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The automorphism group of the moduli space

arXiv:alg-geom/9306001v1 2 Jun 1993The automorphism group of the moduli spaceof semi stable vector bundlesAlexis Kouvidakis∗Tony PantevUniversity of PennsylvaniaDepartment of Mathematics, DRLPhiladelphia, PA 19104-6395alex@math.upenn.edupantev@math.upenn.edu0IntroductionTo every smooth curve C one can associate a natural variety U(r, d) - the moduli space of semi stablevector bundles of rank r and degree d on C. This canonical object has a rich geometrical structurewhich reflects in a beautiful way the geometry of the curve C and its deformations. In the abeliancase r = 1 this relationship is classical and goes back to the Riemann inversion problem, the theoryof Jacobi theta-functions and the Torelli theorem.

Understanding the subtleties of the non-abeliancase has resulted over the years in a multitude of new ideas and unexpected connections with otherbranches of mathematics. One of the remarkable similarities between the Jacobian varieties andthe higher rank moduli spaces is the existence of theta line bundles on them, see [D-N].

These areample determinantal line bundles, naturally arising from the interpretation of U(r, d) as a modulispace and provide valuable information about the projective geometry of U(r, d).In this paper we describe the groups of automorphisms and of polarized automorphisms ofU(r, d) for r > 1. To understand the problem better, consider first the case r = 1.

It is well knownthat for any integer d, the group of automorphisms of the Jacobian Jd(C) = U(1, d) is isomorphicto the semi-direct productJ0(C) ⋊Autgroup(J0(C)) ∼= Aut(Jd(C)),where Autgroup(J0(C)) is the group of group automorphisms of J0. The above isomorphism dependson the choice of a point L0 ∈Jd(C) and is given explicitly by (ξ, φ) →Tξ ◦TL0 ◦φ◦T −1L0 .

Moreover,0New address: Department of Mathematics, University of Crete, Iraklion 71409, Greece.1

the subgroup of automorphisms preserving the class of the theta bundle is justAutθ(Jd(C)) ∼=(J0(C) ⋊((±id) × Aut(C))C−not hyperelliptic,J0(C) ⋊Aut(C)C−hyperelliptic.Some natural automorphisms of U(r, d) can be obtained by analogy with this case. For example,the Jacobian J0(C) acts by translations on the moduli space and, as it turns out, it actuallycoincides with the identity component of the automorphism group.

Furthermore, we have a naturalmorphismdet : U(r, d) −→Jd(C)(0.1)sending a bundle E to its determinant det(E). The fibers of this morphism are moduli spaceson its own right, e.g.

the fiber det−1(L0) over a point L0 is just the moduli space SU(r, L0) ofsemistable bundles of rank r and fixed determinant L0. One can use the fibration (0.1) to constructsome natural automorphisms of U(r, d), that is, certain automorphisms of the fiber SU(r, L0) canbe glued with certain automorphisms of the base Jd(C) to produce global automorphisms of themoduli space.

After choosing a point L0 ∈Jd(C) we can trivialize the bundle (0.1) by pulling itback to an ´etale cover of Jd(C):SU(r, L0) × J0(C)U(r, d)J0(C)Jd(C)✲(E,L)→E⊗L❄p2❄det✲L→L⊗r⊗L0(0.2)The top and the bottom rows of the diagram (0.2) are Galois covers with Galois group the groupof r-torsion points J0(C)[r].Consider the subgroup Aut[r](J0(C)) ⊂Aut(J0(C)) consisting ofgroup automorphisms of J0(C) which act trivially on the torsion points J0(C)[r]. Then given atranslation, Tµ, by a torsion point µ ∈J0(C)[r] and ϕ ∈Aut[r](J0(C)), the automorphism Tµ × ϕof SU(r, L0) × J0(C) commutes with the action of the Galois group and hence descends to anautomorphism of U(r, d).

In addition, every symmetry of the curve C will induce an automorphismof U(r, d).In the special case r|2d, by choosing ν ∈J2dr (C) with property ν⊗r = L⊗20 , we can constructan additional automorphism δ of order two:δ :U(r, d)−→U(r, d)E−→E∨⊗νTo combine these, consider the subgroups UG◦[r] ⊆UG[r] ⊆Aut(J0(C)) × Aut(C) defined by2

Definition 0.1UG◦[r] = {(φ, σ) | φ−1 ◦σ∗∈Aut[r](J0(C))},andUG[r] = {(φ, σ) | φ−1 ◦(±1) ◦σ∗∈Aut[r](J0(C))}.The group UG◦[r] is of index two in UG[r] and we have a natural homomorphism:J0(C) ⋊UG◦[r]−→Aut(U(r, d))(ξ, φ, σ)−→(E 7→σ∗E ⊗ξ ⊗φ(η) ⊗σ∗(η−1))(0.3)where η ∈J0(C) is an arbitrary line bundle with the property η⊗r = det E ⊗L−10 . If in additionr | 2d, then the homomorphism (0.3) can be extended toJ0(C) ⋊UG[r]−→Aut(U(r, d))(ξ, φ, σ)−→E 7→(σ∗E ⊗ξ ⊗φ(η) ⊗σ∗(η−1)if φ−1 ◦σ∗∈Aut[r](J0(C))σ∗E∨⊗ξ ⊗φ(η) ⊗σ∗(η)if φ−1 ◦(−1) ◦σ∗∈Aut[r](J0(C))(0.4)The commutative diagram (0.2) suggests that the subgroups (0.3) (or (0.4) in the case r | 2d)will not differ too much from the full automorphism group, if the moduli space SU(r, L0) doesn’thave excess automorphisms.

The variety SU(r, L0) has a lot of advantages. It is a Fano variety ofPicard number one and by a theorem of Narasimhan and Ramanan, [N-R 2], H0(SU(r, L0), TSU(r, L0)) =0 unless g = 2, r = 2 and d even.

Therefore, in general, Aut(SU(r, L0)) is finite and contains thegroup J0(C)[r]. In the case r = 2, odd degree and g = 2 the group Aut(SUs(2, L0)) was describedby Newstead [Ne] as a consequence of his proof of the Torelli theorem for the variety SUs(2, L0).To generalize his result we adopt a different viewpoint - we use Hitchin’s abelianization to proveour main theoremTheorem A Let C be a curve without automorphisms.Then the automorphism group of themoduli space SU(r, L0) can be described as follows1.

If r ∤2d, then the natural mapJ0(C)[r]−→Aut(SU(r, L0))µ−→(E 7→E ⊗µ)is an isomorphism, and2. If r | 2d, then the natural mapJ0(C)[r] ⋊Z/2Z−→Aut(SU(r, L0))(µ, ε)−→(E 7→δε(E) ⊗µ)is isomorphism for r ≥3 and has kernel Z/2Z for r = 2.3

The strategy of the proof is to lift the automorphism Φ to a birational automorphism of the modulispace of Higgs bundles and then study the induced automorphism ˜Φ ˜d of the family Prym ˜d( ˜C, C)of smooth spectral Pryms.To determine the map ˜Φ ˜d explicitly we study the ring of relativecorrespondences over the universal spectral curve and show that the family of groups Prym( ˜C, C)does not have non-trivial global group automorphisms. Furthermore, we show that all the sectionsof Prym ˜d( ˜C, C) come from pull-back of line bundles on the base curve C and conclude that ˜Φ ˜d mustbe a multiplication by ±1 along the fibers of Prym ˜d( ˜C, C) followed by a translation by a pull-backbundle.

To finish the proof, we use the rational map from a spectral Prym to the moduli spaceSU(r, L0) to recover from ˜Φ ˜d the original automorphism Φ.For curves with automorphisms the statement of the main theorem has to be modified appro-priately. To avoid some technical complications which occur in the case g = 2, we assume thatg ≥3 throughout the paper.

Consider the subgroups SG◦[r] ⊆SG[r] ⊆J0(C) × Aut(C) defined asDefinition 0.2SG◦[r] = {(ξ, σ) | ξr = L0 ⊗σ∗(L−10 )},andSG[r] = {(ξ, σ) | ξr = L0 ⊗σ∗(L±10 )}.Again SG◦[r] is a normal subgroup of index two in SG[r] and we haveTheorem B Let C be any smooth curve of genus g ≥3. We have1.

If r ∤2d, then the natural mapSG◦[r]−→Aut(SU(r, L0))(ξ, σ)−→(E 7→σ∗E ⊗ξ)is surjective, and2. If r | 2d, then the natural mapSG[r]−→Aut(SU(r, L0))(ξ, σ)−→E 7→(σ∗E ⊗ξif (ξ, σ) ∈SG◦[r]σ∗E∨⊗ξif (ξ, σ) ∈SG[r] \ SG◦[r]is surjective.Remark 0.1 The maps in the above Theorem B are actually isomorphisms.

Our proof of theinjectivity involves some standard arguments from Prym theory similar to those in [N-R 1]; theproof is not included here because of its length.4

The first part of Theorem B gives a positive answer to a question posed by Tyurin in the surveypaper [Ty].The main theorem, together with the observation that every automorphism of U(r, d) must liftto an automorphism of SU(r, L0) × J0(C), see Lemma 6.3, givesTheorem C Let C be a smooth curve of genus g ≥3. Then the automorphisms of the full modulispace U(r, d) can be described as follows1.

If r ∤2d, then the map (0.3)J0(C) ⋊UG◦[r]֒→Aut(U(r, d))(ξ, φ, σ)→(E 7→σ∗E ⊗ξ ⊗φ(η) ⊗σ∗(η−1))is surjective, and2. If r|2d, then the map (0.4)J0(C) ⋊UG[r]֒→Aut(U(r, d))(ξ, φ, σ)→E 7→(σ∗E ⊗ξ ⊗φ(η) ⊗σ∗(η−1)if φ−1 ◦σ∗∈Aut[r](J0(C))σ∗E∨⊗ξ ⊗φ(η) ⊗σ∗(η)if φ ◦σ∗∈Aut[r](J0(C))is surjective.The description of the polarized automorphisms of U(r, d) follows easily from Theorem C:Theorem D For any curve C of genus g ≥3 the automorphisms of the moduli space U(r, d)which preserve the class of the theta bundle are those belonging to the image of the subgroupJ0(C) ⋊((±id) × Aut(C)) ⊆J0(C) ⋊UG[r] under the map (0.4).As an application of the technics we use in this work, we prove in the Appendix at the end ofthe paper the Torelli Theorem for the moduli space of vector bundles:Theorem E Let C1, C2 be smooth curves of genus g ≥3 and L1, L2 line bundles of degree d onC1, C2 respectively.

If SUC1(r, L1) ≃SUC2(r, L2), then C1 ≃C2.The paper is organized as follows. In the first section we gather all the facts about the modulispace of Higgs bundles, the linear system of spectral curves and the Hitchin map which we are goingto use later on.

We have included the proofs of some statements which seem to be well known tothe experts in the field but which can not be found in the standard sources, as well as, the proofsof some facts about the discriminant locus in the Hitchin base which help us simplify our mainargument. In the second section we construct the induced automorphism ˜Φ ˜d and study its first5

properties. The third section contains a discussion of the relative Picard of the universal spectralcurve and its sections over a Zariski open set in the Hitchin base.

In section four we analyze thering of relative correspondences on the fibers of the universal spectral curve whose description isthe key ingredient in the proof of the main theorem. In the fifth section we give the proof of themain theorem and discuss the case of curves with symmetries.

In the sixth section we derive thedescription of Aut(U(r, d)).Acknowledgments: We are very grateful to Ron Donagi for his constant support and for manyvaluable discussions and suggestions. We would like to thank Eyal Markman for explaining to usthe proof of Lemma 1.4 and George Pappas for helpful conversations.Notation and ConventionsSU(r, L0) : The moduli space of semi stable vector bundles of rank r and fixed determinant L0 ofdegree d.SUs(r, L0) : The moduli space of stable vector bundles of rank r and fixed determinant L0 of degreed.X(r, L0) : The total space of the cotangent bundle T ∗SUs(r, L0).M(r, L0) : The moduli space of semi stable Higgs pairs.H : M(r, L0) −→W : The Hitchin map.C : A smooth curve of genus g ≥3.ωC : The canonical bundle on C.S◦: The total space of the canonical bundle ωC.S : The space P(O ⊕ωC).α : S −→C : The canonical map.Y ∈H0(S, OS(1)) : The infinity section of S.X ∈H0(S, OS(1) ⊕α∗ωC) : The zero section of S.x = X/Y : The tautological section of S◦.Y∞: The divisor at infinity div(Y ) ⊂S.X0 : The zero divisor div(X) ⊂S◦.W = H0(C, O) ⊕H0(C, ω⊗2C ) ⊕· · · ⊕H0(C, ω⊗rC ).Wk = H0(C, ω⊗kC ).W ≃W2 ⊕· · · ⊕Wr, embedded in W as {1} ⊕W2 ⊕· · · ⊕Wr.W ∞≃W2 ⊕· · · ⊕Wr, embedded in W as {0} ⊕W2 ⊕· · · ⊕Wr.D : The discriminant divisor.W reg = W \ D.˜Cs : The spectral curve of genus ˜g = r2(g −1) + 1 associated to s ∈W.6

πs : ˜Cs −→C : The projection to the base curve C.π : ˜C −→C : The universal spectral curve and its projection to C.β : ˜C −→S◦: The map to S◦.Prym( ˜Cs, C) : The prymian of the map πs : ˜Cs −→C.Prym( ˜C, C) : The universal prymian associated to the map π : ˜C −→C.Prym ˜d( ˜Cs, C) : The “prymian” of degree ˜d = d+r(r−1)(g−1), i.e. the set {˜L ∈J ˜d( ˜Cs) | det π∗˜L =L0}.Prym ˜d( ˜C, C) : The universal spectral “prymian” of degree ˜d.1Spectral curves1.1Linear systems of spectral curvesLet C be a smooth curve of genus g. Let S◦be the total space of the line bundle ωC →C and letS = P(ωC ⊕OC) be the projective extension of ωC.

