1 Overview and definitions

Overview

LetC be a nonsingular projective curve over a fieldk. We would like to define an abelian varietyJ, called the Jacobian variety ofC, such thatJ.k/DPic⁰.C /(functorially). Un-fortunately, this is not always possible: clearly, we would want thatJ.k^sep/DPic⁰.C_k^sep/;

but then

J.k^sep/ DJ.k/DPic⁰.Ck^sep/ , DGal.k^sep=k/;

and it is not always true thatPic⁰.Ck^sep/ DPic⁰.C /. However, this is true whenC.k/¤

;.

ASIDE1.1. LetCbe a category. An objectX ofCdefines a contravariant functor hXW^C!^Set,T 7!Hom.T; X /:

MoreoverX 7!hX defines a functorC!Fun.C;Set/(category of contravariant functors from Cto sets). We can think of hX.T /as being the set of “T-points” ofX. It is very easy to show that the functorX 7! hX is fully faithful, i.e.,Hom.X; Y /DHom.hX; hY/

— this is the Yoneda Lemma (AG 1.39). ThusC can be regarded as a full subcategory of Fun.C;Set/:Xis known (up to a unique isomorphism) once we know the functor it defines, and every morphism of functors hX ! hY arises from a unique morphismX ! Y. A contravariant functorFW^C ! ^Setis said to berepresentableif it is isomorphic tohX for some objectX ofC, andX is then said torepresentF.

Definition of the Jacobian variety.

For varietiesV andT overk, setV .T /DHom.T; V /Dh_V.T /. For a nonsingular variety T,

P_C⁰.T /DPic⁰.C T /=qPic⁰.T /

85

(families of invertible sheaves of degree zero on C parametrized by T, modulo trivial families–cf. (4.16)). This is a contravariant functor from the category of varieties over kto the category of abelian groups.

THEOREM1.2. AssumeC.k/ ¤ ;. The functorP_C⁰ is represented by an abelian variety J.

From (1.1), we know thatJ is uniquely determined. It is called theJacobian varietyof C.

Apointed variety overk is a pair.T; t / withT a variety over k andt 2 T .k/. We always regard an abelian variety as a pointed variety by taking the distinguished point to be 0. Adivisorial correspondence between two pointed varieties .S; s/and .T; t /is an invertible sheafLonST whose restrictions toS ftgandfsg T are both trivial.

PROPOSITION1.3. LetP 2 C.k/, and letJ DJac.C /. There is a divisorial correspon-denceMonCJ that is universal in the following sense: for any divisorial correspondence LonC T (some pointed varietyT /such thatLt is of degree0for allt, there is a regular map'WT !J sending the distinguished point ofT to0and such that.1'/ML.

REMARK1.4. (a) The Jacobian variety is defined even whenC.k/ D ;; however, it then doesn’t (quite) represent the functorP (because the functor is not representable). See below.

(b) The Jacobian variety commutes with extension of scalars, i.e.,Jac.C_k0/D.Jac.C //_k0

for any fieldk⁰k.

(c) Let Mbe the sheaf in (1.3); as x runs through the elements of J.k/, Mx runs through a set of representatives for the isomorphism classes of invertible sheaves of degree 0onC.

(d) Fix a pointP0inJ.k/. There is a regular map'P₀WC ! J such that, on points, '_P0sendsP toŒP P0; in particular,'_P0sendsP0to0. The map'_Q0differs from'_P0 by translation byŒP0 Q0(regarded as a point onJ /.

(e) The dimension ofJ is the genus ofC. IfC has genus zero, thenJac.C /D0(this is obvious, becausePic⁰.C /D0, even when one goes to the algebraic closure). IfC has genus1, thenJac.C /DC (providedC has a rational point; otherwise it differs fromC — becauseJac.C /always has a point).

Construction of the Jacobian variety.

Fix a nonsingular projective curve over k. For simplicity, assumek D k^al. We want to construct a variety such thatJ.k/is the group of divisor classes of degree zero onC. As a first step, we construct a variety whose points are the effective divisors of degree r, some r > 0. LetC^rDCC:::C (rcopies). A point onC^r is an orderedr-tuple of points onC. The symmetric group onr letters,Sr, acts onC^r by permuting the factors, and the points on the quotient variety C^.r/ D^df C^r=Sr are the unorderedr-tuples of points on C. But an unorderedr-tuple is just an effective divisor of degreer,P

i Pi. Thus C^.r/ DDiv^r.C /D f^df effective divisors of degree r on Cg: Writefor the quotient mapC^r !C^.r/,.P1; :::; Pr/7!P

Pi. LEMMA1.5. The varietyC^.r/is nonsingular.

