Infinitesimal study of a factorization of the Prym map

Digital Object Identifier (DOI) 10.1007/s10231-003-0099-8

Annali di Matematica 183, 401–419 (2004)

Roy Smith · Robert Varley

Infinitesimal study of a factorization of thePrym map

Dedicated in memory of Fabio Bardelli

Received: March 20, 2003; in final form: June 11, 2003Published online: January 28, 2004 – Springer-Verlag 2004

Abstract. The Prym map Rg → Ag−1 factors through a space Φ = X of intrinsicallypolarized varieties, namely (C/C) → X → (P, Ξ), where π : C → C is a connectedetale double cover of a smooth curve C of genus g, X ⊂ C(2g−2) consists of the set of evenprecanonical effective divisors on C, and (P,Ξ) is the principally polarized Prym varietyassociated to π. X is a connected, reduced local complete intersection of (pure) dimensiong − 1, and when C is non hyperelliptic X is normal and irreducible. By analogy with theproofs of the classical Torelli theorem for curves by Andreotti and by Andreotti-Mayer andGreen, which factor the Jacobi map through a symmetric product of the curve, the presentfactorization may be used to attack the Torelli problem for Prym varieties. In [19] we haveshown that X determines the Prym variety (P,Ξ), as the Albanese variety of X, and thatX also determines the double cover C/C, at least when C is non hyperelliptic and thecodimension of singΞ in P is at least 5. The next challenge in this approach to the Torelliproblem is to analyze the infinitesimal structure of these maps.

The goal of the present paper is to show the first map (C/C) → X is unramified whenC is non hyperelliptic, i.e. that a first order deformation of C/C which induces the trivialfirst order deformation of X, is already trivial on C/C. (This question was studied for g = 3by H. Yin [23].) We do this as follows for g ≥ 3. There is a map p → Dp from C to a curveof effective Cartier divisors on X. We prove that if C is non hyperelliptic, this map is anisomorphism from C onto a smooth connected component S of the Hilbert scheme of X.This is an analogue of Prop. 4.1. b), p. 334, in [7], (that the set Dp : p ∈ C, is a connectedcomponent of Hilb(C(g−1)) isomorphic to C).

Then we deduce that if a first order deformation of C/C induces the trivial deformationof X, the deformation of C is isomorphic to the trivial deformation of the curve S in Hilb(X).It follows that the original deformation of C/C is trivial. The complementary question ofwhether every first order deformation of X comes from a first order deformation of C/C,analogous to Thm. 3.6 of [7], is proved in [23] for g = 3 and C non hyperelliptic, but remainsopen for g ≥ 4 at the time of writing. We will work throughout over the complex numbers,and will generally assume the base curve C is smooth and non hyperelliptic, although someresults are true more generally.

Mathematics Subject Classification (2000). 14H40, 14K

R. Smith, R. Varley: Department of Mathematics, University of Georgia, Athens, GA 30602,USA, e-mail: [email protected]; [email protected]

402 R. Smith, R. Varley

0. Definitions and notations

0.1. The Prym variety and the Prym difference surface

Given a connected etale double cover π : C → C of a smooth curve C of genusg ≥ 3, the kernel of the associated norm map Nm : Pic0(C) → Pic0(C) on linebundles has two connected components Nm−1(0) = P0 ∪ P1, [14, bottom p. 329,where Nm = ψ, cf. (b), bottom p. 341]. If P0 is the component containing 0,then the principal polarization of the Jacobian of C, considered as a cohomologyclass ϑ in H2(Pic0(C),Z), restricts on P0 to twice a principal polarization ξ ,i.e. ϑ|P0 = 2ξ . The resulting pair (P0, ξ) is by definition the principally polarizedPrym variety determined by π. Since C has genus g, the double cover C has genus2g − 1, and since Nm is surjective P0 has dimension g − 1. We often denote theimage π(p) on C, of a point of C, by π(p) = p.

If ι : C → C is the fix point free involution associated to the double cover π,then for each point p on C let p′ = ι(p), and define the Prym difference mapa : C × C → P0 by a(p, q) = (1 − ι)(p − q) = p − q − p′ + q′, where we writep − q for OC(p − q) when there is no likelihood of confusion between a divisorand the line bundle it determines. This map has image in P0 since a(p, p) = 0. Theimage of this map is called the Prym difference surface. When q is fixed, and C isnon hyperelliptic, the map p → a(p, q) is an embedding of C onto a curve in P0

called an Abel Prym curve. The natural map p → (p − p′) is a translate in P1, ofany one of these embeddings, hence is also an embedding. Its image is also calledan Abel Prym curve.

0.2. The Abel map α, and the Prym theta divisor Ξ

The polarized Prym variety (P0, ξ) has a distinguished “theta divisor” Ξ determinedup to translation as follows. The inverse image Nm−1(ωC ) of the canonical linebundle ωC of C, under the norm map Nm : Pic2g−2(C) → Pic2g−2(C) also has twoconnected components [14, pp. 341–2], Nm−1(ωC ) = P ∪ P−, where P = L :Nm(L) = ωC and h0(L) is even and P− = L : Nm(L) = ωC and h0(L) is odd.Since 2g − 2 = g(C) − 1, the image of the Abel map α : C(2g−2) → Pic2g−2(C),(where α(D) = OC(D)), is the natural model Θ ⊂ Pic2g−2(C) of the theta divisorfor the Jacobian of C, and by [14, Prop.(a), p. 342], P · Θ = 2Ξ , where Ξ istherefore a natural model for the theta divisor of the Prym variety determined byC/C. Analogously to representing the Jacobian variety (Pic0(C), ϑ) by the pair(Pic2g−2(C), Θ), we may consider the pair (P,Ξ) to represent the Prym variety(P0, ξ).

0.3. The parametrization ϕ : X → Ξ , of Ξ by the divisor variety X

We define the divisor variety as X = α−1(P), where α : C(2g−2) → Pic2g−2(C)

is the Abel map parametrizing the theta divisor of C, and P ⊂ Pic2g−2(C) is the“even” half of the set of precanonical line bundles on C with respect to π, i.e. [22,

Infinitesimal study of a factorization of the Prym map 403

p. 99] X = D ∈ C(2g−2) such that Nm(O(D)) ∼= ωC and h0(C,O(D)) is even,is the set of effective, even, “precanonical” divisors on C with respect to π. WhenC is non hyperelliptic X is normal and irreducible [3, Cor. of Prop. 3, p. 365].Denoting by ϕ : X → P the restriction of the Abel map α to X, the image setΞ = ϕ(X) ⊂ P of effective even precanonical line bundles on C with respect to π,is the natural model of the theta divisor for the Prym variety P.

By Riemann’s singularities theorem Ξ ⊂ singΘ, and since P · Θ = 2Ξ , allsmooth points of Ξ are double points of Θ. Hence by Abel’s theorem the fiber ofthe Abel map α : C(2g−2) → Pic2g−2(C) over a smooth point of Ξ is isomorphicto P1, and X is a Zariski P1 bundle over Ξsm . Thus when C is non hyperelliptic,X is a normal irreducible variety fibered by ϕ over smooth points of Ξ by copiesof P1, and over singular points of Ξ by projective spaces of odd dimension ≥ 1.It can happen that ϕ−1(L) ∼= P1 for some singular points L of Ξ , by [14, Prop.,bottom p. 343].