Denote by α : S →C the natural projectionand let OS(1) be the relative hyperplane bundle on S corresponding to the vector bundle ωC ⊕OC.Definition 1.1 An r-sheeeted spectral curve is an element eC of the linear system |α∗ω⊗rC ⊗OS(r)|having the property eC ⊂S◦The adjective spectral refers to another interpretation of the curve eC which we recall next. For anyvector bundle E of rank r over C and any ωC-twisted endomorphism θ ∈H0(C, EndE ⊗ωC) of Ethere is a suitable notion of a characteristic polynomial.

Define the i-th characteristic coefficientsi ∈H0(C, ω⊗iC ) of θ as si = (−1)i+1tr(∧iθ).The characteristic polynomial P(x) of θ can bewritten formally as P(x) = xr + s1xr−1 + . .

. + sr. Geometrically the polynomial P(x) correspondsto a subcheme eCs ⊂S◦which via the projection α is an r-sheeted branch cover of C. Indeed, letX ∈H0(C, α∗ωC ⊗OS(1)) be the zero section of the ruled surface S and let Y ∈H0(C, OS(1)) bethe infinity section of S. By setting x := XY , i.e.

x is the tautological section of α∗ωC →S◦, wecan reinterpret Y ⊗rP(x) = Xr + s1Xr−1Y + . .

. srY r as a section of α∗ω⊗rC ⊗OS(r) whose divisoris contained in S◦, that is - a spectral curve, for more details see [B-N-R], [Hi 1].Remark 1.1 We can view the space fW := ⊕ri=1H0(C, ω⊗iC ) as the space of characteristic polyno-mials of ωC-twisted endomorphisms of rank r vector bundles.

The vector space fW can be identifiedexplicitly as the locus of spectral curves in |α∗ω⊗rC ⊗OS(r)| as follows. The push forward map α∗induces an isomorphismα∗: H0(S, α∗ω⊗rC ⊗OS(r)) −→H0(C, α∗(α∗ω⊗rC ⊗OS(r))).7

On the other hand,α∗(α∗ω⊗rC ⊗OS(r))=ω⊗rC ⊗α∗(OS(r)) = ω⊗rC ⊗(Symr(OC ⊕ω−1C ) ==OC ⊕ωC ⊕. .

. ⊕ωrC.Therefore we get an isomorphismα∗: H0(S, α∗ω⊗rC ⊗OS(r)) −→⊕ri=0H0(C, ω⊗iC ),whose inverse is⊕ri=0H0(C, ω⊗iC )−→H0(S, α∗ω⊗rC ⊗OS(r))(s0, s1, .

. .

, sr)−→s0Xr + s1Xr−1Y + . .

. + srY r.Here we have identified the spaces H0(C, ω⊗iC ) and α∗H0(C, ω⊗iC ) ⊂H0(C, α∗ω⊗iC ) via α.For any s = (s0, s1, .

. .

, sr) denote by eCs the divisor Div(s0Xr + s1Xr−1Y + . .

. + srY r) onS.

Let Div(X) = X0 and Div(Y ) = Y∞. By definition the curve eCs is spectral if eCs ⊂S◦, orequivalently, if eCs∩Y∞= ∅.

Since X0 ∩Y∞= ∅it follows that eCs∩Y∞̸= ∅if and only if Y∞⊂eCs,i.e. if and only if s0 = 0.Thus the locus of all spectral curves in the projective space |α∗ω⊗rC ⊗OS(r)| is the affine openset fW = { eCs | s0 ̸= 0}.For the purposes of this paper we will be mainly concerned with the slightly smaller spaceW := H0(C, ω⊗2C ) ⊕H0(C, ω⊗3C ) ⊕.

. .

⊕H0(C, ω⊗rC ),consisting of the characteristic polynomials of traceless twisted endomorphisms of rank r vectorbundles. Let h : ˜C →W be the family of spectral curves parametrized by W. To construct thevariety ˜C consider the line bundle p∗S(α∗ω⊗rC ⊗OS(r)) →W × S. There is a natural tautologicalsection c ∈H0(W × S, p∗S(α∗ω⊗rC ⊗OS(r))) given byc(((s2, .

. .

, sr); p)) = Xr(p) + s2(p)Xr−2(p)Y 2(p) + . .

. + sr(p)Y r(p),and ˜C = Div(c) is just the divisor of this section.The universal family ˜C is proper over W and admits a natural compactification to a projectivevariety C - the universal family for the linear system P(H0(C, OC)⊕W).

Set W := H0(C, OC)⊕W.Next we are going to study the linear system P(W) and the total spaces of the universal families ˜Cand C.Lemma 1.1 The linear system P(W) is base point free.Furthermore, if r ≥3 the morphismfP(W) : S →P(W)∨is an inclusion when restricted to S◦and contracts the infinity section Y∞.8

Proof. Let p ∈S.

If p ∈Y∞, then the curve Div(Xr) ∈P(W) does not pass through p. If p /∈Y∞,then choose a section sr ∈H0(C, ω⊗rC ) not vanishing at α(p). The curve Div(srY r) ∈P(W) doesnot pass trough p.Consider the morphismf|W| : S −→P(W)∨.Since a curve eC ∈|α∗ω⊗rC ⊗OS(r)| is either spectral or contains Y∞, see Remark 1.1, it follows thatfP(W) contracts Y∞.If p ̸= q ∈S are points in the same fiber F of α, then they are separated by P(W) because bypulling P(W) back on F we get the linear systemP(Span(xr0, xr−20x21, .

. .

, xr1)) ⊂|OF (r)|,which for r ≥3 separates the points on F ∼= P1.If p ̸= q ∈S are points in two different fibers of α, then by choosing a section sr ∈H0(C, ω⊗rC )satisfying sr(α(p)) = 0 and sr(α(q)) ̸= 0, we get a curve Div(srY r) passing trough p and notpassing through q.The above considerations and the local triviality of S◦→C show that P(W) separates also thetangent directions because for every point p ∈S◦the morphism fP(W) maps the fiber to hrough pand a suitable translate of X0 into two transversal curves in P(W)∨.✷Here are some immediate corollaries from the above lemma which are going to be usefull lateron.Corollary 1.1 The generic spectral curve ˜Cs is smooth.Proof. Follows from the fact that P(W) does not have base points and from Bertini’s theorem.✷Corollary 1.2 The total spaces of the universal families ˜C and C are smooth.Proof.

Since ˜C ⊂C is a Zariski open set it is enough to show that C is smooth.Let π : U →P(W)∨be the universal bundle of linear hypersurfaces in P(W). The total spaceC fits in the fiber product diagramCUSP(W)∨✲❄❄π✲fP(W )9

But both S and U are smooth varieties and moreover the projection π : U →P(W)∨is a smoothmap. Therefore the fiber product C = Sf|W |×π P is also smooth.✷Let W r−1,r (resp.W ∞r−1,r) be the linear subsystem of W consisting of points of the form(s0, 0, .

. .

, sr−1, sr) (resp. (0, 0, .

. .

, sr−1, sr)). According to Remark 1.1, W r−1,r (resp.

W ∞r−1,r)can be embedded naturally in the vector space H0(S, α∗ω⊗rC ⊗OS(r)) and thus, |(W r−1,r| (resp.|(W ∞r−1,r|) can be viewed as a linear subsystem of |α∗ω⊗rC ⊗OS(r)|. The arguments in the proof ofLemma 1.1 yield the following corollary.Corollary 1.3 The linear system W r−1,r (resp.

W ∞r−1,r) is base point free. Furthermore if r ≥3,then the morphism f|W r−1,r| : S →P(W r−1,r)∨(resp.

f|W ∞r−1,r| : S →P(W ∞r−1,r)∨) is an inclusionwhen restricted to S◦and contracts the infinity section Y∞.In the next section we are going to prove the irreducibility of various discriminant loci. Thefollowing fact is an essential ingredient in the proof of Corollary 1.6 and is also of independentinterest.Proposition 1.1 Let eC be a spectral curve of degree r and let Σ ⊂eC be an irreducible componentof eC.

Then Σ is a spectral curve of degree l ≤r.Proof. Since Σ ⊂eC ⊂S◦, we only need to show thatOS(Σ) = OS(l) ⊗α∗ω⊗lC ,for some l ≤r.

But Pic(S) = Z · OS(1) ⊕α∗Pic(C), and henceOS(Σ) = OS(l) ⊗α∗L,for some line bundle L on C. Since Σ ⊂S◦we have Σ · Y∞= 0 which yields0 = OS(Σ) · OS(1) = OS(l) · OS(1) + α∗L · OS(1) = −l(2g −2) + deg L,i.e. deg L = l(2g −2).On the other hand the line bundle OS(l) ⊗α∗L has a section Σ which does not vanish on theinfinity divisor Y∞.

But the sections of OS(l) ⊗α∗L vanishing on Y∞are H0(S, OS(l) ⊗α∗L ⊗OS(−Y∞)) = H0(S, OS(l −1) ⊗α∗L). By pushing forward OS(l −1) ⊗α∗L on C we getH0(S, OS(l −1) ⊗α∗L)=H0(C, α∗(OS(l −1)) ⊗L) = H0(C, Syml−1(OC ⊕ω−1C ) ⊗L) ==H0(C, L ⊕(L ⊗ω−1C ) ⊕.

. .

⊕(L ⊗ω−(l−1)C)) = ⊕l−1i=0H0(C, L ⊗ω−iC ).10

Similarly H0(S, OS(l) ⊗α∗L) = ⊕li=0H0(C, L ⊗ω−iC ) and henceH0(S, OS(l) ⊗α∗L)/H0(S, OS(l −1) ⊗α∗L) ∼= H0(C, L ⊗ω−lC ).The section of the divisor Σ gives a non-zero element in H0(S, OS(l)⊗α∗L)/H0(S, OS(l−1)⊗α∗L)and therefore H0(C, L ⊗ω−lC ) ̸= 0. Since deg(L ⊗ω−lC ) = 0 we get that L ⊗ω−lC ∼= OC.✷Remark 1.2 The meaning of the above proposition becomes transparent if we use the interpreta-tion of the spectral curves as characteristic polynomials of ωC-twisted endomorphisms.Let F →eC be a line bundle.

Then F can be viewed as a sheaf on S supported on eC ⊂S◦.Consider the rank r vector bundle E = α∗F on C. The push-forward of the homomorphismF⊗x−→F ⊗α∗ωCis a ωC-twisted endomorphism of E:θ : E −→E ⊗ωC.In this way the curve eC then can be described as the zero scheme of the sectiondet(α∗θ −x · idα∗E) ∈H0(S, α∗ω⊗rC ⊗OS(r)).Let now Σ ⊂eC be a component of eC. Consider the sheaf F ⊗OΣ on S supported on Σ. LetE′ := α∗(F ⊗OΣ) and letθ′ : E′ −→E′ ⊗ωCbe the push forward of the homomorphismF ⊗OΣ⊗x−→(F ⊗OΣ) ⊗α∗ωC.The commutative diagram of (torsion) sheaves on S◦FF ⊗α∗ωCF ⊗OΣ(F ⊗OΣ) ⊗α∗ωC✲⊗x✻✲⊗x✻11

then pushes down to a commutative diagram of vector bundles on C:EE ⊗ωCE′E′ ⊗ωC✲θ✲θ′✻✻In particular E′ is θ invariant and θ′ = θ|E′. Then Σ is the zero scheme of the sectiondet(α∗θ′ −x · idα∗E′) ∈H0(S, α∗ω⊗lC ⊗OS(l)),where l = rkE′, i.e.

