5 The canonical maps from the symmetric powers of C to its Jacobian variety

Throughout this section C will be a complete nonsingular curve of genusg > 0. Assume there is ak-rational pointP onC, and writef for the mapf^P defined in2.

Letf^r be the mapC^r !J sending.P1; :::; Pr/tof .P1/C Cf .Pr/. On points, f^r is the map .P1; :::; Pr/ 7! ŒP1 C CPr rP . Clearly it is symmetric, and so induces a mapf^.r/WC^.r/ !J. We can regardf^.r/as being the map sending an effective divisorDof degreeronC to the linear equivalence class ofD rP. The fibre of the map f^.r/WC^.r/.k/! J.k/containingDcan be identified with the space of effective divisors linearly equivalent to D, that is, with the linear system jDj. The image ofC^.r/ inJ is a closed subvariety W^r ofJ, which can also be written W^r D f .C /C Cf .C / (r summands).

THEOREM5.1. (a) For allr g, the morphismf^.r/WC^.r/ !W^r is birational; in partic-ular,f^.g/ is a birational map fromC^.g/ontoJ.

(b) Let D be an effective divisor of degree r on C, and let F be the fibre of f^.r/

containingD. Then no tangent vector toC^.r/ atDmaps to zero under.df^.r//D unless it lies in the direction ofF; in other words, the sequence

0!TD.F /!TD.C^.r//!Ta.J /; aDf^.r/.D/;

is exact. In particular,.df^.r//_DWT_D.C^.r//!Ta.J /is injective ifjDjhas dimension zero.

The proof will occupy the rest of this section.

ForDa divisor onC, we writeh⁰.D/for the dimension of H⁰.C;L.D//D ff 2k.C /j.f /CD0g andh¹.D/for the dimension ofH¹.C;L.D//. Recall that

h⁰.D/ h¹.D/Ddeg.D/C1 g;

and thatH¹.C;L.D//^_ DH⁰.C; ˝¹. D//, which can be identified with the set of! 2

˝_{k.C /=k}¹ whose divisor.!/D.

LEMMA5.2. (a) LetD be a divisor onC such thath¹.D/ > 0; then there is a nonempty open subset U of C such that h¹.D CQ/ D h¹.D/ 1 for all points Q in U, and h¹.DCQ/Dh¹.D/forQ …U.

(b) For anyr g, there is an open subsetU ofC^r such thath⁰.P

Pi/ D 1for all .P1; :::; Pr/inU.

PROOF. (a) IfQis not in the support ofD, then

H¹.C;L.DCQ//^_D .C; ˝¹. D Q//

can be identified with the subspace of .C; ˝¹. D//of differentials with a zero at Q.

Clearly therefore we can takeU to be the complement of the zero set of a basis ofH¹.C;L.D//

together with a subset of the support ofD.

(b) LetD0be the divisor zero onC. Thenh¹.D0/Dg, and on applying (a) repeatedly, we find that there is an open subsetU ofC^rsuch thath¹.P

Pi/Dg rfor all.P1; :::; Pr/ inU. The Riemann-Roch theorem now shows thath⁰.P

Pi/DrC.1 g/C.g r/D1

for all.P1; :::; Pr/inU. ₂

In proving (5.1), we can assume thatk is algebraically closed. If U⁰ is the image in C^.r/ of the set U in (5.2b), then f^.r/WC^.r/.k/ ! J.k/ is injective on U⁰.k/, and so f^.r/WC^.r/ ! W^r must either be birational or else purely inseparable of degree> 1. The second possibility is excluded by part (b) of the theorem, but before we can prove that we need another proposition.

PROPOSITION5.3. (a) For allr 1, there are canonical isomorphisms .C; ˝¹/ ^'! .C^r; ˝¹/^{Sr '}! .C^.r/; ˝¹/:

Let! 2 .C; ˝¹/correspond to!⁰ 2 .C^.r/; ˝¹/; then for any effective divisorD of degreeronC,.!/Dif and only if!⁰has a zero atD.

(b) For allr 1, the mapf^.r/W .J; ˝¹/! .C^.r/; ˝¹/is an isomorphism.

