Automorphisms of a Symmetric Product of a Curve

(with an Appendix by Najmuddin Fakhruddin)

Indranil Biswas and Tomás L. Gómez

Received: February 27, 2016 Revised: April 26, 2017

Communicated by Ulf Rehmann

Abstract. Let X be an irreducible smooth projective curve of genus g > 2 defined over an algebraically closed field of characteristic different from two. We prove that the natural homomorphism from the automorphisms of X to the automorphisms of the symmetric product Sym^{d}(X) is an isomorphism if d > 2g − 2. In an appendix, Fakhruddin proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve.

2000 Mathematics Subject Classification: 14H40, 14J50

Keywords and Phrases: Symmetric product; automorphism; Torelli theorem.

1. Introduction

Automorphisms of varieties is currently a very active topic in algebraic geometry; see [Og], [HT], [Zh] and references therein. Hurwitz’s automorphisms theorem, [Hu], says that the order of the automorphism group Aut(X) of a compact Riemann surface X of genus g ≥ 2 is bounded by 84(g − 1). The group of automorphisms of the Jacobian J(X) preserving the theta polarization is generated by Aut(X), translations and inversion [We], [La]. There is a universal constant c such that the order of the group of all automorphisms of any smooth minimal complex projective surface S of general type is bounded above by c · K_{S}^{2} [Xi]. Let X be a smooth projective curve of genus g, with g > 2, over an algebraically closed field of characteristic different from two. Take any integer d > 2g −2. Let Sym^{d}(X) be the d-fold symmetric product of X. Our aim here is to study the group Aut(Sym^{d}(X)) of automorphisms of the algebraic variety Sym^{d}(X). An automorphism f of the algebraic curve X produces an algebraic automorphism ρ(f) of Sym^{d}(X) that sends any {x_{1} , ⋯, x_{d}} ∈ Sym^{d}(X) to {f(x_{1}) , ⋯, f(x_{d})}. This map

ρ : Aut(X) ⟶ Aut(Sym^{d}(X)) , f ⟼ ρ(f)

is clearly a homomorphism of groups. We prove the following:

Theorem 1.1. The natural homomorphism

ρ : Aut(X) ⟶ Aut(Sym^{d} X)

is an isomorphism.

The idea of the proof of Theorem 1.1 is as follows. The homomorphism ρ is evidently injective, so we have to show that it is also surjective. The Albanese variety of Sym^{d}(X) is the Jacobian J(X) of X. So an automorphism of Sym^{d}(X) induces an automorphism of J(X). Using results of Fakhruddin (Appendix A) and Collino–Ran ([Co], [Ra]), we show that the induced automorphisms of J(X) respects the theta divisor up to translation. Invoking the strong form of the Torelli theorem for the Jacobian mentioned above, it follows that such automorphisms are generated by automorphisms of the curve X, translations of J(X), and the inversion of J(X) that sends each line bundle to its dual. Using a result of Kempf we show that if an automorphism α of J(X) lifts to Sym^{d}(X), then α is induced by an automorphism of X, and this finishes the proof. It should be clarified that we need a slight generalization of the result of Kempf [Ke]; this is proved in Section 2. The proof of Theorem 1.1 is in Section 3. In Appendix A by Fakhruddin the following is proved. Let C_{1} and C_{2} be smooth projective curves of genus g ≥ 2 over an algebraically closed field k. If Sym^{d} C_{1} ≅ Sym^{d} C_{2} for some d ≥ 1, then C_{1} ≅ C_{2} unless g = d = 2.