Σ is a spectral curve.Corollary 1.4 The locus R⊂Wconsisting of reducible spectral curves has codimensioncodim(R, W) ≥g −1Proof. Set Wi := H0(C, ω⊗iC ).

Let Rr be the image of R under the natural projection W →Wr.It suffices to show that codim(Rr, Wr) ≥g −1 or equivalently, since Rr is invariant under dilations,that codim(P(Rr), P(Wr)) ≥g −1.For every i = 1, . .

. , r −1 consider the variety Si ⊂P(Wr) × P(Wi) defined bySi = {(D, G) | D ≥G}.Since a component of a spectral curve is a spectral curve we haveP(Rr) ⊂pP(Wr)(S1) ∪.

. .

∪pP(Wr)(Sr−1),and therefore codim(P(Rr), P(Wr))≥min1≤i≤r−1{codim(pP(Wr)(Si), P(Wr))}.Furthermoredim(pP(Wr)(Si)) = dim Si because the map pP(Wr) : Si −→P(Wr) is finite on its image. To computedim Si look at the second projectionpP(Wi) : Si −→P(Wi).

(1.5)The map (1.5) is obviously onto and for any G ∈P(Wi) we get by Riemann-Rochdim p−1P(Wi)(G) = h0(C, ω⊗rC (−G)) −1 = h0(C, ω⊗(r−i)C) −1 ≤(2(r −i) −1)(g −1).Consequently by the fiber-dimesion theorem we get dim Si ≤(2r −2)(g −1) −1 and hencecodim(pP(Wr)(Si), P(Wr)) = g −1,for any i = 1, . .

. , r −1.✷12

1.2Discriminant lociAs we saw in Corollary 1.1 the generic spectral curve is smooth. Therefore the space W of spectralcurves contains a natural divisor parametrizing the singular spectral curves:D := {s ∈W | ˜Cs−is not smooth},which due to Corollary 1.2 is just the discriminant locus for the map h : ˜C →W.The divisor D is irreducible.

To see this consider the full family fW of spectral curves and itsdiscriminant divisor eD. We have a natural inclusion W ֒→fW and clearly D = W ∩eD.

The firststep will be to show that eD is irreducible and the second to deduce the irreducibility of D fromthat. To achieve this we need to introduce some auxilliary objects.Let Affbe the additive group of the vector space H0(C, ωC).

There is a natural affine action ofAffon the bundle ωC which can be considered as an action on its total spaceτ :Aff−→Aut(S◦)γ−→(pτγ7→p + γ(α(p))).The action τ on S◦has a natural lift to an action on the bundle α∗ωC →S◦which gives an affineaction of Affon the space of its global sections: γ →(µ 7→µ + α∗γ). Consequently we obtain apolynomial action on the space of spectral curvesρ :Aff−→Aut(fW)γ−→(xr + s1xr−1 + .

. .

+ srργ7→(x + γ)r + s1(x + γ)r−1 + . .

. + sr).For every γ ∈Affwe have a commutative diagram˜C˜CfWfW✲τγ❄h❄h✲ργ(1.6)where τγ is the automorphism of the universal spectral curve induced by τγ ∈Aut(S◦).Proposition 1.2 The discriminant divisor eD ⊂fW is irrreducible.Proof.

Consider the incidence correspondence eΓ ⊂fW × S◦defined byeΓ := {(s, p) ∈fW × S◦| Ordp ˜Cs ≥2}.Since eD = p eW (eΓ) it suffices to show that eΓ is irreducible. The commutativity of the diagram (1.6)implies that eΓ is ρ-invariant.

Furthermore, if s′ = ργ(s), then s′1 = s1 + rγ and hence ρ is a freeaction. Therefore we can form the quotient variety eΓ/Aff.

Since the fibrationeΓ −→eΓ/Aff13

is a locally trivial affine bundle, the irreducibility of eΓ is equivalent to the irreducibility of eΓ/Aff.The fibers of the projectionpC := α ◦pS◦: eΓ −→Care ρ-invariant and by taking the quotient we obtain a morphismpC : eΓ/Aff−→C.Let Ξ ⊂fW × C be the incidence correspondenceΞ = {(s, a) ∈fW × C | ˜Cs is singular at the point α−1(a) ∩X0}.More explicitly by using the Jacobian criterion for smoothnes one gets, see [B-N-R] Remark 3.5,Ξ = {(s, a) ∈fW × C | Div(sr) ≥2a, Div(sr−1) ≥a}.The fact that ω⊗rCis very ample for any r ≥2, implies that the fibration Ξ →C is a vector bundleof rank dim fW −3. On the other hand we have a natural morphism of fibrationsΞeΓ/AffC✲t❅❅❅❘✠pCwhere t((s, a)) = OrbAff((s, α−1(a)∩X0)).

It is easy to see that t is onto. Indeed, if (s, p) ∈eΓ, thenOrdp( ˜Cs) ≥2.

Choose γ ∈H0(C, ωC) with the property γ(α(p)) = p. Then(ρ−γ, τ−γ) · (s, p) = (ρ−γ(s), α−1(α(p)) ∩X0)),and henceOrbAff((s, p)) = OrbAff((ρ−γ(s), α−1(α(p)) ∩X0))).Consequently t is onto and eΓ/Affis irreducible.✷Remark 1.3 The variety eΓ/Aff(and therefore eΓ) is actually smooth. Indeed, let Ξa be the fiberof the bundle Ξ over the point a ∈C.

Let (s, a), (s′, a) ∈Ξa be such that t((s, a)) = t((s′, a)). ThenOrbAff((s, α−1(a) ∩X0)) = OrbAff((s′, α−1(a) ∩X0)),and therefore s′ = ργ(s) for some γ ∈H0(C, ωC) satisfying γ(a) = 0.14

Let Ξ0 ⊂Ξ be the subbundleΞ0 = {(s, a) ∈Ξ | Div(s1) ≥a}.The morphism t descends to an isomorphism of fibrationsΞ/Ξ0eΓ/AffC✲t❅❅❅❘✠pCand thus eΓ/Aff∼= Ξ/Ξ0 is smooth. Furthermore, observe that p eW : eΓ →eD is birational because forthe generic s ∈eD the curve ˜Cs has a unique ordinary double point.

Therefore eΓ can be viewed asa natural desingularization of eD.Corollary 1.5 The discriminant divisor D ⊂W is irreducible.Proof. Consider the incidence correspondenceΓ = {(s, p) | Ordp( ˜Cs) ≥2}.Let q|Γ : Γ →eΓ/Affbe the restriction to Γ of the natural quotient morphism q : eΓ →eΓ/Aff.

If(s, p) ∈eΓ, then (ρ−1r ·s1(s), τ−1r ·s1(s)) ∈Γ. Combined with the fact that ρ acts freely, this yieldsΓ ∩OrbAff((s, p)) = {(ρ−1r ·s1(s), τ−1r ·s1(s))}.Consequently q|Γ is an isomorhism and in particular Γ and D are irreducible.✷There are two other discriminant loci whose irreducibility will be usefull.Define Wr :=H0(C, ω⊗rC ) and Wr−1,r := H0(C, ω⊗(r−1)C)⊕H0(C, ω⊗rC ).

Let Dr = D∩Wr and Dr−1,r = D∩Wr−1,rbe the discriminant divisors for the families of spectral curves Wr and Wr−1,r respectively.Proposition 1.3 The discriminant divisor Dr ⊂Wr is irreducible.Proof. Consider the incidence correspondence Γr ⊂Wr × C defined byΓr = {(sr, a) | Div(sr) ≥2a}.Using the Jacobian criterion it is easy to check that a curve ˜Cs, s ∈Wr is singular if and only if srhas a double zero.15

Therefore Dr = pWr(Γr) and it suffices to show that Γr is irreducible.The divisor Γr isequivariant with respect to the natural C× action on Wr (extended trivially to Wr × C). Thus theproblem reduces to showing that P(Γr) ⊂P(Wr) × C is irreducible.Consider the projection pC : P(Γr) →C.

By Riemann-Roch pC is onto and codim(p−1C (a), P(Γr))= 2 for every a ∈C. Moreover, p−1C (a) ⊂P(Wr) is a linear subspace and hence all the fibers of pCare equidimensional and irreducible.

Therefore P(Γr) is irreducible.✷Remark 1.4 The divisor P(Dr) is a divisor in the projective space of the complete linear systemP(H0(C, ω⊗rC )). This allows us to give more geometric description of P(Dr):P(Dr) = P({H ⊂P(W ∨r )−hyperplane | H ⊃P(Taf|Wr|(C)) for some a}),that is, P(Dr) = f|Wr|(C)∨is the dual hypersurface of the r-canonical model of the curve C.Proposition 1.4 The divisor Dr−1,r ⊂Wr−1,r is irreducible.Proof.

Consider the incidence varietyΓr−1,r = {(a, b, p) ∈Wr−1,r × S◦| Ordp(xr + ax + b) ≥2}.Again pWr−1,r(Γr−1,r) = Dr−1,r so it suffices to show that Γr−1,r is irreducible.Consider the projectionpS◦: Γr−1,r −→S◦.Let (U, z) be a coordinate chart on C. Then we can choose natural coordinates (w, z) : α−1(U)f→C×U on α−1(U), so that the tautological section x is x = wα∗(dz).Let (z0, w0) ∈α−1(U). Then for any (a, b) ∈Wr−1,r one has in a neighborhood of (z0, w0):a=(a0 + a1(z −z0) + .

. .

)dz⊗(r−1)=f(z)dz⊗(r−1)b=(b0 + b1(z −z0) + . .

. )dz⊗r=g(z)dz⊗r.Therefore in the local coordinates (z, w) the equation of the curve xr + ax + b becomeswr + f(z)w + g(z) = 0.

(1.7)The conditions for (1.7) to have singularity at (z0, w0) areb0=0(rwr−1 + f(z))|(z0,w0)=0(f ′(z)w + g′(z))|(z0,w0)=016

or equivalentlyb0=0rwr−10+ a0=0a1w0 + b1=0. (1.8)The equations (1.8) determine a codimension 3 affine subspace of Wr−1,r and hence Γr−1,r|α−1(U) −→α−1(U) is a trivial affine bundle.

Therefore Γr−1,r is an affine bundle over S◦and hence is irre-ducible.✷Corollary 1.6 Let h : ˜C →W be the universal spectral curve and let hr−1,r : ˜C|Wr−1,r →Wr−1,rand hr : ˜C|Wr →Wr be the subfamilies parametrized by Wr−1,r and Wr respectively. Then thedivisors h−1(D), h−1r−1,r(Dr−1,r) and h−1r (Dr) are irreducible.Proof.

Recall the following standard lemma, see [Sh],Lemma 1.2 Let f : X →Y be a proper map between algebraic varieties of pure dimension. If Yis irreducible, the general fiber of f is irreducible and all the fibers are equidimensional then X isirreducible.As we saw above each of the discriminant loci Dr−1,r, Dr and D is irreducible.

According toLemma 1.2 it suffices to show that the generic fiber of each of the maps h, hr−1 and hr−1,r isirreducible. But, by Corollary 1.4, there exists a point s ∈Dr ⊂Dr−1,r ⊂D such that ˜Cs isirreducible which finishes the proof.✷1.3The Hitchin mapFor a line bundle L0 ∈Picd(C) denote by SU(r, L0) the moduli space of semistable vector bundlesof rank r and determinant L0.

It is well known that SU(r, L0) is a normal projective variety whosesmooth locus coincides with the locus SUs(r, L0) of stable vector bundles, see [Se]. Since C is acurve, the deformations of any vector bundle E →C are unobstructed and the Kodaira-SpencermapT[E]SUs(r, L0) −→H1(C, EndoE),(1.9)is an isomorphism; here EndoE is the bundle of traceless endomorphisms of E.Using the isomorphism (1.9) and Serre’s duality, one gets a canonical identificationT ∗[E]SUs(r, L0) ∼= H0(C, EndoE ⊗ωC),17

of the fiber of the cotangent bundle to SUs(r, L0) at the point [E] and the space of ωC-twistedtraceless endomorphisms. Therefore the collection of characteristic coefficients defined in Section1.2 gives rise to a morphismH :T ∗SUs(r, L0)−→W(E, θ)−→(s2(θ), s3(θ), .

. .

, sr(θ)). (1.10)Hitchin [Hi 1] was the first one to study the morphism (1.10) in various situations.

He discoveredmany of its remarkable properties and in particular he showed that H endows the (holomorphic)symplectic manifold X(r, L0) := T ∗SUs(r, L0) with a structure of an algebraically completely inte-grable hamiltonian system. More specifically, he showed that there exists a partial compactificationX(r, L0)M(r, L0)W✲◗◗◗sH✑✑✑✰H(1.11)with general fiber an abelian variety and such that the fibers of (1.10) embed as Zariski opensets in the fibers of H : M(r, L0) →W.