1. OVERVIEW AND DEFINITIONS 87 PROOF. In general, when a finite group acts freely on a nonsingular variety, the quotient will be nonsingular. In our case, there are points onC^r whose stabilizer subgroup is non-trivial, namely the points.P1; :::; Pr/in which two (or more)Pi coincide, and we have to show that they don’t give singularities on the quotient variety. The worst case is a point Q D.P; :::; P /, and here one can show that

Ob_Q 'kŒŒ1; :::; r;

the power series ring in the elementary symmetric functions1; :::; r in theXi, and this is

a regular ring. See (3.2). ₂

LetPic^r.C /be the set of divisor classes of degree r. For a fixed pointP0 onC, the map

ŒD7!ŒDCrP0WPic⁰.C /!Pic^r.C /

is a bijection (both Pic⁰.C /andPic^r.C /are fibres of the mapdegWPic.C / ! Z). This remains true when we regard Pic⁰.C / and Pic^r.C / as functors of varieties over k (see above), and so it suffices to find a variety representing thePic^r.C /.

For a divisor of degreer, the Riemann-Roch theorem says that

`.D/DrC1 gC`.K D/

whereKis the canonical divisor. Sincedeg.K/D2g 2,deg.K D/ < 0and`.K D/D 0whendeg.D/ > 2g 2. Thus,

`.D/DrC1 g > 0, if r Ddeg.D/ > 2g 2:

In particular, every divisor class of degreer contains an effective divisor, and so the map 'W feffective divisors of degree rg !Pic^r.C /,D7!ŒD

is surjective whenr > 2g 2. We can regard this as a morphism of functors 'WC^.r/ Pic^r.C /:

Suppose that we could find a sectionsto', i.e., a morphism of functorssWPic^r.C /! C^.r/ such that'ıs Did. Thensı' is a morphism of functorsC^.r/ ! C^.r/ and hence by (??) a regular map, and we can form the fibre product:

C^.r/ J⁰

.1;sı'/

?

? y

?

? y C^.r/C^.r/ C^.r/: Then the map from

J⁰.k/D f^df .a; b/2C^.r/C^.r/ jaDb; b Dsı'.a/g

toPic^r.C /sendingb to'.b/is an isomorphism. Thus we will have constructed the Jaco-bian variety; in factJ⁰will be a closed subvariety ofC^.r/. Unfortunately, it is not possible to find such a section: the Riemann-Roch theorem tells us that, forr > 2g 2, each divi-sor class of degreeris represented by an.r g/-dimensional family of effective divisors, and there is no nice functorial way of choosing a representative. However, it is possible to do this “ locally”, and so constructJ⁰as a union of varieties, each of which is a closed subvariety of an open subvariety ofC^.r/. For the details, see4 below.

Definitions and main statements

Recall that for an algebraic spaceS,Pic.S /denotes the groupH¹.S;O_S/of isomorphism classes of invertible sheaves onS, and thatS 7!Pic.S /is a functor from the category of algebraic spaces overkto that of abelian groups.

LetC be a complete nonsingular curve overk. The degree of a divisorDDniPi onC isniŒk.Pi/Wk. Since every invertible sheafLonC is of the formL.D/for some divisorD, andDis uniquely determined up to linear equivalence, we can definedeg.L/ Ddeg.D/.

Then

deg.Lⁿ/Ddeg.nD/Dndeg.D/;

and the Riemann-Roch theorem says that

.C;Lⁿ/Dndeg.L/C1 g:

This gives a more canonical description ofdeg.L/Wwhen.C;Lⁿ/is written as a poly-nomial inn,deg.L/is the leading coefficient. We writePic⁰.C /for the group of isomor-phism classes of invertible sheaves of degree zero onC.