2. Some properties of the Picard bundle

The degree of a line bundle ξ over a smooth projective variety Z is the class of the first Chern class c_{1}(ξ) in the Néron-Severi group NS(Z), so the line bundles of degree zero on Z are classified by the Jacobian J(Z). As before, X is a smooth projective curve of genus g, with g > 2, over an algebraically closed field of characteristic different from two. For any integer d, let P^{d} = Pic^{d}(X) be the abelian variety that parametrizes the line bundles on X of degree d. It is a torsor for J(X). A branding of P^{d} is a Poincaré line bundle Q on X × P^{d} [Ke, p. 245]. Two brandings differ by tensoring with the pullback of a line bundle on P^{d}. A normalized branding is a branding such that Q|_{{x}×P^{d}} has degree zero for one point x ∈ X (equivalently, for all points of X). Two normalized brandings differ by tensoring with the pullback of a degree zero line bundle on P^{d}. The natural projection of X × P^{d} to P^{d} will be denoted by π_{P^{d}}. A normalized branding Q induces an embedding

I_{Q} : X ⟶ J(P^{d}) =: J  (2.1)

Automorphisms of Symmetric Product of a Curve 1183

that sends any x ∈ X to the point of J corresponding to the line bundle Q|_{{x}×P^{d}} on P^{d}. If

Q′ = Q ⊗ π_{P^{d}}^{∗}L_{j} ,

where L_{j} is the line bundle corresponding to a point j ∈ J(P^{d}) = J, then we have I_{Q′} = I_{Q} + j. Assume that d > 2g − 2. Since H^{1}(X, L) = H^{0}(X, L^{∗} ⊗ K_{X})^{∗} = 0 if L is a line bundle with degree(L) > 2g − 2, the direct image π_{P^{d}∗}Q, where Q is a branding, is locally free. A Picard bundle W (Q) on P^{d} is the vector bundle π_{P^{d}∗}Q, where Q is a normalized branding. From the projection formula it follows that two Picard bundles differ by tensoring with a degree zero line bundle on P^{d}. There is a version of the following proposition for d < 0 in [Ke, Corollary 4.4] (for negative degree, the Picard bundle is defined using the first direct image).

Proposition 2.1. Let d > 2g − 2.

(1) H^{1}(P^{d}, W (Q)) is non-zero (in fact, it is one-dimensional if it is nonzero) if and only if 0 ∈ I_{Q}(X).

(2) Let L_{j} be the line bundle on P^{d} corresponding to a point j ∈ J. Then H^{1}(P^{d}, L_{j} ⊗ W (Q)) is non-zero (in fact, it is one-dimensional if it is non-zero) if and only if −j ∈ I_{Q}(X).

Proof. Part (1). If 0 ∉ I_{Q}(X), then H^{1}(P^{d}, W (Q)) = 0 by [Ke, p. 252, Theorem 4.3(c)]. Fix a line bundle M on X of degree one, and consider the associated Abel-Jacobi map

X ⟶ J(X), x ⟼ M^{−1} ⊗ O_{X}(x) .

Let N_{X/J(X)} be the normal bundle of the image of X under this Abel-Jacobi map. If 0 ∈ I_{Q}(X), then using [Ke, p. 252, Theorem 4.3(d)] it follows that H^{1}(P^{d}, W (Q)) is canonically isomorphic to the space of sections of the skyscraper sheaf on X

K_{X}^{−1} ⊗ ∧^{0}N_{X/J} ⊗ M^{d}|_{I_{Q}^{−1}(0)} = K_{X}^{−1} ⊗ M^{d}|_{I_{Q}^{−1}(0)},

where I_{Q} is constructed in (2.1). But the space of sections of this skyscraper sheaf is clearly one-dimensional, because I_{Q}^{−1}(0) consists of one point of X. Part (2) follows from part (1) because L_{j} ⊗ W (Q) = W (Q ⊗ π_{P^{d}}^{∗}L_{j}), and I_{W (Q⊗π_{P^{d}}^{∗} L_{j})} = I_{W (Q)} + j. □

For a point j ∈ P^{d}, by −j we denote the point of P^{−d} corresponding to the dual of the line bundle corresponding to j. Note that for j, j′ ∈ P^{d}, we have −j + 2j′ ∈ P^{d}.