1. The natural divisors D p on X

Definition 1.1. Since X ⊂ C(2g−2), we have for each p in C a natural ample divisorDp = X ∩ Dp ⊂ X, where as sets, Dp = the subset of those D in C(2g−2) withD ≥ p, and Dp = the subset of those D in X with D ≥ p. Dp = p + C(2g−3)

is reduced by definition, and Dp has the structure of Cartier divisor induced byDp = X · Dp. The restriction to Dp of the Abel map ϕ : X → Ξ , is denotedϕp : Dp → Ξ . The map ϕp is surjective for all p.

Since the family Dp of divisors is parametrized by the curve C, they givea model S of the curve C inside the Hilbert scheme of X. We want to show first thatS is a connected component of Hilb(X) which is isomorphic to C. The technicalresult we need is that the spaces H0(Dp,ODP (Dp)) are one dimensional for all p.We will prove the following precise statements, (numbered as they occur later withtheir proofs). The universal assumption is that C is a smooth non hyperellipticcurve, and π : C → C is a connected etale double cover.

Proposition 4.2. If C is non hyperelliptic of genus g ≥ 3, and p any point of C,then h0(Dp,ODP (Dp)) = 1.

Corollary 4.4. If C is non hyperelliptic of genus g ≥ 3, then the image curve S ofthe injective map C → Div(X), p → Dp, is a smooth connected component ofthe Hilbert scheme Hilb(X) isomorphic to C.

Lemma 1.2. If C is non hyperelliptic of genus g ≥ 3, the map C → Div(X) takingp to Dp is injective, hence defines a curve S birational to C of ample divisors on X.

Proof. If p = q are points of C with images p, q, on C, we want to show thatDp = Dq . Choose a hyperplane in canonical space transverse to the embeddedmodel ϕK (C) of C and cutting a canonical divisor D on ϕK (C) which dominatesp but not p + q. Assume first p = q. Since the norm map h : X → |ωC |, (therestriction of Nm : C(2g−2) → C(2g−2) to X), is finite and surjective there is


a precanonical even divisor D in X such that h(D) = D. Then D dominates eitherp or p′ but not q. If D dominates p then D belongs to Dp but not Dq so Dp = Dq

and we are done. If instead D dominates p′, then simply replace p′ by p, andreplace also one other point of D by its conjugate. Since D does not dominate q,we cannot introduce the point q into D this way. By Mumford’s parity trick [13,p. 188, step III], D still belongs to X, and now D dominates p but not q, so againDp = Dq . If q = p, since D dominates p but not 2p, then D dominates eitherp or p′, but not p + p′. Hence D belongs either to Dp or to Dp′ but not both, soDp = Dp′ .

It will follow from Prop. 4.2 that C is isomorphic to the smooth connectedcomponent S of the Hilbert scheme Hilb(X).

1.3. Idea of the proof of Proposition 4.2

We want to work down on Ξ via the birational map ϕp : Dp → Ξ , (the restrictionto Dp of the Abel map ϕ : X → Ξ). Then we will argue roughly as follows:

i) We proved in [18] that on Ξ , ϕ(Dp ∩ Dp′) is a Gauss divisor, hence (on Ξ)ODp(Dp) has the cohomology class of a Gauss divisor, and is effective (sinceit belongs to the family ODp(Dq)), hence should be defined on Ξ by a translateof Ξ which we claim is a non trivial translate. (cf. Claim 4.3)

ii) Since ODp(Dp) ∼= ODp(Dp + Dp′ − Dp′ ) ∼= ODp(Dp′) ⊗ ODp(Dp − Dp′),and ODp(Dp′ ) (on Ξ) is a Gauss divisor for all p, we will show for all p on C,that ODp(Dp − Dp′) is non trivial. To this end we will relate the differencesurfaces defined by S, in Pic0(X), Pic0(Ξ), and Pic0(Dp).

iii) We also must justify working with divisors on Ξ , since the birational mapϕp : Dp → Ξ does generally have an exceptional divisor.

Remark 1.4. In reference to 1.3.i) the formula in Claim 4.3, is closely related to theequation “Spp = Ξ · Ξ[p,p]” argued set theoretically and cohomologically in [4],p. 615. Since with our weaker assumptions, the intersection divisor Ξ · Ξ[p,p] mayno longer be irreducible, set theoretic and cohomological equality need not implyequality as divisors in our case. Since this is an essential ingredient of our theorem,we include a complete proof of 4.3. Some of the present arguments, establishingproperties of the various incarnations of the Prym “difference surface” may alsohave independent interest.

2. The difference surfaces of divisors

For any points p, q ∈ C, and any d ≥ 1, if we defineDp = D : D ≥ p ⊂ C(d) asabove, then O(Dp −Dq) is in Pic0(C(d)) since all the line bundles O(Dp) : p ∈ Con C(d) are algebraically equivalent, and O(Dp −Dq) = O(Dp)⊗ (O(Dq))

∗. Thuswe have morphisms C × C → Pic0(C(d)), (p, q) → O(Dp −Dq), to each abelianvariety Pic0(C(d)). We call the image of any one of these maps a “difference


surface”. The usual difference surface, for d = 1, lives in Pic0(C), but we shownext that the groups Pic0(C(d)) are canonically isomorphic for all d ≥ 1, in sucha way that these surfaces correspond.

The formula we will need is the one in Corollary 3.7, ODp(Dp − Dp′) =ϕ∗

p(Ξa(p) − Ξ), a consequence of the more general one in Lemma 2.4, OX(Dp −Dq) = ϕ∗(Ξa(p,q) − Ξ), where a(p, q) = (1 − ι)(p − q) is the point on the usualdifference surface, and a(p) = a(p, p′). The line bundle OP(Ξa(p,q) −Ξ) is a pointon the difference surface in Pic0(P), while OX(Dp − Dq) is the correspondingpoint of the difference surface in Pic0(X). It is natural to expect that the Abel mapϕ : X → P relates these two. The lemmas establishing these relations are 2.3, 2.4where 2.4 is the Prym version of 2.3.

Lemma 2.1. Let C be a connected smooth projective algebraic curve of genusg ≥ 1, and d any positive integer. Let α : C → Pic1(C) be the canonical Abelmap, p → OC(p), (so that α(p) − α(q) = OC(p − q) ∈ Pic0(C)); and for anyp ∈ C, let Dp = D : D ≥ p be the effective divisor equal to p + C(d−1) on C(d)

for d ≥ 2, and equal to p for d = 1, and let O(Dp) be the associated line bundleon C(d). Then there is a unique isomorphism f : Pic0(C) → Pic0(C(d)) such that,for any p, q ∈ C, α(p) − α(q) maps to O(Dp −Dq).

Proof. Since the image of the difference map spans Pic0(C) there is at most onesuch isomorphism f , so it suffices to find one. Fix a base point b ∈ C. Then itsuffices to prove:

Claim. For all d ≥ 2 the map λ : C(d−1) → C(d), D → D + b, induces anisomorphism on H1,Z, and induces a map λ∗ : Pic0(C(d)) → Pic(C(d−1)) takingOC(d)(Dp − Dq) to OC(d−1)(Dp −Dq).

