Tu, Loring W. Semistable bundles over an elliptic curve. (English) Zbl 0786.14021 Adv. Math. 98, No. 1, 1-26 (1993). 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. Reviewer: E.Arrondo (Madrid) Cited in 2 ReviewsCited in 57 Documents MSC: 14H52 Elliptic curves 14H10 Families, moduli of curves (algebraic) 14H60 Vector bundles on curves and their moduli Keywords:moduli space; symmetric power of a curve; Jacobian of line bundles; Brill-Noether loci; theta divisors Citations:Zbl 0084.173 PDFBibTeX XMLCite \textit{L. W. Tu}, Adv. Math. 98, No. 1, 1--26 (1993; Zbl 0786.14021) Full Text: DOI