Proposition 2.2. Assume that g(X) > 1 and d > 2g − 2. Let j be a point of Pic^{0}(X), and let T_{j} : P^{d} ⟶ P^{d} be the translation by j. Let M be a degree zero line bundle on P^{d}. If

T_{j}^{∗}(M ⊗ W (Q)) ≅ W (Q) ,

then j = 0 and M = O_{P^{d}}.

Let i : P^{d} ⟶ P^{d} be the inversion given by z ⟼ −z + 2z_{0}, where z_{0} is a fixed point in P^{d}. If

i^{∗}T_{j}^{∗}(M ⊗ W (Q)) ≅ W (Q) ,

then X is a hyperelliptic curve.

Proof. The first part is [Ke, Proposition 9.1] except that there it is assumed that d < 0; the proof of Proposition 9.1 uses [Ke, Corollary 4.4] which requires this hypothesis. However, the case d > 2g − 2 can be proved similarly; for the convenience of the reader we give the details. Let y ∈ J be the point corresponding to the line bundle M. The line bundle on P^{d} corresponding to any t ∈ J will be denoted by L_{t}. In particular, M = L_{y}. For every t ∈ J, using the hypothesis, we have

T_{j}^{∗}(L_{t+y} ⊗ W (Q)) = T_{j}^{∗}L_{t} ⊗ T_{j}^{∗}(M ⊗ W (Q)) = L_{t} ⊗ W (Q) ;  (2.2)

note that the fact that a degree zero line bundle on an Abelian variety is translation invariant is used above. Combining (2.2) and the fact that T_{j} is an isomorphism, we have

H^{1}(P^{d}, L_{t} ⊗ W (Q)) ≅ H^{1}(P^{d}, T_{j}^{∗}(L_{t+y} ⊗ W (Q))) ≅ H^{1}(P^{d}, L_{t+y} ⊗ W (Q)) .

Using Proposition 2.1 it follows that t ∈ −I_{Q}(X) if and only if t+y ∈ −I_{Q}(X). Hence I_{Q}(X) = y + I_{Q}(X). If g(X) > 1, this implies that y = 0. Therefore, we have W (Q) = T_{j}^{∗}(W (Q)). Using the fact that c_{1}(W (Q)) = θ, a theta divisor, it follows that θ is rationally equivalent to the translate θ − j, hence j = 0. The proof of the second part is similar. We have

i^{∗}T_{j}^{∗}(L_{y−t} ⊗ W (Q)) = i^{∗}T_{j}^{∗}L_{−t} ⊗ i^{∗}T_{j}^{∗}(M ⊗ W (Q))
= i^{∗}L_{−t} ⊗ W (Q) = L_{t} ⊗ W (Q) ;  (2.3)

the fact that i^{∗}L_{−t} = L_{t} is used above. Consequently,

H^{1}(P^{d}, L_{t}⊗W (Q)) ≅ H^{1}(P^{d}, i^{∗}T^{∗}(L_{y−t}⊗W (Q))) ≅ H^{1}(P^{d}, L_{y−t}⊗W (Q)) ,

and using [Ke, Corollary 4.4] it follows that t ∈ −I_{Q}(X) if and only if y − t ∈ −I_{Q}(X). Hence I_{Q}(X) = −I_{Q}(X) − y. Let

f : X ⟶ X

be the morphism uniquely determined by the condition

I_{Q}(x) = −I_{Q}(f(x)) − y .

We note that f is well defined because −I_{Q} and y + I_{Q} are two embeddings of X in J with the same image, so they differ by an automorphism of X which is f. In other words, if we identify X with its image under I_{Q}, then f is induced from the automorphism T_{−y} ∘ i of J. This automorphism T_{−y} ∘ i is clearly an involution. Let ω ∈ H^{0}(X, Ω_{X}) be an algebraic 1-form on X. Then f^{∗}ω = −ω, because of the isomorphism H^{0}(X, Ω_{X}) = H^{0}(J, Ω_{J}) induced by I_{Q}, and the fact that i^{∗} acts as multiplication by −1 on the 1-forms on J. It now follows by Lemma 2.3 that f is a hyperelliptic involution. □

Lemma 2.3. Let g > 1. Let f : X ⟶ X be an involution satisfying the condition that f^{∗}ω = −ω for every 1-form ω. Then X is hyperelliptic with f being the hyperelliptic involution.