Assuming the Claim, we deduce the lemma as follows. By universal coeffi-cients, H1

Z∼= HomZ(H1,Z,Z), so a map inducing isomorphism on H1,Z, also

induces an isomorphism on H1Z

. To show such a map also induces an isomorph-ism on Pic0, recall the following Hodge theory: for a smooth projective var-iety M over C, Pic0(M) ∼= H1(M,OM )/H1(M,Z) (as complex torus), andH1(M,C) = H1(M,Z) ⊗ C has a Hodge filtration completely specified byF1 = ImH0(M,Ω1

M ) → H1(M,C), the quotient H1(M,C)/F1 is canoni-cally isomorphic to H1(M,OM ), and the quotient H1(M,C)/(F1 + H1(M,Z))

is canonically isomorphic to Pic0(M). Moreover, if f : M → M′ is a morphismof smooth projective varieties over C, then Pic0(M′) and Pic0(M) are projectivevarieties overC, and the induced complex analytic map f ∗ : Pic0(M′) → Pic0(M)

is algebraic by Chow’s theorem. Thus, if a morphism f : M → M′ of smooth pro-jective varieties over C, induces an isomorphism f ∗ : H1(M′,Z) → H1(M,Z),then the map f ∗ : Pic0(M′) → Pic0(M) is an isomorphism of abelian varietiesover C.

As for functoriality, we may consider the exponential sequence and use Cechcocycles to verify commutativity with the natural maps. In particular, pullingback line bundles is compatible with pulling back topological cocycles. I.e. themap H1(M′,OM′)/H1(M′,Z) → H1(M,OM )/H1(M,Z) induced by a morphism


f : M → M′ agrees with the map Pic0(M′) → Pic0(M) given by pulling backline bundles by f .

Proof of Claim. Now for all p = b, λ∗(Dp,C(d)) = (Dp,C(d−1)), hence for all p, q =b, λ∗(OC(d)(Dp − Dq)) = OC(d−1)(Dp − Dq) as claimed. Hence by continuity thisholds for all p, q. To show that λ induces isomorphism on H1,Z, we can invoke theLefschetz hyperplane theorem as long as d ≥ 3, and for d = 2 we get at least thatλ∗ : H1(C,Z) → H1(C(2),Z) is surjective. Then by Macdonald’s proof [10] thatthese groups are torsion free of the same rank, we deduce that this last map toois isomorphic. Another way to see that the groups H1(C(d),Z) for d ≥ 1 are alltorsion free of the same rank, and hence get this last isomorphism without citingother arguments, is to note that H1(C,Z) ∼= H1(Picd(C),Z) ∼= Z2g for all d, andthus since all the groups H1(C(d),Z) for d ≥ 2 are isomorphic by Lefschetz, itsuffices to show H1(C(d),Z) ∼= H1(Picd(C),Z) for some d ≥ 2. In case d ≥ 2g,then α : C(d) → Picd(C) is a surjective fiber bundle with complex projectivespace fibers, so the homology spectral sequence for a fibration, and coefficientgroup Z, shows H1(Picd(C),Z) ∼= H1(C(d),Z). Thus H1(Picd(C),Z) ∼= Z2g ∼=H1(C(d),Z), is torsion free of rank 2g when d ≥ 2g, and hence also for all d ≥ 2.

(for the claim)

Lemma 2.2. If π : C → C is an etale connected double cover of a smooth curveC of genus g ≥ 1, and N : C(d) → C(d) the associated norm map on divisors,then the isomorphisms defined above commute with π∗ and N∗, i.e. the followingdiagram commutes:

Pic0(C)f−−−−→ Pic0(C(d))

π∗

N∗

Pic0(C) −−−−→f

Pic0(C(d))

where f , and f are the unique homomorphisms from 2.1, taking O(p − q) toO(Dp −Dq), and O(p − q) to O(Dp −Dq).

Proof. Since the image of the difference map spans Pic0(C) it suffices to show bothcompositions agree on elements of form p−q. Since π∗(p−q) = p+ p′−(q+q′),hence f π∗(p−q) = Dp+Dp′ −Dq−Dq′ . In the other direction f(p−q) = Dp−Dq

so N∗ f(p − q) = Dp +Dp′ − Dq −Dq′ also.

Next we show when g ≥ 3, and C is non hyperelliptic, the isomorphismf : Pic0(C) → Pic0(C(g−1)) agrees with the composition of the polarizing iso-morphisms and the pullback α∗ induced by the Abel map α : C(g−1) → Picg−1(C),which will imply in particular that α∗ is an isomorphism on Pic0. Let κ be any pointof Picg−1(C) and Θ = α(C(g−1)) ⊂ Picg−1(C) be the natural polarizing divisoron Picg−1(C).


Lemma 2.3. If C is non hyperelliptic of genus g ≥ 3, and f is the homomorphismin 2.1, then the following diagram commutes:

Pic0(C)f−−−−→ Pic0(C(g−1))

λΘ

α∗

Pic0(Pic0(C)) −−−−→ψ

Pic0(Picg−1(C))

where

(i) f is the isomorphism defined by Lemma 2.1,(ii) α∗ is induced by the Abel map α : C(g−1) → Picg−1(C);(iii) λΘ : Pic0(C) → Pic0(Pic0(C)) is the polarization map t → O(Θt−κ −Θ−κ),

where t ranges over Pic0(C), and κ is fixed in Picg−1(C); and(iv) ψ : O(Θt−κ − Θ−κ) → O(Θt − Θ) is induced by translation by κ.

In particular, α∗(O(Θp−q − Θ)) = O(Dp − Dq) = f(p − q). Furthermore, λΘ

and ψ are independent of the choice of κ in Picg−1(C).

Proof. Both λΘ and ψ are independent of the choice of κ by the theorem of thesquare. To show commutativity, we again trace p − q around, so we must showα∗(ψ(O(Θp−q−κ−Θ−κ))) = α∗(O(Θp−q−Θ)) = O(Dp−Dq), where only the lastequality needs proof. This is clearly true when p = q, so assume p = q. Then, as in[15, p. 77], or [21, above 1.19], we claim α∗(Θp−q) = Dp+D ≥ 0 : D ≤ |K −q|,is a union of two irreducible components, at least set theoretically. I.e. if α(D)

belongs to Θp−q , then h0(D) ≥ 1, and also h0(D − p + q) ≥ 1. Hence if Ddoes not contain p, then neither does D + q, so h0(D − p + q) ≥ 1 impliesh0(D + q) ≥ 2, hence by RRT, we get h0(K − D − q) ≥ 1. Conversely, if D ≥ 0and h0(K − D −q) ≥ 1, then RRT implies h0(D +q) ≥ 2 and h0(D − p +q) ≥ 1,so α(D) is in Θp−q . And if D ≥ 0, and D ≥ p, then h0(D− p+q) ≥ 1, so α(D) isin Θp−q . This completes the proof that α∗(Θp−q) = Dp + D ≥ 0 : D ≤ |K − q|as sets.

Let D ≥ 0 : D ≤ |K − q| = Eq = effective divisors E such that E + q isdominated by a divisor in |K |. Since α : C(g−1) → Θ has exceptional locus ofcodimension 2 in C(g−1), α is an isomorphism in codimension one. Since by [15,p. 77], α(Eq) is isomorphic to the Abel image of C(g−2), both Eq and its image inΘ are irreducible. Since Dp is also irreducible, we see α∗(Θp−q) has exactly twocomponents.