LetT be a connected algebraic space overk, and letLbe an invertible sheaf onCT. Then (I 4.2) shows that.Ct;Lⁿ_t/, and thereforedeg.L_t/, is independent oft; moreover, the constant degree ofLt is invariant under base change relative to mapsT⁰ ! T. Note that for a sheafMonC T,.qM/_t is isomorphic toO_Ct and, in particular, has degree 0. Let

P_C⁰.T /D fL2Pic.C T /jdeg.Lt/D0all tg=qPic.T /:

ThusP_C⁰.T /is the group of families of invertible sheaves onC of degree0parametrized by T, modulo the trivial families. Note thatP_C⁰ is a functor from algebraic spaces overk to abelian groups. It is this functor that the Jacobian attempts to represent.

THEOREM1.6. There exists an abelian variety J over k and a morphism of functors WP_C⁰ !J such thatWP_C⁰.T /!J.T /is an isomorphism wheneverC.T /is nonempty.

BecauseC is an algebraic variety, there exists a finiteGaloisextensionk⁰ofksuch that C.k⁰/is nonempty. LetG be the Galois group ofk⁰overk. Then for every algebraic space T overk,C.T_k0/is nonempty, and so.T_k0/WP_C⁰.T_k0/!J.T_k0/is an isomorphism. As

J.T /D^df Mor_k.T; J /'Mor_k0.T_k0; J_k0/^G DJ.T_k0/^G,

we see that J represents the functorT 7! P_C⁰.Tk⁰/^G, and this implies that the pair.J; / is uniquely determined up to a unique isomorphism by the condition in the theorem. The varietyJ is called theJacobian varietyofC. Note that for any fieldk⁰ kin whichC has a rational point,defines an isomorphismPic⁰.C /!J.k⁰/.

WhenC has ak-rational point, the definition takes on a more attractive form. Apointed k-spaceis a connected algebraick-space together with an elements2S.k/. Abelian vari-eties will always be regarded as being pointed by the zero element. Adivisorial correspon-dencebetween two pointed spaces.S; s/and.T; t /overkis an invertible sheafLonST such thatLjS ftgandLjfsg T are both trivial.

THEOREM1.7. Let P be a k-rational point onC. Then there is a divisorial correspon-denceM^P between.C; P /andJ such that, for every divisorial correspondenceLbetween .C; P /and a pointedk-scheme.T; t /, there exists a unique morphism'WT !J such that '.t /D0and.1'/M^P L.

1. OVERVIEW AND DEFINITIONS 89 Clearly the pair.J;M^P/is uniquely determined up to a unique isomorphism by the condition in (1.7). Note that each element ofPic⁰.C /is represented by exactly one sheaf M_a,a2J.k/, and the map'WT !J sendst 2T .k/to the uniqueasuch thatM_a L_t.

Theorem 1.6 will be proved in4. Here we merely show that it implies (1.7).

LEMMA1.8. Theorem 1.6 implies Theorem 1.7.

PROOF. Assume there is ak-rational pointP onC. Then for anyk-spaceT, the projection qWC T !T has a sectionsD.t 7!.P; t //, which induces a map

sD.L7!LjfPg T /WPic.C T /!Pic.T /

such thatsqDid. Consequently,Pic.C T /DIm.q/˚Ker.s/, and soP_C⁰.T /can be identified with

P⁰.T /D fL2Pic.C T /jdeg.Lt/D0all t,LjfPg T is trivialg.

Now assume (1.6). AsC.T /is nonempty for allk-schemesT,J represents the functor P_C⁰ DP⁰. This means that there is an elementMofP⁰.J /(corresponding toidWJ ! J under/such that, for everyk-schemeT andL 2 P⁰.T /, there exists a unique morphism 'WT ! J such that .1 '/M L. In particular, for each invertible sheaf L on C of degree 0, there is a unique a 2 J.k/ such thatMa L. After replacing Mwith .1ta/Mfor a suitablea 2 J.k/, we can assume thatM₀ is trivial, and therefore that Mis a divisorial correspondence between.C; P /andJ. It is clear thatMhas the universal

property required by (1.7). ₂

EXERCISE1.9. Let .J;M^P/ be a pair having the universal property in (1.7) relative to some pointP on C. Show thatJ is the Jacobian ofC.

We next make some remarks concerning the relation betweenP_C⁰ andJ in the case that C does not have ak-rational point.