Proof. Consider the canonical morphism

F : X ⟶ ℙ(H^{0}(X, Ω_{X}))

that sends any x ∈ X to the hyperplane H^{0}(X, Ω_{X}(−x)) in H^{0}(X, Ω_{X}). By definition,

H^{0}(X, Ω_{X}(−x)) = {ω ∈ H^{0}(X, Ω_{X}) | ω(x) = 0} ,

but the hypothesis implies that ω(x) = 0 if and only if ω(f(x)) = f^{∗}(ω)(x) = 0. Therefore, we have

H^{0}(X, Ω_{X}(−x)) = H^{0}(X, Ω_{X}(−f(x))) ,

and it follows that F (x) = F (f(x)), implying that the canonical morphism is not an embedding; note that f is not the identity because there are nonzero algebraic 1-forms. Therefore, X is hyperelliptic, and f is the hyperelliptic involution. □

We note that Lemma 2.3 is clearly false if the characteristic of the base field is two. Hence the proof of Proposition 2.2 needs the assumption that the base field has characteristic different from two.

3. Proof of Theorem 1.1

Using the morphism X ⟶ Sym^{d}(X), y ⟼ dy, it follows that the homomorphism ρ in Theorem 1.1 is injective. Fix a point x ∈ X. Let L be the normalized Poincaré line bundle on X ×J(X), i.e., it is trivial when restricted to the slice {x} × J(X). Let

E := q_{∗}(L ⊗ p^{∗}O_{X}(dx))

be the Picard bundle, where p and q are the projections from X × J(X) to X and J(X) respectively. Since d > 2g − 2, it follows that E is a vector bundle of rank d − g + 1. We will identify Sym^{d}(X) with the projective bundle P (E) = ℙ(E^{∨}). Let θ be the theta divisor of J(X); in particular, we have θ^{g} = g!. The Chern classes of E are given by c_{i}(E) = θ^{i}/i! [ACGH]. Let Z be a smooth projective variety and z_{0} ∈ Z a point. Then there is an abelian variety Alb(Z) and a morphism

a_{Z} : Z ⟶ Alb(Z)

such that a_{Z}(z_{0}) = 0, and given any morphism ϕ : Z ⟶ A, where A is an abelian variety and ϕ(z_{0}) = 0, there is a unique homomorphism h : Alb(Z) ⟶ A such that h ∘ a_{Z} = ϕ. The Alb(Z) is called the Albanese variety for (Z, z_{0}) while a_{Z} is called the Albanese morphism. The Albanese variety of (P (E), dx), where x is the fixed point of X, is the Jacobian J(X), and the Albanese morphism sends an effective divisor ∑_{ℓ=1}^{d} P_{ℓ} of degree d to the degree zero line bundle O_{X}((∑_{ℓ=1}^{d} P_{ℓ}) − dx). Given an automorphism

φ : P (E) ⟶ P (E) ,

the universal property of the Albanese variety yields a commutative diagram

\xymatrix{P (E) \ar[r]_{≅}^{φ} \ar[d] & P (E) \ar[d] \\ J(X) \ar[r]_{≅}^{α} & J(X)}  (3.1)

and this produces an automorphism of projective bundles

\xymatrix{P (E) \ar[rr]^{ψ}_{≅} \ar[dr] &  & P (α^{∗}E) \ar[dl] \\  & J(X)}

Therefore, there is a line bundle L on J(X) such that there is an isomorphism

α^{∗}E ≅ E ⊗ L .  (3.2)