Moreover, by computing the tangent spaces to Θ and Θp−q at a general pointof each component of Θ ∩ Θp−q , we see the intersection divisor is reduced, hencealso α∗(Θp−q) = Dp + Eq is the reduced sum of these two divisors.

Next we wish to find an explicit divisor representing the line bundle OΘ(Θ). Weclaim the following “Gauss divisor” works. (Intuitively we let p approach q in theprevious description.) The Gauss map γ : Θ |KC| is regular on smooth pointsof Θ, hence has no base divisor. Hence the pullback of each hyperplane is a Cartierdivisor whose support is the closure of the set theoretic inverse image (within theregular points) of that hyperplane. If we pull back the hyperplane in |KC | dual to the


point ϕK (q), we get the set Γq = closure of all smooth points D on Θ such that incanonical space we have ϕK (q) in the span of D. Then α∗(Γq) = Dq + Eq, as sets.But each of these sets has the same cohomology class as before. Hence in order forα∗(Γq) to have the same cohomology class as α∗(OΘ(Θ)), each component mustbe reduced. Hence α∗(O(Θp−q − Θ)) = α∗(O(Θp−q − Γq)) = O(Dp −Dq).

Next the difference surfaces of divisors and line bundles in the groups Pic0(C(d))

and Pic0(Picd(C)) may be mapped naturally onto skew symmetric differencesurfaces in corresponding incarnations (P, P0, Pic0(P), Pic0(P0), Pic0(X)) of thePrym variety, and we must examine the compatibility of these maps. We study howthe image under the map f described in 2.2 of the difference surface is related to itsskew symmetric image under 1 − ι in the Prym variety P0, by mapping them bothinto Pic0(X) where they agree with the pull back of the Prym difference surface inPic0(P).

Lemma 2.4. Assume C is non hyperelliptic of genus g ≥ 3, and C → C a con-nected etale double cover. Then we have the following.

2.4.1 There is a unique homomorphism σ : P0 → Pic0(X) such that the followingdiagram commutes:

Pic0(C)f−−−−→ Pic0(C(2g−2))

1−ι

ρX

P0 −−−−→σ

Pic0(X)

where P0 = ker(Nm : Pic0(C) → Pic0(C))0 = im((1 − ι) : Pic0(C) →Pic0(C)) is the usual definition of the Prym variety as an abelian variety,and ρX is the restriction of line bundles to X ⊂ C(2g−2).

2.4.2 The map σ in 2.4.1, satisfies the following relations:i. For all p, q in C, σ((1 − ι)(p − q)) = O(Dp − Dq).ii. For all a in P0, σ(a) = ϕ∗(OP(Ξa − Ξ)) where ϕ : X → P is the

restriction of the Abel map α : C(2g−2) → Pic2g−2(C). I.e. the followingdiagram commutes:

P0λΞ−−−−→ Pic0(P0)

σ

ψ

Pic0(X) ←−−−−ϕ∗ Pic0(P)

where ψ λΞ(a) = OP(Ξa − Ξ).

Proof. If the map σ exists making the diagram commute in 2.4.1, then 2.4.2.i istrue simply by commutativity. Since 1 − ι is surjective, to show such a σ exists andis unique, it suffices to check that ρX f annihilates the kernel of 1 − ι.


Claim. Ker(1 − ι) = Im(π∗ : Pic0(C) → Pic0(C)), where π∗ is induced by thedouble cover π : C → C.

Proof of claim. The kernel of 1−ι equals symmetric line bundles on C and the pointis to show these are all pull backs from C. This is easy for symmetric divisors so thepoint is to show that symmetric line bundles are represented by symmetric divisors.This reduces by an argument of Mumford to Hilbert’s theorem 90 computing thekernel of the norm map in this case for the cyclic quadratic Galois extension offunction fields of these curves. Let L = ι∗L be a symmetric line bundle andL = OC(D), so that L = OC(D) ∼= ι∗L = OC(ιD), and hence D ≡ ιD,and thus D − ιD = ( f ), where f is a meromorphic function on C. Hence 0 =Nm(D − ιD) = Nm( f ) = (Nm f ), so Nm f is a non zero constant which we mayassume is 1. Then by Hilbert 90, we have f = g/ιg for some function g on C, sothat D − ιD = (g) − (ιg), and hence ιD + (g) = D + (ιg) is ι-invariant. ThenL = OC(D) ∼= OC(D + (ιg)). Then since D + (ιg) is ι-invariant, it consists ofpairs of conjugate points

∑n j(p j + p′

j), and if we take E = ∑n jπ(p j) we have

π∗(E) = D + (ιg), and hence L ∼= π∗(OC(E)) as claimed. Hence, to show σ exists it suffices to show ρX f π∗ is zero on Pic0(C).

By Lemma 2.2 it suffices to show that ρX N∗ f is zero, where N∗ f is thecomposition Pic0(C) → Pic0(C(2g−2)) → Pic0(C(2g−2)). We claim that ρX N∗ isalready zero, since it factors through the zero group Pic0(|ωC |) where |ωC | ∼= Pg−1,which can be seen as follows. Since the following diagram of norm maps and theirrestrictions commutes,

X −−−−→ C(2g−2)

h

N

|ωC | −−−−→ C(2g−2)

so also does the corresponding pullback diagram it induces

Pic0(X)ρX←−−−− Pic0(C(2g−2))

h∗

N∗

Pic0(|ωC |) ←−−−−ρ|ω|

Pic0(C(2g−2))

Since the lower left hand group is zero, we are done, and σ exists and is unique. (for parts 2.4.1 and 2.4.2.i.)

Proof of Part 2.4.2.ii. Recall we want to prove commutativity of the followingdiagram, where λ and ψ are defined by polarization and translation as before.

P0λΞ−−−−→ Pic0(P0)

σ

ψ

Pic0(X) ←−−−−ϕ∗ Pic0(P)

First we prove a result that allows us to pull the calculation back to C.


Sublemma 2.5. The following diagram of restrictions and translations commutes:

Pic0(C(2g−2))α∗←−−−− Pic0(Pic2g−2(C))

ψ←−−−− Pic0(Pic0(C))

ρX

ρP

ρP0

Pic0(X) ←−−−−ϕ∗ Pic0(P) ←−−−−

ψPic0(P0)

Proof of Sublemma 2.5. Apply the functor Pic0 to the following commutativediagram which induces the one above, where the vertical maps are inclusions andthe right hand horizontal maps are translations by an element κ in P:

C(2g−2) α−−−−→ Pic2g−2(C) −−−−→ Pic0(C)

X −−−−→ϕ

P −−−−→ P0

(for Sublemma 2.5.)

Now we can prove what we want up to a factor of 2.

Sublemma 2.6. On P0, 2σ = ϕ∗ ψ 2λΞ .

Proof of Sublemma 2.6. Let j denote the inclusion P0 → Pic0(C). Then we claimthat both 2σ and ϕ∗ ψ 2λΞ are equal to ρX α∗ ψ λΘ j , and thus are equalto each other as well.

First consider 2σ : P0 → Pic0(X). Then 2σ = σ (1 − ι) j : P0 →Pic0(C) → P0 → Pic0(X) since (1 − ι) = 2id on P0, and σ (1 − ι) =ρX f : Pic0(C) → Pic0(C(2g−2)) → Pic0(X) by 2.4.1. Now f = α∗ ψ λΘ :Pic0(C) → Pic0(Pic0(C)) → Pic0(Pic2g−2(C)) → Pic0(C(2g−2)) by Lemma 2.3,applied to C. Hence 2σ = σ (1 − ι) j = ρX f j = ρX α∗ ψ λΘ j .