The variety M(r, L0) is again a moduli space whichparametrizes the equivalence classes of semistable Higgs pairs - that is, pairs (E, θ) of a bundle andan ωC-twisted endomorphism such that for every θ-invariant subbundle U ⊂E the usual inequalityµ(U) ≤µ(E) for the slopes of U and E holds. The characteristic coefficients map (1.10) is a welldefined morphism on the whole variety M(r, L0) and is called the Hitchin map of M(r, L0).

Thereis a natural C×-action on the moduli space M(r, L0)(E, θ) −→(E, tθ),which induces via the Hitchin map a weighted C×-action on the vector space W. A number t ∈C×acts on W by multiplying the piece H0(C, ω⊗iC ) with ti.The compactification diagram (1.11)and its generalizations have been studied extensively in the last years, see [B-N-R], [Hi 1],[Hi 2],[Ni], [Si]. For our purposes, the most convenient is the approach in [B-N-R] which we proceed todescribe.Proposition 1.5 (Beauville-Narasimhan-Ramanan) Assume that πs : ˜Cs →C is an integralspectral curve.

Then the push-forward map πs∗induces a canonical bijection between• Isomorphism classes of rank one torsion-free sheaves F on ˜Cs of Euler characteristic d −r(g −1) satisfying det(πs∗F) = L0.• Isomorphism classes of Higgs pairs (E, θ) satisfying det E = L0, trθ = 0, H(θ) = s.18

Remark 1.5 Let s ∈W be such that ˜Cs is integral. Then every Higgs pair satisfying H(θ) = s isstable.

Indeed, if we assume that there exists a proper θ-invariant subbundle U ⊂E, then the spec-tral curve of (U, θ|U) will be contained in ˜Cs which contadicts the integrality of ˜Cs. ConsequentlyE does not have θ-invariant subbundles and in particular (E, θ) is stable.Under the bijection in Proposition 1.5 this translates into the well known fact that the modulifunctor of rank one torsion free sheaves on an integral curve possesing only planar singularitiesis representable by an irreducible fine moduli space - the compactified Jacobian of the curve, see[A-I-K].

Therefore the push-forward map πs∗induces an isomorphismπs∗: Prym ˜d( ˜Cs, C) −→H−1(s),where Prym ˜d( ˜Cs, C) = {F ∈Jd−r(g−1)( ˜Cs) | det(πs∗F) = L0}, and Jd−r(g−1)( ˜Cs) is the generalizedJacobian parametrizing rank one torsion free sheaves of Euler characteristic d −r(g −1) on ˜Cs.Remark 1.6 If the curve ˜Cs is smooth, then the Prymian Prym ˜d( ˜Cs, C) is in a natural way torsorover an abelian variety. Indeed, it is easy to see thatdet(πs∗F) = Nmπs∗(F) ⊗det(πs∗O ˜Cs),for any line bundle F on ˜Cs.

Since πs∗O ˜Cs = OC ⊕ω−1C ⊕. .

. ⊕ω−(r−1)C, see [B-N-R], we getPrym ˜d( ˜Cs, C) = {F ∈J˜d( ˜Cs) | Nmπs∗(F) = L0 ⊗ω⊗r(r−1)2C},with ˜d = d + r(r −1)(g −1).Remark 1.7 We will need more explicit geometric description for the fiber H−1(s) of the Hitchinmap over a generic point s ∈D of the discriminant.

Observe first that there exists a point s ∈Dfor which ˜Cs is irreducible and has a unique ordinary double point as singularity.Indeed, letsr ∈H0(C, ω⊗rC ) be a section having a unique double zero a ∈C. It follows then by the Jacobiancriterion for smoothness that the curve˜Cs : Xr + srY r = 0,has a unique node lying over a. Consequently, by upper semicontinuity there is a non-empty Zariskiopen set Do ⊂D satisfyingDo = {s ∈D | ˜Cs is irreducible and has a unique ordinary double point}.19

Let now s ∈Do and say p ∈˜Cs is its ordinary double point.Set a = πs(p).To studythe geometric properties of Prym ˜d( ˜Cs, C), we recall the construction of the compactified JacobianJd−r(g−1)( ˜Cs) in this simple case. Let˜Cνs˜CsC✲ν❅❅❅❅❘πνs❄πsbe the normalization of the curve ˜Cs.

Let ν−1(p) = {p+, p−} ⊂˜Cνs . For any rank one torsion freesheaf F →˜Cs we have the exact sequence0 −→F −→ν∗ν∗F −→Cp −→0.

(1.12)Therefore χ(F) + 1 = χ(ν∗ν∗F) = χ(ν∗F) and hence the pull-back induces a morphismν∗: Jd−r(g−1)( ˜Cs) −→Jd+1−r(g−1)( ˜Cνs ),(1.13)where Jd+1−r(g−1)( ˜Cνs ) is the component of the Picard group of ˜Cνs consisting of line bundles ofEuler characteristic d + 1 −r(g −1). One can show that (1.13) is a bundle with structure groupC× and with fibers isomorphic to a P1 glued at two points.

To identify the bundle (1.13) explicitly,consider a Poincare bundlePJd+1−r(g−1)( ˜Cνs ) × ˜Cνs❄and set P+ = P|Jd+1−r(g−1)( ˜Cνs )×{p+} and P−= P|Jd+1−r(g−1)( ˜Cνs )×{p−}. The P1-bundleP(P+ ⊕P−) −→Jd+1−r(g−1)( ˜Cνs )does not depend on the choice of the Poincare bundle P and is furnished with two sections X+ andX−corresponding to P+ and P−respectively.

Furthermore, there is a bundle isomorphismJd−r(g−1)( ˜Cs)P(P+ ⊕P−)/X+ ∼X−Jd+1−r(g−1)( ˜Cνs )✲∼=❍❍❍❍❍❍❍❥ν∗✟✟✟✟✟✟✟✟✙To describe the subvariety Prym ˜d( ˜Cs, C) ⊂Jd+1−r(g−1)( ˜Cs) in these terms, we need the followinglemma20

Lemma 1.3 If F1, F2 ∈Jd+1−r(g−1)( ˜Cs) are such thatdet(πs∗F1) = det(πs∗F2),thendet(πνs∗ν∗F1) = det(πνs∗ν∗F2).Proof. For any F ∈Jd+1−r(g−1)( ˜Cs) we have the short exact sequence (1.12).

After taking directimages, we get0πs∗Fπs∗ν∗ν∗Fπs∗CpR1πs∗Fπνs∗ν∗FCa0✲✲✲✲i.e. we have a short exact sequence of sheaves on C:0 −→πs∗F −→πνs∗ν∗F −→Ca −→0.Therefore the bundle πs∗F is a Hecke transform of the bundle πνs∗ν∗F with center at an (r −1)-dimensional subspace of the fiber (πνs∗ν∗F)a. Thusdet(πs∗F) = det(πνs∗ν∗F) ⊗OC(−x),which yields the statement of the lemma.✷According to the above lemma, if a point F ∈Jd+1−r(g−1)( ˜Cs) belongs to the subvarietyPrym ˜d( ˜Cs, C), then every point in the fiber (ν∗)−1(ν∗(F)) also belongs to this subvariety.Define Prym( ˜Cνs , C) := {F ∈Jd+1−r(g−1)( ˜Cνs ) | det(πνs∗F) = L0(−a)}.

Then Prym ˜d( ˜Cs, C) fitsin the fiber squarePrym ˜d( ˜Cs, C)Jd+1−r(g−1)( ˜Cs)Prym( ˜Cνs , C)Jd+1−r(g−1)( ˜Cνs )✲❄ν∗❄ν∗✲In particular, Prym ˜d( ˜Cs, C) is a bundle over the abelian variety Prym( ˜Cνs , C) whose total space issingular along a divisor and has a smooth normalization which is a P1-bundle over Prym( ˜Cνs , C).The total space X(r, L0) of the cotangent bundle of SUs(r, L0) is a Zariski open set in themoduli space of Higgs bundles M(r, L0). It has the advantage that we have morphisms both to21

SUs(r, L0) and W:X(r, L0)SUs(r, L0)W✟✟✟✙Π❍❍❍❍❥Hwhose fibers are generically transversal. The natural projection Π : X(r, L0) →SUs(r, L0) is onto bydefinition and the Hitchin map H is dominant by an argument of Beauville-Narasimhan-Ramanan,see [B-N-R].

In particular, this implies that for the generic element s ∈W the push-forward mapπs∗: H−1(s) ∩X(r, L0) →SUs(r, L0) is dominant. In specific geometric situations it is importantto know what is the codimension of the locus of those s ∈W for which πs∗is not dominant.

Ageneral result to this extend can be deduced easily from the following Lemma 1.4 communicatedto us by E. Markman.First we introduce some notation.Definition 1.2 A vector bundle E ∈U(r, d) is called very stable if it does not have non-zeronilpotent ωC-twisted endomorphism.By a result of Drinfeld and Laumon, see [La], [B-N-R], very stable vector bundles always exist.Let U ⊂SUs(r, L0) be the non-empty Zariski open set consisting of very stable bundles. For anE ∈X(r, L0) denote by XE := T ∗[E]SUs(r, L0) = H0(C, EndoE ⊗ωC) ⊂X(r, L0) the fiber of thecotangent bundle at [E].Lemma 1.4 For any E ∈U the Hitchin mapHE = H|XE : XE −→W(1.14)is surjective.Proof.

Since E is very stable we have H−1E (0) = 0 and hence the dimension of the generic fiber ofHE is zero. On the other hand, dim XE = dim W and therefore HE : XE →W is dominant.Denote by W o ⊂W the Zariski open subsetW o = {s ∈W | dim H−1E (s) = 0}.Clearly the subvariety XE is preserved by the C× action on M(r, L0).Furthermore the C×-equivariance of H implies that W o is a C×-invariant subset of W. Since H−1E (0) = {0}, we havethatHE : XE \ {0} −→W \ {0}(1.15)22

is a C×-equivariant morphism and thus descends to a morphismHE : P(XE) −→Pweight,where Pweight = (W \ {0})/C× is the corresponding weighted projective space.The morphism HE is dominant because its range contains the Zariski open set (W o\{0})/C× ⊂Pweight. But P(XE) is a projective variety and therefore HE(P(XE)) is projective and thus HE isonto.We obtain a commutative diagram of C×-bundles over P(XE) and Pweight respectivelyXE \ {0}W \ {0}P(XE)Pweight✲HE❄❄✲HESince the C× on the fibers of XE \ {0} →P(XE) and W \ {0} →Pweight is simply transitive, weobtain that the map (1.15) (and consequently the map (1.14)) is surjective.✷Corollary 1.7 For any s ∈W the rational mapπs∗= Π|H−1(s) :H−1(s)SU(r, L0)♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣✲is dominant.Proof.

By Lemma 1.4 the range Π(H−1(s)) contains the nonempty Zariski open set U.✷2The abelianization of an automorphism2.1The map ϕWStart with an automorphism Φ of SU(r, L0).We would like to abelianize it, i.e.to lift Φ tothe moduli space of Higgs bundles and induce from it an automorphism of the family of spectralPryms. The hope is that in this way we will get a simpler picture because of the fact that theautomorphisms of the abelian varieties are pretty well understood.

As we will see in the subsequentsections, this hope turns out to be unsubstantiated in that simple form since the spectral Prymsare not generic abelian varieties and can have non-trivial group automorphisms. This is the reason23

why we have to work with the whole family of Pryms rather then the individual Prym varietiesand to make essential use of its “rigidity” as a group scheme over W.To construct the abelianized version of Φ, observe first that the codifferential of Φ provides anatural lift of Φ|SUs(r,L0) to the total space of the cotangent bundledΦ∗: X(r, L0) −→X(r, L0).The automorphism dΦ∗commutes with the projection Π : X(r, L0) →SUs(r, L0) by deffinition.The behaviour of dΦ∗with respect to the Hitchin map is described by the following proposition.Proposition 2.1 There exists an automorphism ϕW ∈Aut(W) making the following diagramcommutativeX(r, L0)X(r, L0)WW✲dΦ∗❄H❄H✲ϕW(2.16)Proof. Let s ∈W \ D. Then the spectral curve ˜Cs is smooth and (see Remark 1.5) H−1(s) =Prym ˜d( ˜Cs, C).

Consider the fiberH−1X (s) := H−1(s) ∩X(r, L0) = {F ∈Prym ˜d( ˜Cs, C) | α∗F−stable}.The argument in Remark 5.2 of [B-N-R] gives codim(H−1X (s), H−1(s)) ≥2.Therefore sincePrym ˜d( ˜Cs, C) = H−1(s) is smooth, the regular map H ◦dΦ∗: H−1X (s) →W extends by Har-togs theorem to a morphism H ◦dΦ∗: H−1(s) →W. But Prym ˜d( ˜Cs, C) is projective and W is anaffine space; henceH ◦dΦ∗(Prym ˜d( ˜Cs, C)) = point in W.Denote this point by ϕW (s).In this way we obtain a rational map ϕW on W, with possiblesingularities along D which makes the diagram (2.16) commutative.This implies that ϕW is C×-equivariant since dΦ∗is linear on the fibers of the cotangent bundleand H is C×-equivariant.