There is a commutative diagram of groups

\xymatrix{Aut(P (E)) \ar[r]^{λ} & Aut(J(X)) \\ Aut(X) \ar[u]^{ρ} \ar[ur]_{μ}}  (3.3)

where λ is constructed as above using the universal property of the Albanese variety given in (3.1), and ρ is the homomorphism in Theorem 1.1. To construct μ, note that the commutativity of the diagram (3.1) implies that μ(f), f ∈ Aut(X), has to send O_{X}((∑_{ℓ=1}^{d} P_{ℓ}) − dx) to O_{X}((∑_{ℓ=1}^{d} f(P_{ℓ})) − dx). A short calculation yields

μ(f) = (f^{−1})^{∗} ∘ T_{dx−df^{−1}(x)} ,  (3.4)

where T_{a}, a ∈ J(X), is translation on J(X) by a. Let θ′ = c_{1}(α^{∗}E) = θ + L. Then

c_{i}(α^{∗}E) = α^{∗}c_{i}(E) = \frac{α^{∗}θ^{i}}{i!} = \frac{θ′^{i}}{i!} ,

and θ′^{g} = α^{∗}θ^{g} = g!. Now we apply Lemma A.2; here the condition g > 2 is used. We obtain that θ^{i} = θ′^{i} for all i > 1. We identify X with the image in J(X) of the Abel-Jacobi map. In particular X is numerically equivalent to θ^{g−1}/(g − 1)!. We calculate the intersection (note that the condition g > 2 is again used, because we need g − 1 > 1)

θ′X = θ′ \frac{θ^{g−1}}{(g − 1)!} = θ′ \frac{θ′^{g−1}}{(g − 1)!} = g .

Invoking a characterization of a Jacobian variety due to Collino and Ran, [Co], [Ra], it follows that (J(X) , θ′ , X) is a Jacobian triple, i.e., θ′ is a theta divisor of the Jacobian variety J(X) up to translation. This means that θ and θ′ differ by translation, in other words, the class of c_{1}(L) in the Néron-Severi group NS(J(X)) is zero. Consequently, α is an isomorphism of polarized Abelian varieties, i.e., it sends θ′ to a translate of it. The strong form of the classical Torelli theorem ([La, Théorème 1 and 2 of Appendix]) tells us that such an automorphism α is of the form

α = F ∘ σ ∘ T_{a} , σ ∈ {1 , ι} ,

where F = (f^{−1})^{∗} for an automorphism f of X, while T_{a} is translation by an element a ∈ J(X) and ι sends each element of J(X) to its inverse. If X is hyperelliptic, then ι is induced by the hyperelliptic involution, so we may assume that σ is the identity map of X when X is hyperelliptic. Let f be an automorphism of X with F = (f^{−1})^{∗} being the induced isomorphism on J(X). Using the definition of E, it is easy to check that

F^{∗}E ≅ T_{dx′−dx}^{∗}E ,

where x′ = f^{−1}(x). We claim that α = F ∘ T_{a}. To prove this, assume that α ≠ F ∘ T_{a}. Then X is not hyperelliptic, and α = F ∘ ι ∘ T_{a}. Hence

α^{∗}E = T_{a}^{∗}ι^{∗}F^{∗}E = T_{a}^{∗}ι^{∗}T_{dx′−dx}^{∗}E ,

and using (3.2),

E ≅ ι^{∗}T_{dx′−dx−a}^{∗}(E ⊗ L) .

Now from Proposition 2.2 it follows that X is hyperelliptic, and we arrive at a contradiction. This proves the claim. Summing up, we can assume that α = F ∘ T_{a}. Using (3.2),

E ≅ T_{dx−dx′−a}^{∗}(E ⊗ L)

From Proposition 2.2 it follows that L is the trivial line bundle, and a = dx − dx′. Therefore,

α = (f^{−1})^{∗} ∘ T_{dx−df^{−1}(x)}

for some automorphism f of X, and hence, by (3.4),

Image(λ) ⊂ Image(μ) .  (3.5)

We will now show that the morphism λ is injective. Suppose α = λ(φ) = Id_{J(X)}. Using (3.2), the morphism φ is induced by an isomorphism between E and E ⊗ L. We have just seen that L is trivial, the morphism φ is induced by an automorphism of E, and this automorphism has to be multiplication by a nonzero scalar, because E is stable with respect to the polarization given by the theta divisor (cf. [EL]). Therefore, the morphism φ is the identity. This proves that the morphism λ is injective. The homomorphism μ is also injective, since it is a composition of a translation and the pullback induced by an automorphism of X.

