The proof that follows is short and elementary but unilluminating (to me, at least).
There are many proofs of Torelli’s theorem, but I don’t know if there is one that is short, elementary, and conceptual. Advice appreciated.
Throughout this section,C will be a complete nonsingular curve of genusg 2over an algebraically closed fieldk, andP will be a closed point ofC. The mapsfPWC !J andf.r/WC.r/!J corresponding toP will all be denoted byf. Thereforef .DCD0/D f .D/Cf .D0/, and iff .D/Df .D0/, thenDD0CrP wherer Ddeg.D/ deg.D0/.
As usual, the image ofC.r/ inJ is denoted byWr. A canonical divisorK onC defines a point onC.2g 2/whose image inJ will be denoted by. For any subvarietyZofJ,Z will denote the image ofZunder the mapx7! x.
LEMMA13.1. For allainJ.k/,.Wag 1/DWg 1a . PROOF. For any effective divisorDof degreeg 1onC,
h0.K D/Dh1.K D/Dh0.D/1;
and so there exists an effective divisorD0such thatK DD0. Then f .D/ aD f .D0/ — a, which shows that .Wag 1/ Wg 1a . On replacinga by a, we get that .Wg 1a /Wag 1, and soWg 1a D.Wg 1a / .Wg 1a /. 2 LEMMA13.2. For anyr such that0r g 1;
War Wbg 1 ” a2Wbg 1 r:
PROOF. (HWIfc Df .D/CawithDan effective divisor of degreer, andaDf .D0/Cb withD0an effective divisor of degreeg 1 r, thenc Df .DCD0/CbwithDCD0 an effective divisor of degreeg 1.
H)W As a 2 Wbg 1, there is an effective divisor A of degreeg 1 such that a D f .A/Cb. LetDbe effective of degreer. The hypothesis states thatf .D/CaDf .D/N Cb for someDN effective of degreeg 1, and sof .D/Cf .A/Df .D/N and
DCA NDCrP:
Choose effective divisors A0 andDN0 of degree g 1such that ACA0andDN C ND0are linearly equivalent toK(cf. the proof of 13.1). Then
DCK A0K DN0CrP and so
DC ND0A0CrP:
As theDs form a family of dimensionr, this shows thath0.A0CrP / rC1. (In more detail,jA0CrPjcan be regarded as a closed subvariety ofC.rCg 1/, and we have shown that it projects onto the whole of C.r/.) It follows from the Riemann-Roch theorem that h0.K A0 rP /1, and so there is an effective divisorANof degreeg 1Crsuch that
A0C NACrP K:
ThereforeANCrP K A0A, and sof .A/N Df .A0/andaDf .A/N Cb 2Wbg 1 r.2
13. TORELLI’S THEOREM: THE PROOF 123 LEMMA13.3. For anyr such that0r g 1;
Wg 1 r D\
fWg 1a ja2Wrgand .Wg 1 r/D\
fWag 1ja2Wrg:
PROOF. Clearly, for a fixedainJ.k/;
Wg 1 r Wg 1a ” Wag 1 r Wg 1; and (13.2) shows that both hold ifa2Wr. Therefore
Wg 1 r \
fWg 1a ja2Wrg:
Conversely, c 2 Wg 1a ” a 2 Wg 1c , and so if c 2 Wg 1a for alla 2 Wr, then Wr Wg 1c andWcr Wg 1. According to (13.2), this implies thatc 2 Wg 1 r, which completes the proof the first equality. The second follows from the first and the equation
\fWag 1ja2Wrg D\
f.Wg 1a /ja2Wrg D\
fWg 1a ja2Wrg
: 2
LEMMA13.4. Letr be such that0rg 2, and letaandbbe points ofJ.k/related by an equationaCx DbCy withx 2 W1andy 2Wg 1 r. IfWarC1 *Wbg 1, then WarC1\Wbg 1 DWarCxS withS DWarC1\.Wy ag 2/.
PROOF. Writex Df .X /andy Df .Y /withX andY effective divisors of degree1and g 1 r. IfY X, then, becausef .X /CaDf .Y /Cb, we will haveaDf .Y X /Cb with Y X an effective divisor of degree g 2 r. Thereforea 2 Wbg 2 r, and so WarC1Wbg 1(by 13.2). Consequently, we may assume thatX is not a point ofY.
Letc 2WarC1\Wbg 1. ThencDf .D/CaDf .D0/Cbfor some effective divisors DandD0of degreerC1andg 1. Note that
f .D/CyDf .D/CaCx bDf .D0/Cx;
and soDCY D0CX.