Thus ϕW descends to a rational mapϕW :PweightPweight♣♣♣♣♣♣♣♣♣♣♣✲.But Pweight is a normal projective variety and therefore ϕW is not defined in a locus of codimension≥2. Finally the C×-equivariance of ϕW together with the fact that C× acts transitively on thefibers of W \ {0} →Pweight yield that ϕW itself is not defined on a locus of codimension ≥2.

SinceϕW is a map from a vector space to a vector space, it extends by Hartogs to the whole W.24

✷Since we want to reduce the study of the automorphism Φ to the study of a suitable fiberwiseautomorphism of the family of spectral Pryms, the best situation for us would have been if wecould say that ϕW = id. Unfortunately, at this stage the only information we can extract from theabove proposition is that ϕW commutes with the C× action on W. In the next lemma we use thisto determine some further properties of ϕW .Lemma 2.1 The map ϕW preserves the subspace Wr−1,r ⊂W and moreover φr−1,r := ϕW |Wr−1,ris of the form φr−1,r = (φr−1, φr) whereφr−1 : H0(C, ω⊗(r−1)C) −→H0(C, ω⊗(r−1)C)andφr : H0(C, ω⊗rC ) −→H0(C, ω⊗rC )are linear maps.Proof.

Consider the algebra of regular functions S := C[W] on W. Decomposing S into isotypicalcomponents with respect to the C× action, we get a new grading on it:S = ⊗n≥0Sn,where Sn consists of all functions on W on which a number t ∈C× acts by a multiplication by tn.More explicitly, if we choose coordinates xi1, . .

. , xini in Wi = H0(C, ω⊗iC ), then the elements in Snare just polynomials in xij’s with admissible multidegrees {aij} satisfyingXi,ji · aij = n.Any automorphism of W commuting with the C× action will preserve this grading and in particularwill preserve the subspaces of the form Wi ⊕Wi+1 ⊕.

. .

⊕Wr for any i. In paticular, since ϕWcommutes with the C× action we haveφr−1,r : Wr−1,r −→Wr−1,r.Moreover, since g.c.d.

(r −1, r) = 1, it follows that φr−1,r has the required form.✷25

2.2Extension to the family of PrymsAs we have seen in Section 1.3 the fiber of the Hitchin map H : M(r, L0) −→W over every points ∈W reg := W \ D is isomorphic to an abelian variety. Moreover, one can show, see [Ni], that Dis exactly the locus of critical values of H. Define Prym ˜d( ˜C, C) to be the preimage of W reg underthe Hitchin map.

Then Prym ˜d( ˜C, C) is a smooth variety, see [Ni], and we have a smooth fibrationH : Prym ˜d( ˜C, C) →W reg. Our next goal is to extend the automorhism dΦ∗to an automorphismof the family Prym ˜d( ˜C, C).

The first result in this direction is the following propositionProposition 2.2 The map ϕW preserves the discriminant divisor D ⊂W.Proof. Let s ∈W.

Corollary 1.7 guarantees that Prym ˜d( ˜Cs, C) ∩X(r, L0) ̸= ∅and hence, due toProposition 2.1, dΦ∗induces a birational automorphism between H−1(s) and H−1(ϕW (s)).Let s ∈Do. If we assume that ϕW (s) ̸∈D, then eCϕW (s) is smooth and Prym( eCϕW (s), C) isisomorphic to an abelian variety.

Therefore Prym ˜d( ˜Cs, C) is birational to an abelian variety. Onthe other hand Prym ˜d( ˜Cs, C) is birationally isomorphic to a P1 bundle over an abelian variety, seeRemark 1.7, which is a contradiction because such a bundle has Kodaira dimension −∞.Consequently ϕW (Do) ⊂D.

But ϕW is an automorphism and hence is a closed map whichyields ϕW (D) = D since the discriminant is irreducible, see Corollary 1.5.✷Corollary 2.1 Let φr be the map defined in Lemma 2.1. Then there exists a number λ ∈C× andan automorphism σ of the curve C such thatφr = mλ ◦σ∗,where mλ is the dilation by λ.Proof.

By Lemma 2.1 the map φr is a linear map. On the other hand, Proposition 2.2 implies inparticular that the map φr preserves the discriminant divisor Dr.

Therefore the induced map φron the projective space P(W) preserves the divisor P(Dr). But, by Remark 1.5, the divisor P(Dr)is the dual variety of the r-canonical model of C and therefore the dual map φ∨r preserves thecurve fP(Wr)(C) ⊂P(Wr)∨.

Let σ be the induced automorphism of C. Then, since fP(Wr)(C) spansP(Wr)∨and φ∨r is linear, it follows that φr = σ∗.✷26

Remark 2.1 If the curve C does not have automorphisms, then the above corollary implies thatφr is just a dilation.The proposition above yields the commutative diagramPrym ˜d( ˜C, C)Prym ˜d( ˜C, C)W regW reg♣♣♣♣♣♣♣✲dΦ∗❄H❄H✲ϕWFurthermore, as we saw in the proof of Proposition 2.2 the map dΦ∗gives a birational isomorphismbetween H−1(s) and H−1(ϕW (s)) for any s. In particular, in the case s ∈W reg we get a birationalautomorphism between abelian varieties and hence dΦ∗|H−1(s) extends to a biregular isomorphismbetween H−1(s) and H−1(ϕW (s)). Therefore dΦ∗extends as a continuous automorphism ˜Φ ˜d of thewhole variety Prym ˜d( ˜C, C) and since Prym ˜d( ˜C, C) is smooth, we get:Proposition 2.3 There exists a biregular automorphism ˜Φ ˜d extending dΦ∗which fits in the com-mutative diagramPrym ˜d( ˜C, C)Prym ˜d( ˜C, C)W regW reg✲˜Φ ˜d❄H❄H✲ϕW3The fibration ˜C −→W regr−1,r3.1The Picard group of ˜CWe start with some notation: For a vector space V , the points of the Grassmanian Gr(k, V ) cor-respond to codimension k linear subspaces of V .

Also, for a linear space of sections W on S and agiven point p on S, Wp denotes the linear subspace of sections of W vanishing at p. We now makethe assumption that r ≥3. We are going first to prove our main Theorem A in the case r ≥3and next, in Section 5.3, we outline the modifications needed for the proof of the rank 2 case.

Forsimplicity denote the space Wr−1,r by B and the spaces W r−1,r and W regr−1,r = Wr−1,r \D by B andBreg respectively. In the following we are going to use repeatedly Proposition 6.5 in [Ha], whichcompares the Picard group of a variety X with that of a Zariski open subset U of X.Notation.From now on we will denote by ˜C the part of the universal spectral curve sittingover Breg, unless otherwise stated.27

We have the diagram:˜CBregS◦C❅❅❅❅❘π❄β✲h✲αProposition 3.1 The Picard group of the variety ˜C is isomorphic to the pull back of the Picardgroup of the base curve C by the map π, i.e. Pic ˜C ≃π∗Pic C.Proof.

The variety ˜C can be constructed as follows. Consider the diagram:f ∗|B|U reg|S◦≃˜CU reg ⊂UGr(1, B) × BS◦⊂SGr(1, B)B ⊇B ⊇D✲❄β✲❄✑✑✑✑✑✑✰p1❄p2✲f|B|(3.17)The map f|B| is the map associated to the base point free linear system |B| i.e.

it sends a point p tothe point [Bp] representing the linear subspace Bp. The bundle U is the universal bundle over theGrassmanian.

The fiber of the bundle U ∩p−12 (B) over a point [H] in the Grassmanian is isomorphicto H ∩B. The variety U ∩p−12 (B) is an affine bundle over f|B|(S◦) with fiber over the point f|B|(p)isomorphic to Bp.

Define U reg df= U ∩p−12 (B \ D) = U ∩p−12 (Breg). Then ˜C ≃f ∗|B|U reg|S◦.

Notethat by Corollary 1.6, the divisor f ∗|B|(p−12 (∆) ∩U) is irreducible. Also it is linearly equivalent tozero since Pic B is trivial.To complete the proof of the proposition, we have:Pic ˜C≃Pic f ∗|B|U reg|S◦≃Pic f ∗|B|(U ∩p−12 (B \ D))|S◦≃Pic f ∗|B|(U ∩p−12 (B))|S◦since U ∩p−12 D is an irreducible divisor lin.

equiv. to 0,≃β∗Pic S◦since f ∗|B|(U ∩p−12 (B))|S◦is an affine bundle over S◦,≃π∗Pic Csince S◦is an affine bundle over C.✷28

3.2The Picard group of ˜CpFor a fixed point p in S◦we denote by ˜Cp the subvariety of ˜C consisting of those curves whose imagein S◦passes through p. ˜Cp sits over the subspace Bregpof Breg, consisting of those sections vanishingat p. Let h : ˜Cp −→Bregpdenote again the restriction of the map h on ˜Cp. We are going to calculatethe Picard group of ˜Cp.

Observe that the above fibration h has a section corresponding to the pointp. Take the restriction of the map β to ˜Cp i.e.

β : ˜Cp −→S◦. Let Hp be the preimage of p via themap β.

By a construction similar to that for the variety ˜C in the above Section 3.1, one can see thatthe fibration β : ˜Cp \Hp −→S◦\{p} is the complement of an irreducible divisor linearly equivalentto zero in an affine fibration. We thus have Pic ( ˜Cp \ Hp) ≃β∗Pic (S◦\ {p}) ≃β∗Pic S◦≃π∗Pic C.Since Hp is an irreducible divisor in ˜Cp, we conclude that Pic ˜Cp is generated by π∗Pic C and Hp.We proceed by showing that we actually have:Proposition 3.2 Pic ˜Cp ≃π∗Pic C ⊕Z[Hp].Proof.

Pick a generic pencil P1 in Bp such that all the fibers of the restriction of the compactifieduniversal spectral curve over the pencil are irreducible, except the one over the infinity point bwhich has exactly two irreducible components, see Corollary 1.4. We may also assume that none ofthe singular points of the fibers lies on the N = 2r2(g−1) base points of the system.

The restrictionof the compactified universal spectral curve over P1, is a smooth surface X which is the blow up ofthe surface S over the N base points p1 = p, . .

. , pN of the pencil.

We have the following picture:29

In the above picture, b is the point at infinity, b1, . .

. , bk correspond to the singular fibers andE1, .

. .

, EN are the exceptional divisors. The curve over b consist of two components ˜Y∞and ˜C′b,one of which, ˜Y∞, is the proper transform of the divisor at infinity on S. The intersection of X withthe variety ˜Cp is X◦= X \ h−1(b, b1, .

. .

, bk). We define E◦i = Ei ∩X◦.

Note that E◦1 = Hp ∩X. Toprove the claim, it is enough to show that on X◦the line bundle E◦is independent from π∗Pic C.We havePic X ≃β∗Pic S ⊕Z[Ei]i=1,...,N ≃π∗Pic C ⊕Z[ ˜Y∞] ⊕Ni=1 (⊕Z[Ei]).

(3.18)Now β∗(rY∞+rα∗ωC) = h−1(point)+PNi=1 Ei, where the equality stands for the linear equivalenceof divisors. Therefore, r ˜Y∞+rπ∗ωC = h−1(point)+ PNi=1 Ei = ˜C′b + ˜Y∞+ PNi=1 Ei.

In other words,˜C′b = (r −1) ˜Y∞+ rπ∗ωC −NXi=1Ei. (3.19)All the fibers of h define linear equivalent divisors on X and the restriction on X \ h−1(b) of theline bundles corresponding to the divisors ˜C′b and ˜Y∞are trivial.

We thus getPic X◦≃Pic X \ h−1(b) ≃Pic X. ( ˜Y∞= 0, ˜C′b = 0)≃π∗Pic C ⊕Ni=1 (⊕Z[E◦i ]).

(rπ∗ωC = PNi=1 E◦i )by (3.18) and (3.19).Assume now that π∗L + mHp = 0 on ˜Cp. Then π∗L + mE◦1 = 0 on X◦.

By the description of thePic X◦we conclude that π∗L = 0 and m = 0 and this completes the proof of the proposition.✷3.3The sections of H : J˜d( ˜C) −→W regr−1,rProposition 3.3 The only sections of the map H : J ˜d( ˜C) −→W regr−1,r = Breg are those comingfrom a pull back of a fixed line bundle on C. In other words, if σ : Breg −→J ˜d( ˜C) is a section ofH, then σ(s) = [π∗sM] where M is a fixed line bundle on C. In particular, if r does not divide ˜d,then the map H has no sections.We start with two lemmas:Lemma 3.1 On J ˜d( ˜C) ×Breg ˜C there exists a line bundle P ˜d such that P ˜d|[L]× ˜Cs ≃L⊗n for someinteger n.Proof. See [M-R].✷30

Remark 3.1 One can actually prove that the minimum such positive integer n is equal to g.c.d. (r, d).Lemma 3.2 If σ is a section of the map H whose image lies in the locus π∗Pic C i.e.

