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**Semistable bundles over an elliptic curve.**
*(English)*
Zbl 0786.14021

Almost all papers on moduli spaces of vector bundles on curves need to assume the genus of the curve to be at least two. The paper under review contains a systematic study of the case of elliptic curves. The starting point is the classical paper on vector bundles over an elliptic curve by M. F. Atiyah [Proc. Lond. Math. Soc., III. Ser. 7, 414-452 (1957; Zbl 0084.173)]. It turns out that the main ingredients in the theory have a very easy and nice description:

Let \({\mathcal M}_{n,d}\) be the moduli space of equivalence classes of semistable rank \(n\) vector bundles of degree \(d\) over an elliptic curve \(C\). Then \({\mathcal M}_{n,d}\) identifies with \(S^ hC\), the \(h\)-th symmetric power of the curve \((h\) being the greatest common divisor of \(n\) and \(d)\). – The determinant map \({\mathcal M}_{n,d}\to J_ d(C)\) (the latter being the Jacobian of line bundles of degree \(d\) on \(C)\) identifies with the Abel-Jacobi map \(S^ hC\to J_ h(C)\). In particular, for any \(L\in J_ d\), the space \({\mathcal M}_{n,L}\) of classes of semistable vector bundles with determinant \(L\) is a projective space of dimension \(h-1\). – The appropriate Brill-Noether loci, \(W^ r_{n,0}(\forall)\) (resp. \(W^ r_{n,0}(\exists)\), defined as the set of equivalent classes with all representatives (resp. at least one representative) having \(r+1\) independent sections, are also identified with symmetric powers of \(C\). The same holds for the theta divisors. – The corresponding Brill-Noether loci in \({\mathcal M}_{n,L}\) and theta divisors can also be identified with linear subspaces of it.

These identifications allow to extend to elliptic curves results of Drezet and Narasimhan on the Picard group of \({\mathcal M}_{n,d}\) and \({\mathcal M}_{n,L}\) for \(n\geq 2\), and formulas by Beauville, Narasimhan and Ramanan, Verlinde, and Bott and Szenes about theta divisors.

Let \({\mathcal M}_{n,d}\) be the moduli space of equivalence classes of semistable rank \(n\) vector bundles of degree \(d\) over an elliptic curve \(C\). Then \({\mathcal M}_{n,d}\) identifies with \(S^ hC\), the \(h\)-th symmetric power of the curve \((h\) being the greatest common divisor of \(n\) and \(d)\). – The determinant map \({\mathcal M}_{n,d}\to J_ d(C)\) (the latter being the Jacobian of line bundles of degree \(d\) on \(C)\) identifies with the Abel-Jacobi map \(S^ hC\to J_ h(C)\). In particular, for any \(L\in J_ d\), the space \({\mathcal M}_{n,L}\) of classes of semistable vector bundles with determinant \(L\) is a projective space of dimension \(h-1\). – The appropriate Brill-Noether loci, \(W^ r_{n,0}(\forall)\) (resp. \(W^ r_{n,0}(\exists)\), defined as the set of equivalent classes with all representatives (resp. at least one representative) having \(r+1\) independent sections, are also identified with symmetric powers of \(C\). The same holds for the theta divisors. – The corresponding Brill-Noether loci in \({\mathcal M}_{n,L}\) and theta divisors can also be identified with linear subspaces of it.

These identifications allow to extend to elliptic curves results of Drezet and Narasimhan on the Picard group of \({\mathcal M}_{n,d}\) and \({\mathcal M}_{n,L}\) for \(n\geq 2\), and formulas by Beauville, Narasimhan and Ramanan, Verlinde, and Bott and Szenes about theta divisors.

Reviewer: E.Arrondo (Madrid)

### MSC:

14H52 | Elliptic curves |

14H10 | Families, moduli of curves (algebraic) |

14H60 | Vector bundles on curves and their moduli |