On the other hand, since 2λΞ : P0 → Pic0(P0) is the restriction to P0 of λΘ ,we have 2λΞ = ρP0 λΘ j . Thus ϕ∗ ψ 2λΞ = ϕ∗ ψ ρP0 λΘ j :P0 → Pic0(C) → Pic0(Pic0(C)) → Pic0(P0) → Pic0(P) → Pic0(X). BySublemma 2.5, ϕ∗ψρP0 = ρXα∗ψ. Hence ϕ∗ψ2λΞ = ϕ∗ψρP0λΘ j =ρX α∗ ψ λΘ j . (for Sublemma 2.6.)

From Sublemma 2.6 we see that 2(σ − ϕ∗ ψ λΞ) = 0, hence the homo-morphism (σ − ϕ∗ ψ λΞ) maps the abelian variety P0 into the discrete set ofpoints of order two in Pic0(X). Since P0 is connected, (σ − ϕ∗ ψ λΞ) maps itinto 0. Thus the diagram in 2.4.2.ii commutes. (for Lemma 2.4.)

3. The zeroth Picard groups of the varieties D p ⊂ X → Ξ ⊂ P

Now that we have verified that all the natural isomorphisms of Picard varietiespreserve the respective models of the difference surfaces, we can profit from thesecompatibilities. We want to understand the maps between the groups Pic0 of thevarieties in the sequence Dp ⊂ X → Ξ ⊂ P, where the first and last maps areinclusions and the middle map is the restricted Abel map ϕ : X → Ξ .


Remark 3.1. If dim P ≥ 3, then Kodaira vanishing applied to the long exact se-quence associated to the sheaf sequence 0 → Ξ → OP → OΞ → 0 givesan isomorphism H1(P,OP) ∼= H1(Ξ,OΞ), and Lefschetz implies an isomorph-ism on H1

Zas in the proof of 2.1, hence Ξ → P induces an isomorphism

Pic0(P) → Pic0(Ξ). If dim P = 2, then (P,Ξ) is a classical Jacobian variety,since Ξ is a smooth genus 2 curve when C is non hyperelliptic of genus 3, so themap is again an isomorphism.

Lemma 3.2. If g ≥ 3, and C is non hyperelliptic, the map ϕp : Dp → P inducesan injection ϕ∗

p : Pic0(P) → Pic0(Dp); i.e. for all a = 0 in P0, ϕ∗p(OΞ(Ξa−Ξ)) =

ODp .

Proof. Given ϕp : Dp → Ξ , let V ⊂ Ξ be the largest open set over which ϕp

restricts to an isomorphism U = ϕ−1p (V ) → V ⊂ Ξ . Since Ξ is normal and

irreducible, then codim((Ξ − V ) ⊂ Ξ) ≥ 2, by ZMT, so applying Pic0 to thecommutative diagram:

Dpϕp−−−−→ Ξ

U −−−−→ϕp

V

gives another commutative diagram in which the right vertical map is injective:

Pic0(Dp)ϕ∗

p←−−−− Pic0(Ξ)

Pic0(U) ←−−−−ϕ∗

p

Pic0(V ) .

Since the bottom map is an isomorphism, the top horizontal map is injective. Thenwe are done by Remark 3.1.

Corollary 3.3. The map ϕ∗ : Pic0(Ξ) → Pic0(X) is an injection when C is nonhyperelliptic of genus g ≥ 3.

Proof. The injection ϕ∗p : Pic0(Ξ) → Pic0(Dp) proved in Lemma 3.2 factors as

ϕ∗p = j∗ ϕ∗, where j : Dp → X is the inclusion map, so ϕ∗ is an injection.

Corollary 3.4. The map σ : P0 → Pic0(X) defined in Lemma 2.4 is an injection.

Proof. By the commutative diagram in Lemma 2.4.2.ii, σ = ϕ∗ ψ λΞ isa composition of injections.

Definition 3.5. For all p in C, let a(p) = 2(p − p′) = (1 − ι)(p − p′) in P0.


Remark 3.6. The involution on C equals the minus map on the Abel Prym curve,i.e. on the image of C under the map p → (p − p′) in P1, (see defn. in (0.1)).Moreover the map p → (p − p′) is an embedding when C is non hyperelliptic.Since the involution is fix point free, it follows that the minus map is fix point freeon the Abel Prym curve, i.e. no point of the Abel Prym curve is a point of ordertwo. Thus for all p in C, a(p) = 2(p − p′) = 0 in P0.

Corollary 3.7. For all p in C, ODp(Dp − Dp′) = ϕ∗p(OΞ(Ξa(p) − Ξ)) = ODp .

Proof. By Lemma 2.4.2, OX(Dp −Dp′) = σ(a(p)) = ϕ∗(OΞ(Ξa(p) −Ξ)), hencerestricting further to Dp, we have ODp(Dp − Dp′ ) = ϕ∗

p(OΞ(Ξa(p) − Ξ)), andRemark 3.6, and Lemma 3.2 imply that ϕ∗

p(OΞ(Ξa(p) − Ξ)) = ODp . Remark 3.8. In fact the injection ϕ∗

p : Pic0(Ξ) → Pic0(Dp) is an isomorphism,for C non hyperelliptic of genus g ≥ 3, since ϕp induces an isomorphism H1

Z(Ξ) →

H1Z(Dp) (by Leray), and Pic0(Ξ) and Pic0(Dp) are compact [6, Thm. 2.1] since

both Dp and Ξ are normal. Since Pic0 ∼= H1(O)/H1Z, compactness implies the

real vector space H1(Dp,O), is generated by H1Z(Dp), hence also by the image

of H1Z(Ξ), and so too by the image of H1(Ξ,O). Since the image of H1(Ξ,O) is

a real subspace of H1(Dp,O), then H1(Ξ,O) surjects onto H1(Dp,O), and thusPic0(Ξ) surjects onto Pic0(Dp). Since X is also normal when C is non hyperellipticof genus g ≥ 3, the same argument then shows that ϕ∗ : Pic0(Ξ) → Pic0(X) isalso an isomorphism. Of course then j : Dp → X also induces an isomorphismon Pic0, and σ is an isomorphism too from P0 to Pic0(X).

4. Irreducibility of D p and dimension of H0(D p,OD p(D p))

Next we want to compute h0(Dp,ODp(Dp)), and we want to work down on Ξ ,via ϕp : Dp → Ξ . Although ODp(Dp) is not necessarily a pullback from Ξ , itbecomes one when we restrict to the open set U ⊂ Dp considered above on whichϕp is an isomorphism. In order to know that open set is dense in Dp, we check thatDp is irreducible as follows.

Lemma 4.1. Assume that C is non hyperelliptic of genus g ≥ 3, and let p be anypoint of C. Then the divisor Dp on X is reduced, irreducible and normal.