A global1-form on a product of projective varieties is a sum of global1-forms on the factors. Therefore .C^r; ˝¹/ D L

ip_i .C; ˝¹/, where thepi are the projection maps onto the factors, and so it is clear that the map ! 7! P

p_i! identifies .C; ˝¹/with .C^r; ˝¹/^Sr. Because WC^r ! C^.r/ is separable, W .C^.r/; ˝¹/ ! .C^r; ˝¹/ is injective, and its image is obviously fixed by the action ofSr. The composite map

.J; ˝¹/! .C^.r/; ˝¹/ ,! .C^r; ˝¹/^Sr ' .C; ˝¹/ sends!to the element!⁰of .C; ˝¹/such thatf^r!D P

p_i!⁰. Asf^r D P

f ıpi, clearly!⁰Df!, and so the composite map isfwhich we know to be an isomorphism (2.2). This proves that both maps in the above sequence are isomorphisms. It also com-pletes the proof of the proposition except for the second part of (a), and for this we need a combinatorial lemma.

LEMMA5.4. Let 1; :::; r be the elementary symmetric polynomials inX1; :::; Xr, and letj D P

X_i^jdXi. Then

m0 m 11C C. 1/^mm DdmC1, all mr 1:

PROOF. Letm.i /be themth elementary symmetric polynomial in the variables X1; :::; Xi 1; XiC1; :::; Xr:

Then

m nDm n.i /CXim n 1.i /;

5. CANONICAL MAPS 103 and on multiplying this by . 1/ⁿX_iⁿ and summing over n (so that the successive terms cancel out) we obtain the identity

m m 1Xi C C. 1/^mX_i^mDm.i /:

On multiplying this withdXi and summing, we get the required identity. ₂ We now complete the proof of (5.3). First let D D rQ. Then OO_Q D kŒŒX  and O

O_D DkŒŒ1; :::; r(see the proof of (3.2); byO_D we mean the local ring at the pointD onC^.r/). If! D.a0Ca1X Ca2X²C /dX,ai 2k, when regarded as an element of

˝¹_O

OQ=k, then!⁰Da00Ca11C . We know thatfd1; :::; drgis a basis for˝¹_O

OQ=k

as an OO_D -module, but the lemma shows that0; :::; r 1 is also a basis. Now .!/ D and!⁰.D/D0are both obviously equivalent toa0 Da1 D D ar 1 D0. The proof for other divisors is similar.

We finally prove the exactness of the sequence in (5.1). The injectivity of.d i /Dfollows from the fact thatiWF ,!C^.r/is a closed immersion. Moreover the sequence is a complex becausef ıi is the constant mapx7!a. It remains to show that

dim Im.d i /D Ddim Ker.df^.r//D:

IdentifyTa.J /^_with .C; ˝¹/using the isomorphisms arising from (2.1). Then (5.3) shows that ! is zero on the image of T_D.C^.r// if and only if .!/ D, that is, ! 2 .C; ˝¹. D//. Therefore the image of .df^.r//D has dimensiong h⁰.˝¹. D// D g h¹.D/, and so its kernel has dimensionr gCh¹.D/. On the other hand, the image of .d i /_D has dimensionjDj. The Riemann-Roch theorem says precisely that these two numbers are equal, and so completes the proof.

COROLLARY5.5. For allrg,f^rWC^r !W^rhas degreerŠ.

PROOF. It is the composite ofWC^r !C^.r/andf^.r/. ₂ REMARK5.6. (a) The theorem shows that J is the unique abelian variety birationally equivalent to C^.g/. This observation is the basis of Weil’s construction of the Jacobian.

(See7.)

(b) The exact sequence in (5.1b) can be regarded as a geometric statement of the Riemann-Roch theorem (see especially the end of the proof). In fact it is possible to prove the Riemann-Roch theorem this way (see Mattuck and Mayer 1963).

(c) As we observed above, the fibre of f^.r/WC^.r/.k/ ! J.k/ containing D can be identified with the linear system jDj. More precisely, the fibre of the map of functors C^.r/ ! J is the functorDiv^D_C of (3.14); therefore the fibre off^.r/ containingD(in the sense of algebraic spaces) is a copy of projective space of dimensionh⁰.D/ 1. Corollary 3.10 of Chapter I shows that conversely every copy of projective space inC^.r/is contained in some fibre off^.r/. Consequently, the closed points of the Jacobian can be identified with the set of maximal subvarieties ofC^.r/isomorphic to projective space.

Note that forr > 2g 2,jDjhas dimensionr g, and so.df^.r//D is surjective, for allD. Thereforef^.r/ is smooth (see Hartshorne 1977, III 10.4), and the fibres off^.r/are precisely the copies ofP^{r g}contained inC^.r/. This last observation is the starting point of Chow’s construction of the Jacobian Chow 1954.