σ(s) =[π∗Ms], where Ms is a line bundle on C, then Ms = M for all s ∈Breg.Proof. Consider the map γ : ˜C −→J ˜d( ˜C) ×Breg ˜C which sends a point q sitting over s ∈Breg to(σ(s), q).

Then [γ∗P ˜d|[L]] = nσ(s). By the description of Pic ˜C, see Proposition 3.1, we get thatγ∗P ˜d ≃π∗M1 for a fixed M1 in Pic C. Hence, n[π∗Ms] = nσ(s) = [π∗M1] for all s ∈Breg.

Sincethe map π∗is one to one, see Remark 3.10 in [B-N-R], we conclude that the map Breg −→Pic Cthat sends s to [Ms] has finite image. Since Breg is connected, the map is constant.✷Proof of Proposition 3.3.

Say that σ is not coming from a pull back. By the above Lemma 3.2,we may assume that there exists an s0 ∈Breg such that σ(s0) is not of the form π∗A for some Ain Pic C. Take a point p on S◦that lies on ˜Cs0 ⊆S◦.

The family of curves ˜Cp has a section andtherefore on J ˜d( ˜Cp) ×Bregp˜Cp there exists a Poincare bundle P ˜dp, see [M-R]. We have[γ∗P˜d| ˜Cs] = nσ(s) and [γ∗P˜dp| ˜Cs] = σ(s) for all s ∈Bregp .Therefore, nγ∗P ˜dp| ˜Cs ≃γ∗P ˜d| ˜Cs for all s ∈Bregp .

Since Pic Bregpis trivial, we conclude by the see-sawprinciple, see [Mu], thatnγ∗P˜dp ≃γ∗P˜d| ˜Cp on ˜Cp. (3.20)On the other hand, by the description of the Picard groups of ˜Cp and ˜C, see Propositions 3.1and 3.2, we haveγ∗P˜dp ≃π∗M + mHp and γ∗P˜d ≃π∗L.

(3.21)Hence, by (3.20) and (3.21), we have on ˜Cp that nπ∗M + nmHp ≃π∗L. Proposition 3.2 impliesthat m = 0 i.e.γ∗P ˜dp ≃π∗M.But then, σ(s0) = [γ∗P ˜dp| ˜Cs0] = [π∗M] which contradicts theassumption on σ(s0).✷31

4The ring of correspondences4.1The map β′ : ˜C h×φ h ˜C −→S◦× S◦Throughout the Sections 4 and 5, we will denote by φ the map φr−1,r : B −→B. Accordingto Lemma 2.1, we have that φr−1,r has the form φr−1,r = (φr−1, φr) where φi : H0(C, ωiC) −→H0(C, ωiC) are linear automorphisms for i = r −1, r. Furthermore, by Proposition 2.2, we havethat φ(D) = D. Therefore, the restriction of the map φ on Breg, which we will denote again by φ,induces an automorphism φ : Breg −→Breg.

In this section we study the fiber product ˜C h×φ h ˜Cdefined by the diagramS◦× S◦˜C h×φ h ˜C˜CBreg˜CBreg✛β′✲❄❄h❄φ✲hWe define φ : B −→B to be the map which extends φ to B as φ = (1, φr−1, φr). To investigate themap β′ , we define in the product S × S the following loci A and Γ.Definition 4.1Adf=S◦× S◦\ Imβ′Γdf={(p, q) ∈S × S such that Bp = φ(Bq)}On S × S \ (A ∪Γ) we can define a map f to the Grassmanian Gr(2, B) of codimension 2 linearsubspaces of B, by sending the point (p, q) to the class of the plane Bp ∩φ(Bq).

We have thefollowing diagram:˜C h×φ h ˜C|S◦×S◦\Γ ≃f ∗U reg|S◦×S◦\ΓU reg ⊂UGr(2, B) × BS◦× S◦\ (A ∪Γ) ⊆S × S \ (A ∪Γ)Gr(2, B)B ⊇B ⊇D✲❄β′✲❄✟✟✟✟✟✟✟✟✟✟✟✟✙p1❄p2✲f(4.22)The notation is in complete analogy with that of the Diagram (3.17). As in the case of ˜C, thevariety ˜C h×φ h ˜C|S◦×S◦\(A∪Γ) is an affine bundle over the image of S◦× S◦\ (A ∪Γ) under the mapf.32

Lemma 4.1 dimA ≤2Proof. Let B∞denote the space of sections W ∞r−1,r, i.e.

the linear subspace of W consisting ofpoints of the form (0, 0, . .

. , 0, sr−1, sr).

We haveA= {(p, q) ∈S◦× S◦such that U|f(p,q) ∩p−12 B = ∅}= {(p, q) ∈S◦× S◦such that Bp ∩φ(Bq) = ∅}= {(p, q) ∈S◦× S◦such that B∞p = φ(B∞q )}By Corollary 1.3, the linear system defined by B∞separates points on S◦. Therefore, given a pointp ∈S◦there exists at most one q ∈S◦with (p, q) ∈A and this proves the lemma.✷We give now a better description of the locus Γ.

Consider the diagramSP(B ∨)P(B ∨)✲f|B|❅❅❅❅❘φ∗f|B|❄φ∗In the above diagram the map φ ∗is the induced automorphism of P(B ∨) by the linear automor-phism φ. We denote by π1 and π2 the two projections of Γ to the surface S. It is easy to seethatπ1(Γ)= {p ∈S such that f|B|(p) ∈φ ∗f|B|(S)}= f −1|B|f|B|(S) ∩φ∗f|B|(S)and similarπ2(Γ)= (φ∗f|B|)−1 f|B|(S) ∩φ∗f|B|(S)We describe the fibers of the map π1.

If p ∈Y∞, then it is easy to see that π−11 (p) consistsof the whole divisor Y∞. If p ∈S◦, then π−11 (p) consists of at most one point since the system Bseparates points on S◦.

By the above discussion we conclude thatLemma 4.2 dim (Γ ∩(S◦× S◦)) ≤2.4.2The Picard group of ˜C h×φ h ˜CWe are going to use the following diagram33

˜C h×φ h ˜CS◦× S◦C × C◗◗◗◗◗sπ′❄β′✲α′to calculate the Picard group of ˜C h×φ h ˜C.Lemma 4.3 Pic ( ˜C h×φ h ˜C|S◦×S◦\Γ) ≃π′∗Pic (C × C).Proof. We have ˜C h×φ h ˜C|S◦×S◦\Γ ≃f ∗U reg|S◦×S◦\Γ ≃f ∗U reg|S◦×S◦\(Γ∪A) since the fiber over Ais empty.

Therefore,Pic ( ˜C h×φ h ˜C|S◦×S◦\Γ) ≃≃Pic f ∗U reg|S◦×S◦\(Γ∪A)≃Pic f ∗(U ∩p−12 (B))|S◦×S◦\(Γ∪A)since the preimage of D is irred. divisor lin.

equiv. to 0,≃β′∗Pic (S◦× S◦\ (Γ ∪A))since it is an affine bundle over S◦× S◦\ (Γ ∪A),≃β′∗Pic (S◦× S◦)since codim(Γ ∪A) ≥2,≃Pic (C × C)since S◦× S◦is a rank two affine bundle over C × C.✷The variety ˜C h×φ h ˜C splits as˜C h×φ h ˜C = ˜C h×φ h ˜C|S◦×S◦\Γ ∐˜C h×φ h ˜C|Γ.To calculate its Picard group, we have to consider the following two cases.Case A: dim Γ ∩(S◦× S◦) ≤1: Then one can easily see that dim ˜C h×φ h ˜C|Γ ≤dim( ˜C h×φ h ˜C) −2.We thus have:Lemma 4.4 In case A, the Pic ( ˜C h×φ h ˜C) ≃Pic ( ˜C h×φ h ˜C|S◦×S◦\Γ) ≃π′∗Pic (C × C).Case B: dim Γ∩(S◦×S◦) = 2: Following the notation we used in the description of Γ in the aboveSection 4.1, we have π1(Γ) = S. This implies that φ∗induces an automorphism of f|B|(S).

Thefollowing summarizes the basic properties of the induced automorphism φ∗.1. The point p∞= f|B|(Y∞) remains fixed.2.

φ∗sends a fiber of the map α to a fiber: otherwise we get a P1 cover of the curve C, whichcontradicts the assumption that genus g(C) > 0.34

3. Since C has no automorphisms, φ∗sends a fiber of the map α to the same fiber.Therefore φ∗induces an automorphism χ : S◦−→S◦which preserves the fibers of the mapα : S◦−→C.Lemma 4.5 Let s ∈B and let ˜Cs denote also the image of the corresponding spectral curve on S◦.Then χ( ˜Cs) = ˜Cφ−1(s).Proof.

Let Hs denote the hyperplane in P(B ∨) corresponding to the section s. Then˜Cs ≃f|B|( ˜Cs) = Hs ∩f|B|(S◦)and˜Cφ−1(s) ≃f|B|( ˜Cφ−1(s)) = Hφ−1(s) ∩f|B|(S◦).It is φ∗(Hs) = Hφ−1(s) and so,f|B|( ˜Cφ−1(s))=φ∗(Hs) ∩f|B|(S◦)=φ ∗(Hs) ∩φ ∗f|B|(S◦)since φ ∗f|B|(S◦) = f|B|(S◦) in P(B∗),=φ∗(Hs ∩f|B|(S◦))since φ∗is an automorphism,=φ ∗f|B|( ˜Cs).✷By the above Lemma 4.5, the automorphism χ on S◦induces an automorphism ψ of ˜C over Bwhich makes the following diagram commutative˜C˜CBB❄h✲ψ❄h✲φ−1Since S◦is the total space of the line bundle ωC on C, the map χ acts by a dilation and a translationby a section of ωC on the fibers i.e. it has the form χ = mλ ◦Ts where λ ∈C∗and s ∈H0(C, ωC).We actually claim that s = 0 and that φ−1 has the form φ−1(sr−1, sr) = (λ−(r−1)sr−1, λ−rsr).Indeed,χ ◦h−1(sr−1, sr) = h−1(φ−1(sr−1, sr)).

(4.23)To prove the first claim, we apply (4.23) to (sr−1, sr) = (0, 0). Then, χ ◦h−1(0, 0) = h−1(0, 0)since φ−1 is a linear map.

But h−1(0, 0) is the curve xr = 0, and so, χ ◦h−1(0, 0) is the curve(λx + s)r = 0, see beginning of Section 1.2. This implies that s = 0.35

For the second claim: h−1(sr−1, sr) is the curve xr + sr−1x + sr = 0 and so, χ ◦h−1(sr−1, sr)is the curve λrxr + λsr−1x + sr = 0 i.e. the curve xr + λ−(r−1)sr−1x + λ−rsr = 0 i.e.

the curveh−1(λ−(r−1)sr−1, λ−rsr) which completes the proof.Corollary 4.1 φ(sr−1, sr) = (λr−1sr−1, λrsr).We now proceed with our discussion about the Picard group of ˜C h×φ h ˜C in the case of dimΓ ∩(S◦× S◦) = 2. We have that ˜C h×φ h ˜C|Γ = ∆λ = {(p, ψ−1(p)), p ∈˜C}.

Note that if λ = 1, then ∆λis exactly the diagonal in the fiber product ˜C h×h ˜C. Combining the latter with Lemma 4.3 we get:Proposition 4.1 In case B, the Pic ( ˜C h×φ h ˜C) is generated by π′∗Pic (C × C) and ∆λ.Remark 4.1 It is easy to see that Pic ( ˜C h×φ h ˜C) ≃π′∗Pic (C × C) ⊕Z[∆λ].5Proof of the main Theorem5.1The form of ˜ΦWe start with a definition.

Let C and C1 are two smooth curves. Given a line bundle L on theproduct C × C1, then L induces a mapψL : Pic C −→Pic C1defined by ψL(O(p)) = α∗(L|Cp1 ), where Cp1 is the fiber in the product over the point p ∈C andα : C1 −→Cp1 the natural isomorphism.

