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Vector Bundles on Curves and Theta Functions

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Vector Bundles on Curves and Theta Functions Advanced Studies in Pure Mathematics 45, 2006 Moduli Spaces and Arithmetic Geometry (Kyoto 2004) pp. 145–156 Vector bundles on curves and theta functions Arnaud Beauville Abstract. This is a survey lecture on the “theta map” from the moduli space of SLr-bundles on a curve C to the projective space of r-th order theta functions on JC. Some recent results and a few open problems about that map are discussed. Introduction These notes survey the relation between the moduli spaces of vector bundles on a curve C and the spaces of (classical) theta functions on the Jacobian J of C. The connection appears when one tries to describe the moduli space Mr of rank r vector bundles with trivial determinant as a projective variety in an explicit way (as opposed to the somewhat non-constructive way provided by GIT). The Picard group of the moduli space is infinite cyclic, generated by the determinant line bundle L ;thus the natural maps from Mr to projective spaces are those defined by the k ∗ linear systems |L |, and in the first instance the map ϕL : Mr |L| . The key point is that this map can be identified with the theta map θ : Mr |rΘ| which associates to a general bundle E ∈Mr its theta divisor ΘE,an element of the linear system |rΘ| on J – we will recall the precise defi- nitions below. This description turns out to be sufficiently manageable to get some information on the behaviour of this map, at least when r or g are small. We will describe the results which have been obtained so far – most of them fairly recently. Thus these notes can be viewed as a sequel to [B2], though with a more precise focus on the theta map. For the con- venience of the reader we have made this paper independent of [B2], by recalling in §1 the necessary definitions. Then we discuss the indetermi- nacy locus of θ (§2), the case r =2(§3), the case g =2(§4), and the Received March 9, 2005 Revised March 29, 2006 146 A. Beauville higher rank case (§5). Finally, as in [B2] we will propose a small list of questions and conjectures related to the topic (§6). §1. The moduli space Mr and the theta map (1.1) Throughout this paper C will be a complex curve of genus g ≥ 2. We denote by J its Jacobian variety, and by J k the variety (isomorphic to J = J 0) parametrizing line bundles of degree k on C. For r ≥ 2, we denote by Mr the moduli space of semi-stable vector bundles of rank r and trivial determinant on C. It is a normal, projec- 2 tive, unirational variety, of dimension (r − 1)(g − 1). The points of Mr correspond to isomorphism classes of vector bundles with trivial deter- minant which are direct sums of stable vector bundles of degree zero. The singular locus consists precisely of those bundles which are decom- posable (with the exception of M2 in genus 2, which is smooth). The corresponding singularities are rational Gorenstein – that is, reasonably mild. (1.2) The Picard group of Mr has been thoroughly studied in [D-N]; let us recall the main results. Fix some L ∈ J g−1, and consider the reduced subvariety 0 ∆L := {E ∈Mr | H (C, E ⊗ L) =0 } . M L O Then ∆L isaCartierdivisorin r; the line bundle := Mr (∆L), called the determinant bundle, is independent of the choice of L and −2r generates Pic(Mr). The canonical bundle of Mr is L . ∗ (1.3) To study the rational map ϕL : Mr |L| associated to the determinant line bundle, the following construction is crucial. For a vector bundle E ∈Mr, consider the locus g−1 0 ΘE := {L ∈ J | H (C, E ⊗ L) =0 } . g−1 Since χ(E ⊗ L)=0forL in J , it is readily seen that ΘE is in a natural way a divisor in J g−1 – unless it is equal to J g−1. The latter case (which may occur only for special bundles) is a serious source of trouble – see §2 below. In the former case we say that E admits a theta divisor; this divisor belongs to the linear system |rΘ|, where Θ is the canonical Theta divisor in J g−1. In this way we get a rational map θ : Mr |rΘ| . Proposition 1.4. [BNR] There is a canonical isomorphism |L|∗ −→∼ | rΘ| which identifies ϕL to θ. Vector bundles on curves and theta functions 147 As a consequence, the base locus