Proof. We will use the criterion in [3, Cor. to Prop. 3, p. 365]. If f −1(y) = ∑niqi

is a fiber of a (nonconstant) morphism f : C → Pr , where all qi are distinct and

ni ≥ 1, we call ni the multiplicity of qi in the fiber, and call qi a ramificationpoint of f if ni ≥ 2. Then it suffices to show that the linear system |K − p| = gr

dhas no base points, and defines a morphism in which each fiber has at most oneramification point and that of multiplicity ≤ 3, and that the morphism is birationalonto its image when r ≥ 2. With our hypothesis that C is non hyperelliptic, thesystem |K − p| is always base point free. For g = 3, |K − p| is a g1

3, hence satisfiesthe criterion. If g ≥ 4 and C is not trigonal, the system |K − p| is very ample,hence satisfies the criterion. If g ≥ 4 and C is trigonal, the system |K − p| definesa birational morphism which maps q, r to the same point if and only if p + q + r


belongs to a g13 on C. If g ≥ 5, the g1

3 is unique, so there is at most one ramificationpoint, and that of multiplicity at most 2. If g = 4, there are at most two g1

3’s and atmost two ramification points, each of multiplicity at most 2, and in different fibers.

Proposition 4.2. Assume that C is non hyperelliptic of genus g ≥ 3, and p anypoint of C. Then h0(Dp,ODp(Dp)) = 1.

Proof. First, since Dp appears in the 1-parameter family Dq|q ∈ C of effectiveCartier divisors on X, h0(Dp,ODp(Dp)) ≥ 1. Thus it will suffice to show thath0(Dp,ODp(Dp)) ≤ 1. By Lemma 4.1, Dp is reduced and irreducible. Since C isnon hyperelliptic, Ξ is normal, Dp is irreducible and the map ϕp : Dp → Ξ isbirational, there exists an open subset V ⊂ Ξ such that U = (ϕp)

−1(V ) is a denseopen subset of Dp, ϕp induces an isomorphism from U onto V , and N = (Ξ − V )

has codimension at least 2 in Ξ . (It suffices to let N ⊂ Ξ be the locus of pointswith positive dimensional ϕp-fibers and then let V = Ξ − N.) Then, the restrictionmap H0(Dp,ODp(Dp)) → H0(U,ODp(Dp)) is injective since U is dense in Dp

(and Dp is reduced and ODp(Dp) is a line bundle on Dp, so the support of anynon zero section of ODp(Dp) must meet U). Thus it will suffice to show thatH0(U,OU(Dp)) has dimension at most 1.

Under the isomorphism U ∼= V ⊂ Ξ , we have a line bundle Mp on Vcorresponding to OU(Dp).

Claim 4.3. Mp is isomorphic to the restriction to V of the line bundle OΞ(Ξa(p))

on Ξ .

Proof of Claim. By Corollary 3.7, we know ODp(Dp−Dp′) = ϕ∗p(OΞ(Ξa(p)−Ξ)),

hence this also holds on V . Since we know also ([18], Theorem, part (1), for g ≥ 4,and by [20], Prop., Section 3, for g = 3), that on V , OV (Dp′) = OV (Ξ), we getthe result by adding. (for the claim)

Now we can conclude the proof as follows. Since Ξ is normal andΞ − V has codimension at least 2 in Ξ , the restriction map H0(Ξ,OΞ(Ξa(p))) →H0(V,OΞ(Ξa(p))) = H0(V,Mp) is an isomorphism and hence it suffices to checkthat H0(Ξ,OΞ(Ξa(p))) is 1-dimensional. But for all p, we know that a(p) = 0 byRemark 3.6 above, so we are looking at H0(Ξ,OΞ(Ξa ∩ Ξ)), for a = 0, and sucha line bundle on a theta divisor has a 1-dimensional space of sections. I.e. using[9, Thm. 3.3, p. 57, and Lemma 5.1, p. 63], we see that Hq(P,O(Ξa − Ξ)) = 0for all q, and a = 0. Then the long exact cohomology sequence associated tothe sheaf sequence 0 → OP(Ξa − Ξ) → OP(Ξa) → OΞ(Ξa) → 0, givesh0(Ξ,OΞ(Ξa ∩ Ξ)) = h0(P,OP(Ξa)) = 1. (for Proposition 4.2)

Corollary 4.4. If C is non hyperelliptic of genus g ≥ 3, then the image curve S ofthe injective map C → Div(X), p → Dp, is a smooth connected component ofthe Hilbert scheme Hilb(X) isomorphic to C.

Proof. It suffices to show the tangent space to Hilb(X), at every point Dp of S, isone dimensional. By [5, Cor. 5.4, p. 23], [8, Thm. 2.8, p. 31], this tangent space isisomorphic to H0(Dp,ODp(Dp)), so this follows from Prop. 4.2.


5. Deformations of the divisor variety X

We have shown in Cor. 4.4 above, when C is non hyperelliptic of genus g ≥ 3,that C embeds as a connected component of the Hilbert scheme of X via the mapdefined on points by p → Dp. I.e. we have an injective map C → Hilb(X) whoseimage is contained in a connected component, and we computed that the tangentspace of that component at each point Dp of the image is one dimensional when Cis non hyperelliptic. Hence the image component is a smooth curve and our mapis a proper injection, hence also a surjection, i.e. a bijective morphism of smoothcurves. For smooth proper complex curves such a map is an isomorphism.

Next we want to deduce that the map taking C/C → X is unramified, when Cis non hyperelliptic. We will proceed as follows:

5.1. A first order deformation of the double cover π : C → C yields a first orderdeformation X of X, i.e. a deformation of X over D = Spec(C[ε]), whereε2 = 0.

5.2. If the first order deformation X → D of X is trivial, then the relative Hilbertscheme Hilb(X/D) is the product Hilb(X) ×D.

5.3. Since Hilb(X) contains a special (connected) component isomorphic to C,the scheme Hilb(X/D) ∼= Hilb(X) × D contains the trivial deformation ofthis component.

5.4. The original first order deformation of C maps into Hilb(X/D).5.5. The map in 5.4, is an isomorphism onto the trivial deformation of the special

component of Hilb(X). Hence the original deformation of C is trivial.

(5.1, 5.2, 5.4 seem general, but 5.3, 5.5 use C non hyperelliptic.)

Proof of 5.1. Recall that for a scheme X proper over C, a deformation of X overa (connected, noetherian, complex) scheme S with base point 0, is defined bya pair (q, j), where q : X → S is a proper, flat map and j : X → q−1(0)

is an isomorphism. A first order deformation is a deformation over the schemeD = Spec(C[ε]) (with the base point 0 defined by the maximal ideal (ε)). Thedeformation functor DefX is then the contravariant functor from (Schemes/Cwithbase point) to Sets, defined as follows: for each (S, 0), DefX(S, 0) = isomorphismclasses of deformations of X over (S, 0). The set DefX(D) of (isomorphism classesof) first order deformations carries a natural C-vector space structure. The vectorspace structure is constructed in [16, Lemma 2.10, Remark 2.13, and 3.7, pp. 220–221]. We would also like to highlight Fabio Bardelli’s insightful discussion of firstorder deformations in section IV of [2]. Deformations of the double cover C/C,are defined similarly, i.e. by deformations of C and C, and a map between themwhich induces the double cover π over the base point, except that we will insisthere that the proper morphisms C → S and C → S are actually smooth (and notjust flat).

Claim 5.1.1. There is a map of deformation functors DefC/C → DefX , and inparticular a linear map on first order deformations, DefC/C(D) → DefX(D), forD = Spec(C[ε]).