The extension of the definition to a point [L] ∈Pic Cis given by taking a meromorphic section of the bundle L. In the same way, whenever we have aline bundle on a fiber product of two families of curves we get a map between their relative PicardgroupsConsider the diagram˜C h×φ h ˜CBregCC × CC❅❅❅❅❅❘π1❄π′✠π2✲h′Let L be a line bundle on C × C. Then π′∗L is a line bundle on the fiber product ˜C h×φ h ˜C.Following the above notation, it is easy to see that36

Lemma 5.1 ψπ′∗L = π∗2 ψL Nm1, where Nm1 is the norm map of π1.Note that on the level of fibers we haveψπ′∗L :Pic ˜Cs−→Pic ˜Cφ(s)O(p)7−→π∗φ(s) ψL Nms(O(p))where the subscripts refer to the restriction on the corresponding fiber over a point of Breg.Remark 5.1 We recall the following fact about maps of abelian torsors and induced maps ofabelian schemes. Let p : G −→Z be an abelian scheme and π : T −→Z a G-torsor.

Given twoelements t1 and t2 in the same fiber of T over z ∈Z, we denote by t1 −t2 the unique elementg ∈G over z ∈Z, with the property: Tgt2 = t1. Let now φT : T −→T be an automorphismof the abelian torsor T i.e.

an automorphism that sends a fiber of the map π to another fiberand preserves the action of G. To that, one can associate a group automorphism φG of the abelianscheme G as follows. Given g ∈Gz i.e.

an element of G sitting over z ∈Z, choose an elementtz ∈Tz and defineφG(gz) = φT (Tgztz) −φT (tz).It is easy to check that this is independent from the choice of tz in the fiber Tz and that it definesa group automorphism. Note that given t1 and t2 two elements in the same fiber Tz, thenφT (t1) −φT (t2) = φG(t1 −t2).Following the notation of Proposition 2.3, let ˜Φ : Prym( ˜C, C) −→Prym( ˜C, C) be the groupautomorphism associated to the automorphism ˜Φ ˜d : Prym ˜d( ˜C, C) −→Prym ˜d( ˜C, C) of Prym( ˜C, C)-torsors.

We determine now the form of the map ˜Φ. Consider the mapµ :˜C h×φ h ˜C−→Prym( ˜C, C) H×h ˜C(ps, qφ(s))7−→˜Φ(rps −π∗sNms(ps)), qφ(s)where the notation for a point on a curve, stands also for the line bundle which the point defines.According to Lemma 3.1, on the product Prym( ˜C, C) H×h ˜C we have a line bundle P with theproperty P|[L]× ˜Cφ(s) ≃n L for some integer n. Hence,µ∗P|[ps]× ˜Cφ(s) ≃n˜Φ(rps −π∗sNms(ps)).

(5.24)By using (5.24) and the knowledge of the Picard group of ˜C h×φ h ˜C we will derive the form of themap ˜Φ.37

Case A: dim Γ∩(S◦×S◦) ≤2. Then, by Lemma 4.4, we have that Pic ( ˜C h×φ h ˜C) ≃π′∗Pic (C ×C).We thus getµ∗P ≃π′∗L and so, µ∗P|[ps]× ˜Cφ(s) ≃π′∗L|[ps]× ˜Cφ(s),i.e.˜Φ(nrps −nπ∗sNms(ps)) ≃π∗φψLNms(ps).

(5.25)Take the map ˜Φs : Prym( ˜Cs, C) −→Prym( ˜Cφ(s), C) Now given a line bundle ˜Ls ∈Prym( ˜Cs, C),choose a line bundle ˜Ms in Prym( ˜Cs, C) such that ˜Ls = nr ˜Ms. By choosing a meromorphic section,we can write ˜Ms = O(Pi(p1i −p2i )) where Nm(Pi(p1i −p2i )) = 0.

We have˜Φs(Ls)=˜Φs(rnMs) = ˜ΦsPi rn(p1i −p2i ) ==Pi˜Φs(rnp1i −nπ∗sNms(p1i )) −˜Φs(rnp2i −nπ∗sNms(p2i ))+ Pi ˜Φs(nπ∗sNms(p1i −p2i ))=Pi π∗φ(s)ψLNms(p1i ) −Pi π∗φ(s)ψLNms(p2i )by (5.25),=π∗φ(s)ψLNms(Pi(p1i −p2i ))=0since Nm(Pi(p1i −p2i )) = 0.The later contradicts the fact that ˜Φ is an isomorphism, which means that case A cannot occur.Therefore dimΓ ∩(S◦× S◦) = 2, i.e. we are in case B which we examine bellow:Case B: dimΓ ∩(S◦× S◦) = 2.

By the discussion in Section 4.2, the Pic ( ˜C h×φ h ˜C) is gener-ated by π′∗Pic (C × C) and the divisor ∆λ. Hence,µ∗P ≃π′∗L + n∆λ.Working as before and using the definition of ∆λ, we conclude that ˜Φ(L) = nψ∗(L).

Since ˜Φ andψ∗are isomorphisms, we get that n = ±1. To summarize,Proposition 5.1 ˜Φ = ±ψ∗, where ψ is the map defined in Section 4.2.5.2The conclusion of the proofWe now conclude the proof of the main Theorem A in the case of r ≥3 by examining the two cases˜Φ = ±ψ∗.

We start with some notation. We denote by ψ∗˜d : Prym ˜d( ˜C, C) −→Prym ˜d( ˜C, C) the pullback of the map ψ : ˜C −→˜C.

Note that ψ∗= ψ∗0. We have the diagram:Prym ˜d( ˜C, C)Prym ˜d( ˜C, C)BregBreg❄H✲ψ∗˜d❄H✲φ38

Lemma 5.2 Let x denote an element in Prym ˜d( ˜C, C). We have that1.

If ˜Φ = ψ∗, then ψ∗−1˜d˜Φ ˜d(x) −x is independent from x on the fibers of the map H.2. If ˜Φ = −ψ∗, then ψ∗−1˜d˜Φ ˜d(x) + x is independent from x on the fibers of the map H.Proof.

For the first: Let y be apoint in the fiber of H through x. Since ψ∗−1 ˜Φ = 1, it is enoughto show that ψ∗−1˜d˜Φ ˜d(x) −ψ∗−1˜d˜Φ ˜d(y) = ψ∗−1 ˜Φ(x −y).

Since ψ∗−1 ˜Φ is the group homomorphismassociated to the map ψ∗−1˜d˜Φ ˜d of abelian torsors, the later is true by Remark 5.1 For the second: Itis enough to show that ψ∗−1˜d˜Φ ˜d(x)−ψ∗−1˜d˜Φ ˜d(y) = y−x. But since ˜Φ = −ψ∗we have ψ∗−1 ˜Φ(x−y) =y −x and this case follows as well.✷Proposition 5.2 Let −1 denote the inversion along the fibers of Prym( ˜C, C).

Then1. If ˜Φ = ψ∗, then ˜Φ ˜d = Tπ∗µ ◦ψ∗˜d, where µ is an r-torsion line bundle on the base curve C.2.

If ˜Φ = −ψ∗, then r|2d and ˜Φ ˜d = Tπ∗ν1 ◦(ψ∗˜d) ◦(−1), where ν is a line bundle on the basecurve C which satisfies ν⊗r1= L⊗20⊗ω⊗r(r−1)C.Proof. For the first: According to Lemma 5.2, on each fiber H−1(s) = Prym ˜d( ˜Cs, C) of the mapH, the maps ˜Φ ˜d and ψ∗˜d differ by a translation by a unique element a(s) ∈Prym( ˜Cs, C).

Thisdefines a section of the map H : J0( ˜C) −→Breg and therefore by Proposition 3.3, it must have theform π∗M for some fixed line bundle µ on C. Since the image of the section is in the Prym( ˜C, C)we get that µ is an r-torsion line bundle.For the second: According to Lemma 5.2, we can construct a section of the map H : J2 ˜d( ˜C) −→Breg by assigning to the point s ∈Breg the line bundle ψ∗−1˜d˜Φ ˜d(x) + x for some point x ∈Prym ˜d( ˜Cs, C). By Proposition 3.3, we get that r|2 ˜d.

Since ˜d = d + r(r −1)(g −1) this is equiv-alent to r|2d. The same proposition implies that the above section must have the form π∗ν1 forsome fixed line bundle ν1 on C.To complete the proof of the second part of the proposition,observe that the map ψ∗commutes with the translations by an element of the form π∗ν1 and thatNm π∗ν1 = L⊗20⊗ω⊗r(r−1)C.✷We are ready now to complete the proof of the Theorem A in the case r ≥3.

Pick an ele-ment element s ∈Breg. Then, by Corollary 1.7, the prymian Prym ˜d( ˜Cs, C) maps dominantly to39

SU(r, L0). Let V be its image.

It is enough to prove the theorem for the restriction of the map Φon V. We have the following commutative diagram:Prym ˜d( ˜Cs, C)Prym ˜d( ˜Cφ(s), C)VV❄πs∗✲˜Φ ˜d❄πφ(s)∗✲Φwhere the maps πs∗and πφ(s)∗are rational maps.Assume first that ˜Φ = ψ∗. Let E a vector bundle in V. Then, there exists a line bundle ˜Ls inPrym ˜d( ˜Cs, C) such that πs∗(˜Ls) = E. By Proposition 5.2, the above diagram and the fact that themap ψ commutes with the projections πs∗and πφ(s)∗to the base curve C, we conclude thatΦ(E) = Φπs∗(˜Ls) = πφ(s)∗˜Φ ˜d(˜Ls) = πφ(s)∗(ψ∗˜d(˜Ls) ⊗πφ(s)∗µ) = πφ(s)∗ψ∗˜d(˜Ls) ⊗µ = E ⊗µ.Assume next that ˜Φ = −ψ∗.

Then, following the above notation, we haveΦ(E) = Φπs∗(˜Ls) = πφ(s)∗˜Φ ˜d(˜Ls) = πφ(s)∗(ψ∗˜d(˜L−1s )⊗πφ(s)∗ν1) = πφ(s)∗ψ∗˜d(˜L−1s )⊗ν1 = πs∗(˜L−1s )⊗ν1.We claim that πs∗(˜L−1s ) = E∨⊗ω−(r−1)C. Indeed, consider the map πs : ˜Cs −→C. By relativeduality we haveR0πs∗(˜L−1s ) ≃R0πs∗(ωπs ⊗˜Ls)∨.By the adjunction formula we have, see e.g.

[Hi 1], that ωπs ≃π∗sωr−1C. We thus getπs∗(˜L−1s ) ≃E∨⊗ω−(r−1)C.By choosing ν = ν1 ⊗ω−(r−1)C, we get that φ(E) = E∨⊗ν where ν⊗r = L⊗20 .Thus, we have shown the surjectivity of the maps (1.) and (2.) in the statement of Theorem A.To show the injectivity of the map (1.

), pick up a point µ ̸= 0 ∈J0[r] and consider the set of fixedpoints SU(r, L0)⟨Tµ⟩. A vector bundle E is fixed under the action of Tµ, if we have an isomorphismκ : E −→E ⊗µ.If p | r is the order of the torsion point µ, then the µ-twisted Higgs bundle (E, κ) gives a p-sheetedunramified spectral cover πµ : Cµ →C and a semistable vector bundle Fκ of rank r/p on it with theproperty πµ∗(Fκ) = E, see [N-R 1] for details.

Therefore the locus SU(r, L0)⟨Tµ⟩can be identifiedwith the image of the moduli space SUCµ(r/p) under the pushforward map πµ∗and hence is aproper subvariety.40

We will sketch the proof for the injectivity of the map (2.) in the case r ≥3 and when L0 = O.The modifiations of the argument for general L0 are minor and are left to the reader.

If L0 = O,then it suffices to check that the map E →E∨is not the identity on SU(r, O). But if E is astable vector bundle satisfying E ∼= E∨, then the bundle E⊗2 has a unique (up to scaling) non-zerosection t. But H0(C, E⊗2) = H0(C, Sym2E) ⊕H0(C, ∧2E) and therefore either H0(C, E⊗2) = 0 orH0(C, ∧2E) = 0.

This implies that the isomorphism t between E and E∨is either symmetric orskew-symmetric. Thus, E is either orthogonal or symplectic and hence it lies either in the modulispace of stable SO(r)-bundles or in the moduli space of stable Sp(r)-bundles.

But for r ≥3, thoseare proper subvarieties of SU(r, O) and this yields that the map (2.) is injective for r ≥3.Remark 5.2 For r = 2, the moduli space SU(2, O) coincides with the moduli space of Sp(2)-bundles; if E ∈SU(2, O), then E ≃E∨.5.3The rank 2 caseThe proof in the rank 2 case is a modification of the proof of the rank ≥3 case.The maindifference is that the linear system defined by B = W2 = H0(C, ω2C) does not separate the pointson the surface S◦: A section s ∈B corresponds to the curve x2 + s = 0 on S◦.

If m−1 is thedilation by −1 on S◦, then a curve in the linear system |B| that passes through a point p, passesalso through the point m−1(p). Therefore the map f|B| sends S◦in a 2 : 1 way to the projectivespace.