REMARK1.10. For allk-spacesT,.T /WP_C⁰.T / ! J.T /is injective. The proof of this is based on two observations. Firstly, becauseC is a complete varietyH⁰.C;O_C/ D k, and this holds universally: for any k-schemeT, the canonical mapO_T !qO_CT is an isomorphism. Secondly, for any morphismqWX !T of schemes such thatO_T !qO_X, the functorM7!qMfrom the category of locally freeO_T -modules of finite-type to the category of locally freeO_X-modules of finite-type is fully faithful, and the essential image is formed of those modulesF onX such thatqF is locally free and the canonical map q.qF/!F is an isomorphism. (The proof is similar to that of I 5.16.)

Now letLbe an invertible sheaf onC T that has degree0on the fibres and which maps to zero in J.T /; we have to show thatL qMfor some invertible sheafMon T. Letk⁰ be a finite extension ofksuch thatC has ak⁰-rational point, and letL⁰be the inverse image ofL on.C T /k⁰. ThenL⁰ maps to zero inJ.Tk⁰/, and so (by definition ofJ /we must haveL⁰ qM⁰ for some invertible sheafM⁰onT_k0. ThereforeqL⁰ is locally free of rank one onT_k0, and the canonical mapq.qL⁰/!L⁰is an isomorphism.

ButqL⁰is the inverse image ofqLunderT⁰ !T (see I 4.2a), and elementary descent theory (cf. 1.13 below) shows that the properties of L⁰in the last sentence descend to L;

thereforeLqMwithMDqL.

REMARK1.11. It is sometimes possible to compute the cokernel to WP_C⁰.k/ ! J.k/.

There is always an exact sequence

0!P_C⁰.k/!J.k/!Br.k/

whereBr.k/is the Brauer group ofk. Whenkis a finite extension ofQp,Br.k/DQ=Z, and it is known (see Lichtenbaum 1969, p130) that that the image of J.k/ in Br.k/ is P ¹Z=Z, whereP (the period ofC /is the greatest common divisor of the degrees of the k-rational divisors classes onC.

REMARK1.12. Regard P_C⁰ as a presheaf on the large ´etale site over C; then the pre-cise relation between J and P_C⁰ is that J represents the sheaf associated with P_C⁰ (see Grothendieck 1968,5).

Finally we show that it suffices to prove (1.6) after an extension of the base field. For the sake of reference, we first state a result from descent theory. Let k⁰be a finite Galois extension of a fieldkwith Galois groupG, and letV be a variety overk⁰. A descent datum forV relative tok⁰=k is a collection of isomorphisms'WV ! V, one for each 2 G, such that' D ' ı ' for all and. There is an obvious notion of a morphism of varieties preserving the descent data. Note that for a varietyV overk,V_k0 has a canonical descent datum. If V is a variety overk andV⁰ D V_k0, then a descent datum on an O_V0 -moduleMis a family of isomorphisms'WM! Msuch that' D ' ' for all and.

PROPOSITION1.13. Letk⁰=kbe a finite Galois extension with Galois groupG.

(a) The map sending a varietyV overktoV_k0endowed with its canonical descent datum defines an equivalence between the category of quasi-projective varieties overkand that of quasi-projective varieties overk⁰endowed with a descent datum.

(b) LetV be a variety overk, and letV⁰DV_k0. The map sending anO_V-moduleMto M⁰ DOV⁰ ˝Mendowed with its canonical descent datum defines an equivalence between the category of coherent O_V -modules and that of coherentO_V0-modules endowed with a descent datum. Moreover, ifM⁰is locally free, then so also isM.

PROOF. (a) See AG 16.23.

(b) See Serre 1959, V.20, or Waterhouse 1979,17. (For the final statement, note that being locally free is equivalent to being flat, and thatV⁰is faithfully flat overV.) ₂ PROPOSITION1.14. Letk⁰be a finite separable extension ofk; if (1.6) is true forC_k0, then it is true forC.

PROOF. After possibly enlargingk⁰, we may assume that it is Galois overk(with Galois groupG, say) and thatC.k⁰/is nonempty. LetJ⁰be the Jacobian ofC_k0. ThenJ⁰represents P_C⁰

k0, and so there is a universalMinP_C⁰.J⁰/. For any 2 G,M2 P_C⁰.J⁰/, and so there is a unique map 'WJ⁰ ! J⁰ such that.1'/M D M(in P_C⁰.J⁰//. One checks directly that ' D ' ı '; in particular, '' ¹ D '_id, and so the ' are isomorphisms and define a descent datum on J⁰. We conclude from (1.13) thatJ⁰ has a

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