Combining these it follows that the morphism ρ is also injective (this can also be checked directly), and hence all the homomorphisms in the diagram (3.3) are injective. This, combined with (3.5), shows that ρ is an isomorphism.

Acknowledgements

We thank the referee and the editors for helpful suggestions. The second author wants to thank Tata Institute for Fundamental Research and the University of Warwick, where part of this work was done. The authors want to thank the support by MINECO (ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554 and grants MTM2013-42135-P and MTM2016-79400-P) and the Spanish Ministry of Education for the grant “Salvador de Madariaga” PRX14/00508. We acknowledge the support of the grant 612534 MODULI within the 7th European Union Framework Programme.

Appendix A. Torelli’s theorem for high degree symmetric products of curves

Najmuddin Fakhruddin

Let k be an algebraically closed field and C_{1} and C_{2} two smooth projective curves of genus g > 1 over k . It is a consequence of Torelli’s theorem that if Sym^{g−1} C_{1} ≅ Sym^{g−1} C_{2}, then C_{1} ≅ C_{2}. The same holds for the d-th symmetric products, for 1 ≤ d < g−1 as a consequence of a theorem of Martens [Mar]. We shall show that with one exception the same result continues to hold for all d ≥ 1, i.e., we have the following

Theorem A.1. Let C_{1} and C_{2} be smooth projective curves of genus g ≥ 2 over an algebraically closed field k. If Sym^{d} C_{1} ≅ Sym^{d} C_{2} for some d ≥ 1, then C_{1} ≅ C_{2} unless g = d = 2.

It is well known that there exist non-isomorphic curves of genus 2 over C with isomorphic Jacobians. Since the second symmetric power of a genus 2 curve is isomorphic to the blow up of the Jacobian in a point, it follows that our result is the best possible.

Proof of Theorem. Let C_{1}, C_{2} be two curves of genus g > 1 with Sym^{d} C_{1} ≅ Sym^{d} C_{2} for some d ≥ 1. Since the Albanese variety of Sym^{d} C_{i}, d ≥ 1, is isomorphic to the Jacobian J(C_{i}), it follows that J(C_{1}) ≅ J(C_{2}). If d ≤ g − 1, the theorem follows immediately from [Mar], since the image of Sym^{d} C_{i} in J(C_{i}) (after choosing a base point) is W_{d}(C_{i}). Note that in this case it suffices to have a birational isomorphism from Sym^{d} C_{1} to Sym^{d} C_{2}. Suppose g ≤ d ≤ 2g − 3. Then the Albanese map from Sym^{d} C_{i} to J(C_{i}) is surjective with general fiber of dimension d − g. Interpreting the fibers as complete linear systems of degree d on C_{i}, it follows by Serre duality that the subvariety of J(C_{i}) over which the fibers are of dimension > d − g is isomorphic to W_{2g−2−d}(C_{i}). Therefore if Sym^{d} C_{1} ≅ Sym^{d} C_{2}, then W_{2g−2−d}(C_{1}) ≅ W_{2g−2−d}(C_{2}), so Martens’ theorem implies that C_{1} ≅ C_{2}.

Now suppose that d > 2g − 2 and g > 2. By choosing some isomorphism we identify J(C_{1}) and J(C_{2}) with a fixed abelian variety A. If ϕ : Sym^{d} C_{1} → Sym^{d} C_{2} is our given isomorphism, from the universal property of the Albanese morphism we obtain a commutative diagram

