Throughout this sectionC will be a complete nonsingular curve of genusg > 0over a field k, andJ will be its Jacobian variety.
PROPOSITION6.1. Let P be a k-rational point on C. The map fPWC ! J has the following universal property: for any map'WC !AfromC into an abelian variety sending P to0, there is a unique homomorphism WJ !Asuch that'D ıfP.
PROOF. Consider the map
.P1; :::; Pg/7! X
i
.Pi/WCg !A:
Clearly this is symmetric, and so it factors throughC.g/. It therefore defines a rational map WJ ! A, which (I 3.2) shows to be a regular map. It is clear from the construction that ıfP D ' (note that fP is the composite ofQ 7! Q C.g 1/PWC ! C.g/ with f.g/WC.g/ ! J /. In particular, maps 0to 0, and (I 1.2) shows that it is therefore a homomorphism. If 0is a second homomorphism such that 0ıfP D', then and 0 agree onfP.C /C CfP.C /(gcopies), which is the whole ofJ. 2 COROLLARY6.2. LetN be a divisorial correspondence between.C; P /andJ such that .1fP/N LP; thenN MP (notations as in2 and (1.7)).
PROOF. Because of (I 5.13), we can assume k to be algebraically closed. According to (1.7) there is a unique map'WJ !J such thatN .1'/MP. On points'is the map sendinga2J.k/to the uniquebsuch thatMPjC fbg NjC fag. By assumption,
NjC ffPQg LPjC fQg MPjC ffPQg,
and so.'ıfP/.Q/DfP.Q/for allQ. Now (6.1) shows thatf is the identity map. 2 COROLLARY6.3. LetC1andC2be curves overkwithk-rational pointsP1andP2, and let J1andJ2be their Jacobians. There is a one-to-one correspondence betweenHomk.J1; J2/ and the set of isomorphism classes of divisorial correspondences between .C1; P1/ and .C2; P2/.
PROOF. A divisorial correspondence between.C2; P2/and.C1; P1/ gives rise to a mor-phism.C1; P1/ !J2(by 1.7), and this morphism gives rise to homomorphismJ1 ! J2
(by 6.1). Conversely, a homomorphism WJ1 ! J2 defines a divisorial correspondence .1.fP1ı //MP2between.C2; P2/and.C1; P1/. 2 In the case thatC has a pointP rational overk, defineFWC C !J to be the map .P1; P2/ 7!fP.P1/ fP.P2/. One checks immediately that this is independent of the choice ofP. Thus, ifP 2C.k0/for some Galois extensionk0ofk, andFWCk0Ck0 !Jk0
is the corresponding map, then F D F for all 2 Gal.k0=k/ ; thereforeF is defined overk whether or notC has ak-rational point. Note that it is zero on the diagonalof C C.
6. THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY 105 PROPOSITION6.4. LetAbe an abelian variety overk. For any map'WC C !Asuch that'./D0, there is a unique homomorphism WJ !Asuch that ıF D'.
PROOF. Let k0 be a finite Galois extension of k, and suppose that there exists a unique homomorphism WCk0 ! Jk0 such that ıFk0 D 'k0. Then the uniqueness implies that D for all in Gal.k0=k/, and so is defined over k. It suffices therefore to prove the proposition after extendingk, and so we can assume thatC has ak-rational pointP. Now (I 1.5) shows that there exist unique maps'1and'2fromC toAsuch that '1.P /D0D'2.P /and'.a; b/D'1.a/C'2.b/for all.a; b/2CC. Because'is zero on the diagonal,'1 D '2. From (6.1) we know that there exists a unique homomorphism fromJ toAsuch that'1D ıf, and clearly is also the unique homomorphism such
that' D ıF. 2
REMARK6.5. The proposition says that.A; F /is the Albanese variety ofC in the sense of Lang 1959, II 3, p45. Clearly the pairs .J; fP/ and.J; F / are characterized by the universal properties in (6.1) and (6.4).