Proof. We will omit the direct calculation, from the definitions of the vector spacestructures on DefC/C(D) and DefX(D), that the map DefC/C(D) → DefX(D) is C-linear. Then of course one can check injectivity on first order deformations simplyby showing that the kernel is 0, i.e. by showing that, for a first order deformationof the double cover C/C, if the associated deformation of the divisor variety X istrivial, then the deformation of the double cover is trivial.

Given a (connected, noetherian, complex) scheme S with base point 0, considera deformation C/C over (S, 0) of the double cover C/C. Then the proper, smoothmap p : (C/S)(2g−2) → S on the relative symmetric product, gives a deformationover (S, 0) of the symmetric product C(2g−2). Now consider the relative normmap (C/S)(2g−2) → (C/S)(2g−2) and the subscheme Y ⊂ (C/S)(2g−2) definedby the relative canonical system ωC/S of C → S. Then define the subschemeX ⊂ (C/S)(2g−2) as the appropriate “half” of the pullback of Y (in fact theconnected component of the pullback that coincides over 0 with X ⊂ C(2g−2)).Then certainly the induced map q : X → S is proper and there is j : X→q−1(0).A map (S′, 0) → (S, 0) determines, in a functorial manner, a pullback of (q, j).Finally, for the flatness of q, we can apply Lemma 5.1.2 below. Note that since Yand (C/S)(2g−2) are both smooth over S, then for any point y in Y, there exists anopen neighborhood of y in (C/S)(2g−2) in which the subscheme Y is defined byg − 1 regular functions. Then pulling back these regular functions by the relativenorm map gives g − 1 local equations for the codimension g − 1 subschemeX ⊂ (C/S)(2g−2).

Lemma 5.1.2. Let p : U → S be a smooth map of relative dimension n. Fora point x in U, let f1, ..., fr be regular functions on an open neighborhood U0 ofx and let X ⊂ U0 be the subscheme defined by them. Let Fx = p−1(p(x)) be thefiber of p through x. Suppose that Fx ∩ X has codimension r in Fx at x. Then therestriction of p, X → S, is flat at x.

Proof of Lemma 5.1.2. It suffices to show that if (A, m) is the local ring of S at p(x),and f1, ...., fr are elements of the maximal ideal m[[x1, ...., xn]] ⊂ A[[x1, ...., xn]]such that, modulo m, f1, ...., fr in C[[x1, ...., xn]] form a regular sequence, thenA[[x1, ...., xn]]/( f1, ...., fr) is a flat A algebra. Here we have passed to a (relatively)formal algebraic model of the smooth morphism p, and we note that in a Cohen–Macaulay local ring, such as C[[x1, ..., xn]], r elements of the maximal idealgenerate a height r ideal if and only if they form a regular sequence [11, 16.B,p. 105]. Then, since the local ring A is noetherian, by induction on r it suffices toshow that if a noetherian local A-algebra B is flat over A, f in B is such that f inB/m B is a non zero divisor, then B/( f ) is flat over A; see [11, 20.F, p. 151].

(for Lemma 5.1.2, Claim 5.1.1, and 5.1)

Proof of 5.2. In fact for any (locally noetherian, complex) scheme T , for the productX × T as a scheme over T , the relative Hilbert scheme Hilb((X × T )/T ) isisomorphic over T to the product scheme Hilb(X) × T . This property is actuallya special case of a basic general formula. Namely, given X → S and a base changeS′ → S, then Hilb(X ′/S′) ∼= Hilb(X/S) ×S S′, where X ′ = X ×S S′; takingS = Spec(C) and S′ = T , gives the special case in which X ′/S′ is (X × T )/T .


To prove the general formula we first recall the definition of the (relative) Hilbertfunctor.

Definition 5.2.1. (The “relative” Hilbert functor for X/S on the category oflocally noetherian schemes over S): given X/S, then for any scheme S′/S, letHilbX/S(S′) = closed subschemes of X ×S S′ that are flat over S′. Then HilbX/Sis a contravariant functor from S-Schemes to Sets. (See [5], pp. 10–11, for the caseof the Quot functor, and use QuotOX/X/S, or see [5], p. 17, directly for the Hilbertfunctor, (but with a small typographical error at the end), or [8], p. 9, 1.3.)

Claim 5.2.2. If HilbX/S is representable by an S-scheme H = Hilb(X/S) over S,(together with a universal closed subscheme U ⊂ X ×S Hilb(X/S) that is flat overHilb(X/S)), then for any scheme S′/S, if we set X ′ = X ×S S′, the contravariantfunctor HilbX′/S′ from S′-schemes to Sets is representable by the S′-scheme H ′ =H ×S S′ (together with the closed subscheme U′ = U ×S S′ of X ′ ×S′ H ′).

Proof. We are given, by the definition of representability of HilbX/S, that forevery scheme T/S, the natural map MorS(T,H ) → HilbX/S(T ) is bijective. Nowconsider a scheme T/S′ and the natural map MorS′(T,H ′) → HilbX′/S′(T ); wewant to check that this map is bijective. In fact, we claim that there are bijectionsMorS′(T,H ′) ∼= MorS(T,H ) and HilbX′/S′(T ) ∼= HilbX/S(T ), so that the presentmap MorS′(T,H ′) → HilbX′/S′(T ) is identified with the bijection MorS(T,H ) →HilbX/S(T ).

Since H ′ is the fiber product of H and S′ over S, an S′-morphism from T toH ′ uniquely corresponds to an S-morphism from T to H , i.e. MorS′(T,H ′) ∼=MorS(T,H ). Also, by definition of the functor HilbX′/S′ , HilbX′/S′(T ) = closedsubschemes of X ′ ×S′ T that are flat over T; but X ′ ×S′ T ∼= X ×S T , soHilbX′/S′(T ) ∼= HilbX/S(T ). For the compatibility of the 2 bijections here withthe other 2 maps, start with an element ϕ′ in MorS′(T,H ′). Then the elementof HilbX′/S′(T ) is obtained as (ϕ′)∗(U′) = U′ ×S′ T (using ϕ′ : T → H ′),while the element ϕ in MorS′(T,H ) corresponding to ϕ′ is simply the compositionT → H ′ → H (where the map H ′ → H is projection of the fiber product H ′ =H ×S S′ on the H factor), and then the resulting element of HilbX/S(T ) is obtainedas ϕ∗(U). Then the compatibility is that U′ ×S′ T = (U ×S S′) ×S′ T = U ×S T(under the identification of H ′ ×S′ T and H ×S T ). (for Claim 5.2.2)

Now to prove our original statement 5.2, suppose that the first order deformationX → D is trivial, i.e. X ∼= X × D as schemes over D (compatibly with the usualisomorphism X|0 ∼= X × 0 over the base point). Note that Hilb(X) does exist asa (locally noetherian) scheme because X is projective. Therefore, Hilb(X/D) ∼=Hilb((X × D)/D) is isomorphic over D to the product scheme Hilb(X) × D, byusing S′ = D, and S = point, in the discussion above. (for 5.2)

Proof of 5.3. The statement in 5.3 follows from 4.4 and 5.2, since C × D is thecorresponding connected component of Hilb(X) × D. (for 5.3)

Proof of 5.4. First we want a version of the embedding theorem above with param-eters. I.e. given a deformation C of C, and an induced deformation X of X, wewant to map C into Hilb(X/D).