\xymatrix{Sym^{d} C_{1} \ar[r]^{ϕ} \ar[d]_{π_{1}} & Sym^{d} C_{2} \ar[d]_{π_{2}} \\ A \ar[r]^{f} & A}

where the π_{i}’s are the Albanese morphisms corresponding to some base points and f is an automorphism of A (not necessarily preserving the origin). By replacing C_{2} with f^{−1}(C_{2}) we may then assume that f is the identity. Since d > 2g − 2, the maps π_{i}, i = 1, 2 make Sym^{d} C_{i} into projective bundles over A. By a theorem of Schwarzenberger [Sc], Sym^{d} C_{i} ≅ ℙ(E_{i}), where E_{i} is a vector bundle on A of rank d − g + 1 with c_{j}(E_{i}) = [W_{g−j}(C_{i})], i = 1, 2, 0 ≤ j ≤ g −1, in the group of cycles on A modulo numerical equivalence. Since ϕ is an isomorphism of projective bundles, it follows that there exists a line bundle L on A such that E_{1} ≅ E_{2} ⊗ L. Let θ_{i} = [W_{g−1}(C_{i})], so by Poincaré’s formula [W_{g−j}(C_{i})] = θ_{i}^{j}/j!, i = 1, 2, i ≤ j ≤ g − 1. Lemma A.2 below implies that θ_{i}^{g−1} = θ_{2}^{g−1} in the group of cycles modulo numerical equivalence on A. Since θ_{i}^{g} = g!, this implies that θ_{1} · [C_{2}] = g. By Matsusaka’s criterion [Mat], it follows that W_{g−1}(C_{1}) is a theta divisor for C_{2}, which by Torelli’s theorem implies that C_{1} ≅ C_{2}. If d = 2g − 2 and g > 2, then we can still apply the previous argument. In this case we also have that Sym^{d}(C_{i}) ≅ ℙ(E_{i}), i = 1, 2 but E_{i} is a coherent sheaf which is not locally free. However on the complement of some point of A it does become locally free and the previous formula for the Chern classes remains valid. The above argument clearly does not suffice if g = 2. To handle this case we shall use some properties of Picard bundles for which we refer the reader to [Mu]. Suppose that d > 2 and C_{i}, i = 1, 2 are two non-isomorphic curves of genus 2 with Sym^{d} C_{1} ≅ Sym^{d} C_{2}. Using the same argument (and notation) as the g > 2 case, it follows that there exist embeddings of C_{i}, i = 1, 2, in A and a line bundle L on A such that E_{1} ≅ E_{2} ⊗ L and L^{⊗d−1} ≅ O(C_{1} − C_{2}) (we identify C_{i}, i = 1, 2 with their images). For i ≥ 1, let G_{i} denote the i-th Picard sheaf associated to C_{2}, so that ℙ(G_{i}) ≅ Sym^{i}(C_{2}). (G_{i} is the sheaf denoted by F_{2−i} in [Mu] and G_{d} ≅ E_{2}). There is an exact sequence ([Mu, p. 172]):

0 → O_{A} → G_{i} → G_{i−1} → 0  (A.1)

for all i > 1. We will use this exact sequence and induction on i to compute the cohomology of sheaves of the form E_{1} ⊗ P ≅ E_{2} ⊗ L ⊗ P, where P ∈ Pic^{0}(A). Consider first the cohomology of G_{1}, which is the pushforward of a line bundle of degree 1 on a translate of C_{2}. Since we have assumed that C_{1} ≅ C̸_{2}, it follows that C_{1} · C_{2} > 2. Since C_{1}^{2} = C_{2}^{2} = 2, deg(L|_{C_{2}}) = (C_{1} − C_{2}) · C_{2}/(d − 1) > 0. By Riemann-Roch it follows that h^{j}(A, G_{1} ⊗ L ⊗ P), j = 1, 2 is independent of P, except possibly for one P if deg(L|_{C_{2}}) = 1, and h^{2}(A, G_{1} ⊗ L ⊗ P) = 0 since G_{1} is supported on a curve. Now C_{1} · C_{2} > 2 also implies that c_{1}(L)^{2} < 0. By the index theorem, it follows that h^{0}(A, L ⊗ P) = h^{2}(A, L ⊗ P) = 0 and h^{1}(A, L ⊗ P) is independent of P. Therefore by tensoring the exact sequence (A.1) with L ⊗ P and considering the long exact sequence of cohomology, we obtain an exact sequence