Assume again thatC has ak-rational pointP, and let D Wg 1. It is a divisor on J, and ifP is replaced by a secondk-rational point,is replaced by a translate. For any effective divisorDonJ, write
L0.D/DmL.D/˝pL.D/ 1˝qL.D/ 1DL.m 1.D/ DJ_J D/.
Recall (I 8.1 et seqq.), that D is ample if and only if'L.D/WJ ! J_ is an isogeny, and then .1'L.D//.P/ D L0.D/, whereP is the Poincar´e sheaf on J J_. Write for the image of under the map. 1/JWJ ! J, anda forta D Ca,a 2 J.k/.
Abbreviate. /abya.
THEOREM6.6. The map'L./WJ ! J_ is an isomorphism; therefore, 1'L./ is an isomorphism.J J;L0.//!.J J_;P/.
PROOF. As usual, we can assumekto be algebraically closed. Recall (Milne 1986, 12.13) that'L. /D. 1/2'L./ D'L./, and that'L.a/D'L./for alla2J.k/. 2 LEMMA6.7. LetU be the largest open subset ofJ such that
(i) the fibre off.g/WC.g/!J at any point ofU has dimension zero, and
(ii) ifa2U.k/andD.a/is the unique element ofC.r/.k/mapping to a, thenD.a/is a sum ofgdistinct points ofC.k/.
Thenf 1.a/DD.a/(as a Cartier divisor) for alla2U.k/, wheref DfPWC !J. PROOF. Note first that U can be obtained by removing the subset over which the fibres have dimension> 0, which is closed (AG 10.9), together with the images of certain closed subsets of the formCg 2. These last sets are also closed becauseCg !J is proper (AG Chapter 7), and it follows thatU is a dense open subset ofJ.
Let a 2 U.k/, and let D.a/ D P
iPi, Pi ¤ Pj for i ¤ j. A point Q1 of C maps to a point of a if and only if there exists a divisor Pg
iD2Qi on C such that fP.Q/ D P
ifP.Qi/Ca. The equality impliesPg
iD1Qi D, and the fact that
jDjhas dimension0implies that P
iQi DD. It follows that the support off 1.a/is fP1; :::; Pgg, and it remains to show thatf 1.a/has degreegfor alla.
Consider the map WC ! J sending.Q; b/tof .Q/Cb. As the composite of with1fg 1WC Cg 1 ! C isfgWCg ! J, and these maps have degrees .g 1/ŠandgŠrespectively (5.5), has degreeg. Also is projective becauseC is a projective variety (see Hartshorne 1977, II, Ex. 4.9). Considera2U; the fibre of overa isf 1.a/(more accurately, it is the algebraic subspace ofC associated with the Cartier divisora/. Therefore the restriction of to 1.U /is quasi-finite and projective, and so is finite (AG 8.19). AsU is normal, this means that all the fibres of over points ofU are finite schemes of rankg(AG 10.12). This completes the proof of the lemma. 2 LEMMA6.8. (a) Leta2J.k/, and letf.g/.D/Da; thenfL.a/L.D/.
(b) The sheaves.f . 1/J/L0. /andMP onC J are isomorphic.
PROOF. Note that (6.7) shows that the isomorphism in (a) holds for allain a dense open subset ofJ. Note also that the map
C Q7!.Q;a/! C fag f. 1/! J J m! J equalst aıf, and so
.f . 1//mL. /jC fag 'L.t a1 /jf .C /'L.a/jf .C /fL.a/:
Similarly
.f . 1//pL. /jC fag 'fL. /, and
.f . 1//qL. /jC fag
is trivial. On the other hand,MP is an invertible sheaf onC J such that
(i)MPjC fag L.D gP /ifDis an effective divisor of degreegonC such that f.g/.D/DaI
(ii)MPjfPg J is trivial.