Recall that our map of C into Hilb(X) is defined by a flat family of Cartierdivisors on X parametrized by C. I.e. we take the universal family of divisors onC of degree 2g − 2, defined by the incidence divisor I in C(2g−2) × C and definea divisor D in X × C by restricting the incidence divisor. Then this family is alsoa flat family over C of divisors on X, i.e. the projection D → C is also flat, andhence defines the morphism C → Hilb(X).

We want to do this construction with parameters. Given the deformation Cover D, we accept that the relative Hilbert scheme of C/D with a certain Hilbertpolynomial, is the relative symmetric product (C/D)(d), so that there is a universalsubscheme U ⊂ (C/D)(2g−2) ×D C flat over (C/D)(2g−2). This subscheme U isour parametrized version of the incidence divisor. We claim it is also a family ofCartier divisors on (C/D)(2g−2) flat over C. Then we will restrict it further to givea subscheme of X ×D C which we show is also flat over C, and hence definesa morphism from C to Hilb(X/D).

First we check that U is a Cartier divisor in (C/D)(2g−2) ×D C, i.e. that it islocally principal, using Nakayama, and is defined by a non zero divisor. To checkU locally principal, we use the fact it is a deformation of a Cartier divisor. First ofall the universal subscheme U of (C/D)(2g−2) ×D C, is flat over (C/D)(2g−2), andin turn (C/D)(2g−2) is flat over D, by the smoothness of C and C(2g−2). Thus bythe flatness of the composition of flat maps, U is flat overD. Then the special fiberof the map U → D is the incidence divisor I ⊂ C(2g−2) × C in a smooth ambientvariety, so I is a Cartier divisor, defined locally in the ring A of an affine set V × Wsay by f .

Now there is a corresponding affine open set in (C/D)(2g−2) ×D C, isomorphicto (V ×D) ×D (W ×D) ∼= V × W ×D, hence with affine ring isomorphic to A[ε].Now let J be the ideal in A[ε], corresponding to the intersection of the subschemeU with this affine open set. We wish to prove J is a principal ideal generated bysome element F which specializes to f mod ε. First we simply show that someelement F of J specializes to f .

Since f generates the ideal of the specialization of the subscheme corres-ponding to J , we have A[ε]/(ε, J ) ∼= A/( f ). Now look at the sequence 0 →J → A[ε] → A[ε]/J → 0, and tensor it with ⊗C[ε]C, to specialize it tothe central fiber, obtaining (since A[ε]/J is flat over C[ε]), the exact sequence0 → J/(εJ ) → A → A[ε]/(ε, J ) → 0. Now since f generates the ideal ofthe specialization of A[ε]/J , we have A[ε]/(ε, J ) ∼= A/( f ), and the sequencebecomes 0 → J/(εJ ) → A → A/( f ) → 0. Consequently, J/(εJ ) and ( f ) arekernels of the same map hence isomorphic. Then there is an element F of J whichbecomes equal to f in J/(εJ ), hence generates J mod ε. But by the Nakayamalemma (nilpotent ideal form, [11, p. 11]), then J = (F), so this F also generates J .

Next we check that F is not a zero divisor in A[ε]. But F specializes to f in A,so F = f + λε, for some λ in A. Thus if FG = 0, for G = g + µε, we have0 = ( f + λε)(g + µε) = fg + fµε + gλε, hence fg = 0 so g = 0, since f is nota zero divisor by hypothesis. Then also fµε = 0, so fµ = 0, hence µ = 0.

Thus U, as a flat deformation of a Cartier divisor, is itself a Cartier divisor. Nowwe want to show U is flat over C, via the other projection U ⊂ (C/D)(2g−2) ×D C→ C. By [11], p. 151, since U specializes over every closed point p of C to the


Cartier divisorDp (of divisors of degree 2g −2 on C that contain p), it follows thatU is flat over C. Next we want to restrict this family of divisors on (C/D)(2g−2), toa family of divisors on X.

I.e. restrict U to define D = U ∩ (X ×D C), getting a locally principalhypersurface D ⊂ X ×D C, and project D → C. We claim this D is a Cartierdivisor, and is flat over C as well. By [11] again, p. 151, it suffices to check thatover every closed point p of C, D specializes to a Cartier divisor. We know itspecializes over p to Dp which is a principal hypersurface in X (restriction of Dp

to X). Thus when C is non hyperelliptic, since both X and Dp are then irreducible,Dp is a Cartier divisor in X. More generally, X is reduced of pure dimensiong(C) − 1, and Dp is a finite cover of |ωC − p| + p, hence Dp is of pure dimensiong − 2. So again Dp is a Cartier divisor in X. Thus D is flat over C, and definesa morphism C → Hilb(X/D). (for 5.4)

Proof of 5.5. Next assuming X ∼= X × D (over D), we have Hilb(X/D) ∼=Hilb(X)×D, and we want to show the map C → Hilb(X)×D is an isomorphismof C onto the product with D of the component ∆ of Hilb(X) isomorphic to C.We know this is true over 0. Then since C is connected, it maps into ∆ ×D, whichis the connected component of Hilb(X) × D containing ∆ × 0. Now use thefollowing argument.

Consider a morphism g : C → C × D, over D, which reduces to the identityover 0 in D. Then we claim that g is an isomorphism. Namely, consider thecorresponding algebraic situation: a homomorphism h : A → B of C[ε] algebraswhich is an isomorphism mod ε. Letting K = ker(h) and L = cok(h), we havean exact sequence 0 → K → A → B → L → 0, of C[ε] modules. Then mod ε

(tensoring by ⊗C[ε]C), we get an exact sequence A/εA → B/εB → L/εL → 0.And since h induces an isomorphism mod ε, we have L/εL = 0, and henceby the nilpotent Nakayama lemma, L = 0. Thus we have an exact sequence0 → K → A → B → 0 of C[ε] modules.

Now using flatness of B asC[ε] module, TorC[ε]1 (B,C) = 0, (first Tor module

over C[ε]). Then we get an exact sequence after tensoring with C over C[ε], 0 →K/εK → A/εA → B/εB → 0. Since by hypothesis, h induces an isomorphismmod ε, A/εA → B/εB is injective, so K/εK = 0. Then by nilpotent Nakayamaagain, K = 0, and h : A → B is an isomorphism of C[ε] algebras.

To summarize this argument in geometric terms, consider any morphismg : Y1 → Y2 over D. Then, if g is a closed embedding over 0, g must bea closed embedding (without using that Y1 or Y2 is flat over D). If g reduces to anisomorphism over 0, and Y1 is flat overD, then g must be an isomorphism (withoutusing that Y2 is flat). (for 5.5)

This concludes the proof of the following result, which was the main goal ofthe paper. Informally, for any g ≥ 3, consider the factorization Rg → Φ → Ag−1,C/C → X → (P,Ξ), of the Prym map. Then the map Rg → Φ is unramified atall points C/C of Rg for which C is non hyperelliptic. (To be technically correct,Rg and Φ should be interpreted as moduli stacks instead of coarse moduli spaces.)Here is the precise statement in terms of first order deformations.


Theorem. Let C be a connected, smooth, projective, complex curve of genus g ≥ 3,let π : C → C be a connected etale double cover, and let X = D ∈ C(2g−2) :Nm(D) is a canonical divisor on C and h0(D) is even, the special variety ofdivisors associated to the double cover. Then, if C is non hyperelliptic, the map offirst order deformations, DefC/C(D) → DefX(D), is injective.

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Infinitesimal study of a factorization of the Prym map

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