0 → H^{0}(A, G_{i} ⊗ L ⊗ P) → H^{0}(A, G_{i−1} ⊗ L ⊗ P) → H^{1}(A, L ⊗ P)

→ H^{1}(A, G_{i} ⊗ L ⊗ P) → H^{1}(A, G_{i−1} ⊗ L ⊗ P) → 0 (A.2)

and isomorphisms H^{2}(A, G_{i} ⊗ L ⊗ P) → H^{2}(A, G_{i−1} ⊗ L ⊗ P) for all i > 1. By induction, it follows that H^{2}(A, G_{i} ⊗ L ⊗ P) = 0 for all i > 0. Since the Euler characteristic of G_{i}⊗L⊗P is independent of P, the above exact sequence (A.2) along with induction shows that for all i > 0 and j = 0, 1, 2, h^{j}(G_{i} ⊗ L ⊗ P) is independent of P, except for possibly one P. In particular, this holds for i = d hence h^{j}(A, E_{1} ⊗ P) is independent of P except again for possibly one P. We obtain a contradiction by using the computation of the cohomology of Picard sheaves in Proposition 4.4 of [Mu]: This implies that h^{1}(A, E_{1} ⊗ P) is one or zero depending on whether the point in A corresponding to P does or does not lie on a certain curve (which is a translate of C_{1}). □

Lemma A.2. Let X be an algebraic variety of dimension g ≥ 3 and let E_{i}, i = 1, 2 be vector bundles on X of rank r. Suppose c_{1}(E_{i}) = θ_{i}, c_{j}(E_{i}) = θ_{i}^{j}/j! for i = 1, 2 and j = 2, 3 (j = 2 if g = 3), and E_{1} ≅ E_{2} ⊗L for some line bundle L on X. Then θ_{1}^{j} = θ_{2}^{j} for all j > 1 (j = 2 if g = 3).

Proof. Since E_{1} ≅ E_{2} ⊗L, c_{1}(E_{1}) = c_{1}(E_{2})+rc_{1}(L), hence c_{1}(L) = (θ_{1} −θ_{2})/r. For a vector bundle E of rank r and a line bundle L on any variety, we have the following formula for the Chern polynomial ([Fu], page 55):

c_{t}(E ⊗ L) = ∑_{k=0}^{r} t^{k}c_{t}(L)^{r−k}c_{i}(E).

Letting E = E_{1}, E ⊗ L = E_{2}, and expanding out the terms of degree 2 and 3, one easily sees that θ_{1}^{j} = θ_{2}^{j} for j = 2 and also for j = 3 if g > 3. (Note that this only requires knowledge of c_{j}(E_{i}) for j = 1, 2, 3.) Since any integer n > 1 can be written as n = 2a + 3b with a, b ∈ N, the lemma follows. □

Remark A.3. Using Theorem 1.1, one sees that Theorem A.1 holds over all perfect fields k (of characteristic > 2) if d > 2g − 2: For projective varieties X, Y over a field let \underline{Isom}(X, Y) denote the scheme of isomorphisms. For any d > 0, there is a natural map

\underline{Isom}(C_{1}, C_{2}) → \underline{Isom}(Sym^{d} C_{1}, Sym^{d} C_{2}) of finite schemes over k which one sees is a bijection on geometric points by combining Theorem 1.1 and Theorem A.1. If k is perfect^{1} this implies that the map on k-rational points is also a bijection.

Acknowledgements

I thank A. Collino for some helpful correspondence, in particular for informing me about the paper [Mar].

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Indranil Biswas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400005 India indranil@math.tifr.res.in Tomás L. Gómez Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) C/ Nicolas Cabrera 15 28049 Madrid Spain tomas.gomez@icmat.es

Najmuddin Fakhruddin School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Mumbai 400005 India naf@math.tifr.res.in