IfDCY DD0CX, thenDX, and soc Df .D/Ca Df .D X /CxCa; in this casec 2WarCx.
IfDCY ¤D0CX, thenh0.DCY /2, and so for any pointQofC.k/,h0.DC Y Q/1, and there is an effective divisorQN of degreeg 1such thatDCY QC NQ.
Then
c Df .D/CaDf .Q/N Ca yCf .Q/;
and soc 2 T
fWa yg 1Cdjd 2W1g D.Wg 2/a y (by 13.3). As.Wg 2/a y D.Wy ag 2/ andcis inWarC1by assumption, this completes the proof thatWarC1P
Wbg 1WarCxS.
The reverse inclusion follows from the obvious inclusions: WarCx WarC1;WarCx D WbrCy Wbg 1;.Wy ag 2/.Wy a xg 1 /DWbg 1. 2
LEMMA13.5. Leta 2 J.k/ be such thatW1 * Wag 1; then there is a unique effective divisorD.a/of degreegonC such that
f .D.a//DaC (5)
andW1Wag 1, when regarded as a divisor onC, equalsD.a/.
PROOF. We use the notations of6; in particular, DWg 1. ForaD0, (13.1) says that . /D. Therefore, on applying (6.8), we find thatW1Wag 1 Df .C /. /aC df
D f 1.. /aC/DD, whereDis a divisor of degreegonC such thatf.g/.D/DaC.
This is the required result. 2
We are now ready to prove (12.1a). We useˇto identifyJ withJ0, and writeVr for the images ofC0.r/inJ. AsWg 1andVg 1define the same polarization ofJ, they give the same element ofNS.J /(see I,10), and therefore one is a translate of the other, say Wg 1DVcg 1,c 2J.k/. To prove (12.1a), we shall show thatV1is a translate ofW1or of.W1/.
Let r be the smallest integer such that V1 is contained in a translate of WrC1 or .WrC1/. The theorem will be proved if we can show that r D0. (Clearly,r < g 1:/
Assume on the contrary thatr > 0. We may suppose (after possibly replacingˇ by ˇ/
thatV1 WarC1. Choose anxinW1and ayinWg 1 r, and setbDaCx y. Then, unlessWarC1Wbg 1, we have (with the notations of 13.4)
V1\Wbg 1DV1\WarC1\Wbg 1 D.V1\WarCx/[.V1\S /:
Note that, for a fixed a,WarCxdepends only onxandS depends only ony.
Fix anx; we shall show that for almost ally,V1*Wbg 1, which implies thatWarC1 * Wbg 1 for the samey. Asy runs overWg 1 r, b runs overWg 1 r.aCx/. Now, ifV1 Wbg 1for all b inWg 1 r.aCx/, thenV1 WarCx (by 13.3). This contradicts the definition ofr, and so there existbfor whichV1 *Wbg 1. Note thatV1 Wbg 1.DVbg 1Cc/ ” b 2Vcg 2(by 13.2). ThereforeVcg 2*Wg 1 r.aCx/, and so the intersection of these sets is a lower dimensional subset ofWg 1 r.aCx/whose points are the bfor whichV1Wbg 1.
We now return to the consideration of the intersectionV1\Wbg 1, which equals.V1\ WarCx/[.V1\S / for almost all y. We first show that V1\WarCx contains at most one point. If not, then as b runs over almost all points ofWg 1 r.aCx/ (for a fixedx/, the elementD0.b/Ddf f0 1.V0Wbg 1/(cf. 13.5) will contain at least two fixed points (because WarCx Wag 1Cx y D Wbg 1/, and hence f .D0.b// will lie in a translate of Vg 2. As f0.D0.b// D b C0, we would then have.Wg 1 r/ contained in a translate ofVg 2, sayVdg 2, and so
\fVc ug 1ju2Vdg 2g \
fWg 1u ju2.Wg 1 r/g:
On applying (13.3) to each side, we then get an inclusion of V in translate of .Wr/, contradicting the definition ofr.
Keepingy fixed and varyingx, we see from (5) thatV1\WarCx must contain at least one point, and hence it contains exactly one point; according to the preceding argument, the point occurs inD0.b/with multiplicity one for almost all choices ofy.
14. BIBLIOGRAPHIC NOTES 125 13.4). According to (13.3), we now get an inclusion of some translate ofVg 2inWg 2or .Wg 2/. Finally (13.3) shows that
V1 D\
fV eje2Vg 2g
which is contained in a translate ofW1orW1according asVg 2is contained in a trans-late ofWg 2or.Wg 2/. This completes the proof.