Therefore (a) is equivalent to.f. 1//mL. /jCfagbeing isomorphic toMP˝ pL.gP /jC fagfor alla. As we know this is true for allain a dense subset ofJ, (I 5.19) applied to
MP ˝pL.gP /˝.f . 1//mL. / 1
proves (a). In particular, on taking a D 0, we find that fL. / L.gP /, and so .f . 1//pL. /pL.gP /. Now (I 5.16) shows that
.f . 1//.mL. /˝pL. / 1/MP ˝qN
for some invertible sheafN ofJ. On computing the restrictions of the sheaves tofPg J,
we find thatN . 1/L. /, which completes the proof. 2
Consider the invertible sheaf.f 1/P onCJ_. Clearly it is a divisorial correspon-dence, and so there is a unique homomorphismf_WJ_ !J such that.1f_/MP .f 1/P. The next lemma completes the proof of the theorem.
LEMMA6.9. The maps f_WJ_ !J and'L./WJ !J_are inverse.
6. THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY 107 PROOF. Write D 'L./D 'L. /. We have
.1 /.1f_/MP .1 /.f 1/P .f /P
.f . 1//.1'L.//P .f . 1//L0. / MP:
Therefore,f_ı is a map˛WJ ! J such that.1˛/MP MP; but the only map
with this property is the identity. 2
REMARK6.10. (a) Lemma 6.7 shows that f .C / and cross transversely at any point of U. This can be proved more directly by using the descriptions of the tangent spaces implicitly given near the end of the proof of (5.1).
(b) In (6.8) we showed thatMP .f . 1//L0. /. This implies MP .f . 1//.1'L. //P
.f . 1//.1'L.//P .f . 1//L0./:
Also, becauseD 7!'L.D/is a homomorphism,'L. /D 'L./, and so MP .f . 1//.1'L.//P
.f 1/.1'L. //P .f 1/L0. /:
(c) The map on points J_.k/ ! J.k/ defined byf_ is induced by fWPic.J / ! Pic.C /.
(d) Lemma 6.7 can be generalized as follows. An effective canonical divisorKdefines a point on C.2g 2/ whose image inJ will be denoted. Letabe a point ofJ such that a is not in.Wg 2/ , and writeaD P
if .Pi/withP1; :::; Pgpoints onC. ThenWr and.Wg r/a intersect properly, andWr.Wg r/a DP
.wi1:::ir/where wi1:::ir Df .Pi1/C Cf .Pir/
and the sum runs over the.gr/combinations obtained by takingrelements fromf1; 2; :::; gg. See Weil 1948b,39, Proposition 17.
SUMMARY6.11. Between.C; P / and itself, there is a divisorial correspondence LP D L. fPg C C fPg/:
Between.C; P / andJ there is the divisorial correspondenceMP; for any divisorial correspondenceL between.C; P /and a pointedk-scheme.T; t /, there is a unique mor-phism of pointedk-schemes'WT !J such that.1'/MP LP. In particular, there is a unique mapfPWC !J such that.1fP/MP LP andf .P /D0.
Between J and J_ there is a canonical divisorial correspondence P (the Poincar´e sheaf); for any divisorial correspondenceLbetweenJ and a pointedk-scheme.T; t /there is a unique morphism of pointedk-schemes WT !J such that.1 /P L.
BetweenJ andJ there is the divisorial correspondenceL0./. The unique morphism J ! J_such that.1 /P L0./is'L./, which is an isomorphism. Thus'L./
is a principal polarization of J, called the canonical polarization. There are the following formulas:
MP .f . 1//L0./.f 1/L0./ 1: Consequently,
LP .f f /L0./ 1:
If f_WJ_ ! J is the morphism such that.f 1/P .1f_/MP, then f_ D 'L./1 .
EXERCISE6.12. It follows from (6.6) and the Riemann-Roch theorem (I 11.1) that.g/D gŠ. Prove this directly by studying the inverse image of (and its translates) by the map Cg !J. (Cf. AG 12.10 but note that the map is not finite.) Hence deduce another proof of (6.6).