We now show briefly the adjustments for the rank 2 case of the argument we used in therank ≥3 case.At first one can see e.g. by Corollary 2.1, that the map φ = φ2 is a multiplication by a λ ∈C∗.Following a similar argument with that of Section 3.1, we can prove that the Picard group of˜C −→Breg is againPic ˜C = π∗Pic C.We also havePic ˜Cp = Pic π∗C ⊕Z[Hp](5.26)but in this case the technicalities of the proof are slightly different: Let p′ = m−1(p).

On the fiberof S◦over c = α(p) = α(p′), only the points p and p′ belong in the image of the map β. We write˜Cp = ˜Cp \ (Hp ∪Hp′) ∐(Hp ∪Hp′).

Then Pic ˜Cp is generated by Pic (C \ {c}) and Hp, Hp′. Observethat π∗(c) = Hp + Hp′.

We thus have that Pic ˜Cp is generated by Pic π∗C and Hp. To prove thatthis is a direct sum, we again take a pencil as in the proof of Proposition 3.2, but now the curveat infinity embedded on S, consists of the Y∞and a bunch of N1 fibers over the points c1, .

. .

, cN1on C, which pass through the N = 2N1 base points of the pencil. Therefore the fiber over the41

infinity point b on X consists of the infinity divisor ˜Y∞and a bunch of N1 divisors A1, . .

. , AN1which satisfy π∗(ci) = Ai + Ei + E′i.

Using that we getPic X◦= π∗Pic C ⊕N1i=1 (⊕Z[Ei]) ⊕N1i=1 (⊕Z[E′i]). (π∗(ci) = Ei + E′i)i=1,...N1 .Working as in the rank ≥3 case, this proves relation (5.26).

To prove the analogue of Proposition3.3, we proceed in the same way as in the rank ≥3 case.For the Picard group of ˜C h×φ h ˜C: Let again ψ : ˜C −→˜C be the analogue of the map introducedin Section 4.2. Then the locus Γ consists of two components namely Γ1 = {(p, ψ(p)) for p ∈˜C}and Γ2 = {(p, ψ(m−1(p)))forp ∈˜C}.

By observing that π′∗(Diagonal inC × C) = Γ1 + Γ2we conclude that Pic ( ˜C h×φ h ˜C) is generated by Pic (C × C) and Γ1. The rest of the argument,combined with Remark 5.2, proceeds as in the rank r ≥3 case.5.4Curves with automorphismsFor the rank r ≥3 case, the only modification that has to be done, is in the argument of Section4.2, about the induced automorphism of the embedded surface f|B|.

For the rank 2 case, the onlymodification is in the use of Corollary 2.1. We leave to the reader to fill up the details for the proofof Theorem B.6The automorphisms of U(r, d)Proposition 6.1 Let Φ be an automorphism of U(r, d).

Then Φ factors through the determinantmap i.e. we have the following commutative diagramU(r, d)U(r, d)Jd(C)Jd(C)✲Φ❄det❄det✲φd(6.27)Proof.

We would like to show that the map det ◦φ : det−1(L0) −→Jd(C) is constant for allL0 ∈Jd(C). By definition, det−1(L0) = SU(r, L0) and dim SU(r, L0) = (r2 −1)(g −1) > g, g ≥2.Therefore the map det ◦φ : SU(r, L0) −→Jd(C) has positive dimentional fibers and so, no pullback of a line bundle from Jd(C) can be ample.

According to [D-N], Pic SU(r, L0) = Z[Θ]. SinceSU(r, L0) is a projective variety, Θ is an ample divisor.

Let L be a line bundle on Jd(C). We thushave (det ◦φ)∗L ≃nΘ for some integer n. We claim that n = 0: If not, then say first that n > 0.Then nΘ is ample and so (det ◦φ)∗L is ample, which is a contradiction.

Next, if n < 0, then the42

same argument applied to the line bundle L−1 leads to a contradiction. Therefore (det ◦φ)∗L = Ofor all L ∈Pic Jd(C).

Since SU(r, L0) is irreducible, this implies that the map is constant. Indeed,if the image has positive dimension, then there exists a positive divisor on that - the hyperplanesection of its embedding in the projective space.

But then the pull back of that divisor on SU(r, L0)by the map det ◦φ defines a non - trivial line bundle, which is a contradiction.✷As in the case of the automorphisms of SU(r, L0), an automorphism Φ of U(r, d) induces anautomorphism ˜Φ ˜d of the Jacobian fibration H : J ˜d( ˜C) −→W reg which sends a fiber of the Hitchinmap H to a fiber of H. Let N1 : J ˜d( ˜C) −→Jd(C) be the map defined by N1 = det ◦π∗. Note thatNm = T r(r−1)2ωC ◦N1.Lemma 6.1 Let s be a point in W reg.

Then the following diagram commutesJ˜d( ˜Cs)J˜d( ˜CφW (s))Jd(C)Jd(C)✲˜Φ ˜d❄N1❄N1✲φdProof. For the proof we are going to use the following commutative diagramX(r, d)X(r, d)U(r, d)U(r, d)Jd(C)Jd(C)✲dΦ∗❄π∗❄π∗✲Φ❄det❄det✲φd(6.28)It is enough to show that N1 ˜Φ ˜d = φdN1 on a Zariski open U in J ˜d( ˜Cs).

Choose U to be theintersection of the cotangent bundle X(r, d) to U(r, d) with the Jacobian J ˜d( ˜Cs). According toCorollary 1.7, this is a non empty Zariski open in U(r, d).The proof of the Lemma is now aconsequence of the commutativity of the above diagram.43

✷Let ˜Φ and φ be the group maps associated to ˜Φ ˜d and φd, see Remark 5.1. By using the aboveLemma 6.1, it is easy to see thatCorollary 6.1 The following diagram is commutativeJ0( ˜Cs)J0( ˜CφW (s))J0(C)J0(C)✲˜Φ❄Nm❄Nm✲φ(6.29)where Nm is the norm map.Lemma 6.2 Following the notation of Corollary 6.1 , the diagram bellow is commutativeJ0( ˜Cs)J0( ˜CφW (s))J0(C)J0(C)✲˜Φ✻π∗✲φ✻π∗Proof.

Note first that˜Φπ∗(L) = π∗(M) for some fixed line bundle M.(6.30)Indeed, ˜Φ is a global automorphism on J0( ˜C) and so, ˜Φ π∗(L) defines a section of the map H :J0( ˜C) −→W reg. Hence, by Proposition 3.3, it must have the form π∗(M) for some fixed line bundleM on C. By applying the norm map to (6.30), we get Nm˜Φπ∗(L) = Nm π∗(M).

Corollary 6.1implies that rφ(L) = rM. Let nr denote the multiplication by r. By composing both sides of(6.30) by nr and by using the last relation we get that nr ˜Φπ∗(L) = nrπ∗φ(L).

Since the map nris onto, this completes the proof.✷Lemma 6.3 For any E ∈U(r, d) and M ∈J0(C) we have Φ(E ⊗M) = Φ(E) ⊗φ(M).44

Proof. It suffices to prove the relation for all E in a Zariski open V of U(r, d).

Take ˜Cs a smoothspectral curve and let V be the image of X(r, d) ∩J ˜d( ˜Cs) on U(r, d). According to Corollary 1.7,this is a Zariski open of U(r, d).

Then E = π∗˜L. We haveΦ(E ⊗M)=Φ(π∗˜L ⊗M)=Φ(π∗(˜L ⊗π∗M))by the projection formula,=π∗˜Φ ˜d(˜L ⊗π∗M)by diagram (6.28),=π∗(˜Φ ˜d(˜L) ⊗˜Φ(π∗M))by the definition of the group map ˜Φ,=π∗(˜Φ ˜d(˜L) ⊗π∗φ(M))by Lemma 6.2,=π∗˜Φ ˜d(˜L) ⊗φ(M)by the projection formula,=Φπ∗(˜L) ⊗φ(M)=Φ(E) ⊗φ(M).✷Proof of Theorem C.Let Φ : U(r, d) −→U(r, d) be a given automorphism and let φd theinduced automorphism on Jd(C) as in Lemma 6.1.

Given a vector bundle E ∈U(r, d), we can finda vector bundle EL0 ∈SU(r, L0) and a line bundle η ∈J0(C) such that E = EL0 ⊗η. Note thatη⊗r = detE ⊗L−10 .

The above decomposition is unique up to a choice of an r-torsion point. Letξ1 be a line bundle such that ξ⊗r1= φd(L0) ⊗L−10 .

Then, Tξ−11◦Φ induces an automorphism ofSU(r, L0). By the main Theorem B, we have two cases.Case 1: Tξ−11◦Φ(EL0) = σ∗EL0 ⊗µ, where σ is an automorphism of the curve C and µ a linebundle which satisfies µ⊗r = L0 ⊗σ∗L−10 .

We choose now ξ = ξ1 ⊗µ and so, ξ⊗r = φd(L0)⊗σ∗L−10 .We haveΦ(E)=Φ(EL0 ⊗η)=Φ(EL0) ⊗φ(η)by Lemma 6.3,=Tξ1(Tξ−11 Φ(EL0)) ⊗φ(η)=σ∗EL0 ⊗ξ ⊗φ(η)=σ∗(E ⊗η−1) ⊗ξ ⊗φ(η)by the definition of EL0,=σ∗E ⊗ξ ⊗φ(η) ⊗σ∗(η−1).Therefore,Φ(E) = σ∗E ⊗ξ ⊗φ(η) ⊗σ∗(η−1) where η⊗r = detE ⊗L−10and ξ⊗r = φd(L0) ⊗σ∗L−10 .Note that, due to Lemma 6.3, the map φ−1◦σ∗is the identity on the set of r-torsion points in J0(C).Case 2: Tξ−11 ◦Φ(EL0) = σ∗E∨L0 ⊗ν, where σ is an automorphism of the curve C and ν is a line bun-dle which satisfies ν⊗r = L0 ⊗σ∗L0. In this case we choose ξ = ξ1 ⊗ν and so, ξ⊗r = φd(L0)⊗σ∗L0.45

We haveΦ(E)=Φ(EL0) ⊗φ(η)by Lemma 6.3,=Tξ1(Tξ−11 Φ(EL0)) ⊗φ(η)=σ∗E∨L0 ⊗ξ ⊗φ(η)=σ∗(E∨⊗η) ⊗ξ ⊗φ(η)by the definition of EL0,=σ∗E∨⊗ξ ⊗φ(η) ⊗σ∗(η).Therefore,Φ(E) = σ∗E∨⊗ξ ⊗φ(η) ⊗σ∗(η) where η⊗r = detE ⊗L−10and ξ⊗r = φd(L0) ⊗σ∗L0.Again, due to Lemma 6.3, the map φ ◦σ∗has to be the identity map on the set of r-torsion pointsin J0(C).AAppendix: A proof of the Torelli Theorem for the moduli spaceof vector bundlesUsing the technics of the paper we sketch in the following the proof of Theorem E - the TorelliTheorem for vector bundles.Remark A.1 In the case (r, d) = 1 the theorem has been proven in [M-N], [N-R 2] and [Ty]. Theway they prove it is by showing that the intermediate Jacobian of the moduli space is canonicallyisomorphic to the principally polarized Jacobian of the curve; the theorem then follows from theusual Torelli for Jacobian of curves.

In the case (r, d) ̸= 1, the moduli space of vector bundlesis singular and the construction of the intermediate Jacobian does not apply. In the special caser = 2 and trivial determinant, Balaji has proven a similar result for a desingularization of the spaceconstructed by Seshadri, see [Ba].

Here we prove the theorem for any r, d.Proof of Theorem E. The notation we use is in analogy with the one used in the paper. Thesubscripts 1, 2 in the notation refer to the curves C1, C2 respectively.

Let Φ : SUC1(r, L1) −→SUC2(r, L2) be the given isomorphism. After lifting to the cotangent bundle we obtain in a similarway as in the paper the following commutative diagram:Prym( ˜C1, C1)Prym( ˜C2, C2)Breg1Breg2✲˜Φ❄H1❄H2✲φ46

where ˜Φ is the induced isomorphism, H1, H2 are the Hitchin maps and the map φ is a linearisomorphism. Consider the following diagramS2P(B∨2 )S1P(B∨1 )✲i2❄φ∗✲i1where the maps i1, i2 are those defined by the linear systems B1, B2 respectively and the isomor-phism φ∗is the one induced by the extension φ of φ.

Using arguments similar to those in Sections4.2 and 5.1, we deduce that the existence of an isomorphism ˜Φ in the above diagram, implies thatφ ∗has to map i2(S2) isomorphically to i1(S1). Thus the curve C2 maps to C1 and since both arecurves of the same genus g ≥3, the map is an isomorphism.✷References[A-I-K]Altman, A., Iarrobino, A., Kleiman, S.: Irreducibility of the Compactified Jacobian.Proc.

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