfDis a relative effective divisor onX=T, then its pull-back to a closed subspaceD0ofX0 is a relative effective divisor onX0=T0.
PROOF. We may assume bothT andT0are affine, sayT DSpecmRandT0DSpecmR0, and then have to check that the conditions (a), (b), and (c) above are stable under the base change R ! R0. Write Ui0 D U T T0; clearly D0\Ui0 D Specm.R0i=giR0i/. The conditions (b) and (c) state that
0!Ri gi
!Ri !Ri=giRi !0
is exact and thatRi=giRi is flat overR. Both assertions continue to hold after the sequence
has been tensored withR0. 2
PROPOSITION3.8. LetDbe a closed subscheme ofX, and assume thatDandX are both flat overT. IfDt df
DDT ftgis an effective divisor onXt=t for all pointst ofT, thenD is a relative effective divisor onX.
3. THE SYMMETRIC POWERS OF A CURVE 97 PROOF. From the exact sequence
0!I.D/!OX !OD !0
and the flatness ofX andDoverT, we see thatI.D/is flat overT. The flatness ofOD
implies that, for anyt 2T, the sequence
0!I.D/˝OT k.t /!OXt !ODt !0
is exact. In particular, I.D/˝k.t / ! I.Dt/. AsDt is a Cartier divisor,I.Dt/(and therefore alsoI.D/˝k.t //is an invertibleOXt-module. We now apply the fibre-by-fibre criterion of flatness: if X is flat overT andF is a coherentOX-module that is flat over T and such that Ft is a flat OXt-module for allt inT, thenF is flat over X (Bourbaki 1989, III, 5.4). This implies that I.D/is a flatOX-module, and since it is also coherent, it is locally free overOX. Now the isomorphismI.D/˝k.t / !I.Dt/shows that it is of rank one. It is therefore locally generated by a single element, and the element is not a zero-divisor; this shows thatDis a relative effective divisor. 2 LetWC ! T be a proper smooth morphism with fibres of dimension one. IfDis a relative effective divisor onC=T, thenDtis an effective divisor onCt, and ifT is connected, then the degree ofDt is constant; it is called the degree ofD. Note thatdeg.D/Drif and only ifODis a locally freeOT -module of degreer.
COROLLARY3.9. A closed subspaceDofCis a relative effective divisor onC=T if and only if it is finite and flat overT; in particular, ifsWT !C is a section to, thens.T /is a relative effective divisor of degree1onC =T.
PROOF. A closed subspace of a curve over a field is an effective divisor if and only if it is finite. Therefore (3.8) shows that a closed subspaceDofCis a relative effective divisor on C=T if and only if it is flat overT and has finite fibres, but such a subspaceDis proper over T and therefore has finite fibres if and only if it is finite overT (see Milne 1980, I 1.10, or
Hartshorne 1977, III, Ex. 11.3). 2
WhenDandD0are relative effective divisors onC=T, we writeDD0ifDD0as subspaces ofC(that is,I.D/I.D0//.
PROPOSITION3.10. IfDt Dt0(as divisors onCt/for allt inT, thenD D0.
PROOF. RepresentDas a pair.s;L/(see 3.6). ThenD D0if and only ifsbecomes zero inL˝OD0 DLjD0. ButL˝OD0 is a locally freeOT-module of finite rank, and so the support ofsis closed subspace ofT. The hypothesis implies that this subspace is the whole
ofT. 2
LetDbe a relative effective divisor of degreer onC=T. We shall say thatDissplitif Supp.D/DS
si.T /for some sectionssi to. For example, a divisorD DP
niPi on a curve over a field is split if and only ifk.Pi/Dkfor alli.
PROPOSITION3.11. Every split relative effective divisorDonC=T can be written uniquely in the formDD P
nisi.T /for some sectionssi .
PROOF. Let Supp.D/DS
isi.T /, and suppose thatDjsi.T /has degreeni. ThenDt D .P
nisi.T //t for allt, and so (3.10) shows thatDD P
nisi.T /. 2
EXAMPLE3.12. Consider a complete nonsingular curveC over a fieldk. For eachi there is a canonical sectionsi toqWC Cr !Cr, namely,.P1; :::; Pr/7!.Pi; P1; :::; Pr/. Let Di to besi.Cr/regarded as a relative effective divisor onCCr=Cr, and letDDP
Di. ThenDis the unique relative effective divisorC Cr=T whose fibre over.P1; :::; Pr/is PPi. ClearlyDis stable under the action of the symmetric groupSr, andDcanDSrnD (quotient as a subscheme ofCCr) is a relative effective divisor onCC.r/=C.r/whose fibre overD2C.r/.k/isD.
ForC a complete smooth curve overkandT ak-scheme, defineDivrC.T /to be the set of relative effective Cartier divisors onC T =T of degreer. Proposition 3.7 shows that DivrC is a functor on the category ofk-schemes.
THEOREM3.13. For any relative effective divisorDon.C T / =T of degreer, there is a unique morphism'WT !C.r/such thatDD.1'/ 1.Dcan/. ThereforeC.r/represents DivrC.
PROOF. Assume first that Dis split, so thatD D P
nisi.T /for some sectionssiWT ! C T. In this case, we defineT !Cr to be the map.pıs1; :::; pıs1; pıs2; :::/, where eachsi occursni times, and we take' to be the compositeT !Cr !C.r/. In general, we can choose a finite flat covering WT0 ! T such that the inverse imageD0ofD on C T0is split, and let'0WT0!C.r/be the map defined byD0. Then the two maps'0ıp and'0ıqfromT0T T0toT0are equal because they both correspond to the same relative effective divisor
p 1.D0/D. ıp/ 1.D/D. ıq/ 1.D/Dq 1.D/
onT0T T0. Now descent theory (Milne 1980, I, 2.17) shows that'0factors throughT.2 EXERCISE3.14. Let E be an effective Cartier divisor of degree r on C, and define a subfunctorDivEC ofDivrC by
DivEC.T /D fD2DivrC.T /jDt Eall t 2Tg:
Show that DivEC is representable byP.V / whereV is the vector space .C;L.E//(use Hartshorne 1977, II 7.12, and that the inclusionDivEC ,!DivrC defines a closed immersion P.V / ,!C.r/).
REMARK3.15. Theorem 3.13 says thatC.r/ is the Hilbert scheme HilbPC =k whereP is the constant polynomialr.