of |L| is the locus of bundles E 0 g−1 in Mr such that H (C, E ⊗ L) =0forall L ∈ J . This is also the indeterminacy locus of θ (because |L| cannot have a fixed component). (1.5) The r-torsion subgroup J[r]ofJ acts on Mr by tensor prod- uct; it also acts on |rΘ| by translation, and the map θ is equivariant with respect to these actions. In particular, the image of θ is J[r]-invariant. (1.6) The case when θ is a morphism is much easier to analyze: we know then that it is finite (since |L| is ample, θ cannot contract any curve), we know its degree by the Verlinde formula, etc. Unfortunately there are few cases where this is known to happen: Proposition 1.6. The base locus of |L| is empty in the following cases: a) r =2; b) r =3, g =2or 3; c) r =3, C is generic. All these results except the case r = g = 3 are due to Raynaud [R]. While a) and the first part of b) are easy, c) and the second part of b) are much more involved. We will discuss the latter in §5below.The proof of c) is reduced, through a degeneration argument, to an analogous statement for torsion-free sheaves on a rational curve with g nodes. §2. Base locus (2.1) Recall that the slope of a vector bundle E of rank r and degree d is the rational number µ = d/r. It is convenient to extend the definition of the theta divisor to vector bundles E with integral slope µ, by putting g−1−µ 0 ΘE := {L ∈ J | H (C, E ⊗ L) =0 }.Ifδ is a line bundle such that ⊗r ∼ −1 δ = det E, the vector bundle E0 := E ⊗ δ has trivial determinant ⊂ g−1 ⊂ g−1−µ and ΘE0 J is the translate by δ of ΘE J . (2.2) We have the following relations between stability and exis- tence of the theta divisor: (2.2 a) If E admits a theta divisor, it is semi-stable; (2.2 b) If moreover ΘE is a prime divisor, E is stable. Indeed let F be a proper subbundle of E.Ifµ(F ) >µ(E), the Riemann- Roch theorem implies H0(C, F ⊗L) = 0, and therefore H0(C, E⊗L) =0, g−1−µ for all L in J .Ifµ(F )=µ(E), one has ΘE =ΘF +ΘE/F ,sothat ΘE is not prime. (2.3) The converse of these assertions do not hold. We will see in (2.6) examples of stable bundles with a reducible theta divisor. The first examples of stable bundles with no theta divisor are due to Raynaud [R]. 148 A. Beauville They are restrictions of projectively flat vector bundles on J.Choosea theta divisor Θ on J. The line bundle OJ (nΘ) is invariant under the n-torsion subgroup J[n]ofJ. The action of J[n]doesnot lift to OJ (nΘ), 0 ∗ but it does lift to the vector bundle H (J, OJ (nΘ)) ⊗C OJ (nΘ). Thus → this vector bundle is the pull back under the multiplication nJ : J J of a vector bundle En on J. Restricting En to the curve C embedded in J by an Abel-Jacobi mapping gives the Raynaud bundle Rn.Itiswell g g defined up to a twist by an element of J,hasrankn and slope n .It 0 has the property that H (C, Rn ⊗ α) =0forall α ∈ J.Thusifn | g Rn has integral slope and no theta divisor. More generally, Schneider has shown that a general vector bundle on C of rank ng,slopeg − 1and containing Rn is still stable [S2]. This gives a very large dimension for |L| − 1 M g the base locus of , approximately (1 n )dim r if r = n .Some related results are discussed in [A]. (2.4) Another series of examples have been constructed by Popa [P]. Let L be a line bundle on C spanned by its global sections. The evaluation bundle EL is defined by the exact sequence → ∗ −→ 0 ⊗ O −→ev → 0 EL H (L) C C L 0; it has the same degree as L and rank h0(L)−1. In particular, if we choose ≥ g deg L = g + r with r g +2,EL has rank r and slope µ =1+r . Then, p for all p such that 2 ≤ p ≤ r − 2 and pµ ∈ Z, the vector bundle Λ EL does not admit a theta divisor (see [S1]). For instance, when r =2g, 2 Λ EL gives a base point of |L| in Mg(2g−1). (2.5) An interesting limit case of this construction is when µ =2; this occurs when L = KC,orr = g. The first case has been studied in p [FMP]. It turns out that the vector bundle Λ EK has a theta divisor, equal to Cg−p−1 − Cp (here Ck denotes the locus of effective divisor classes in J k). While the proof is elementary for p =1,itisextremely involved for the higher exterior powers: it requires going to the moduli space of curves and computing various divisor classes in the Picard group of this moduli space.
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