If W is a Picard bundle on the Jacobian y of a curve C , we have the problem of describing W globally. The theta divisor 0 is ample on J . Thus it is possible to write n*W as the sheaf associated to a graded M over the well-known ring ®m>QT(J,fi'j(m^6)). In this paper we compute the degree of generators and relations for such a module M . There are naturally occurring locally free sheaves called Picard bundles on the Jacobian / of a smooth complete curve C of positive genus g over k = k . These

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... bundles describe the global variation of the sections of invertible sheaves on C with pleasant degree. The inversion problem is to give a description of the Picard bundles globally on J . As such analytic description is lacking, we must content ourselves with two algebraic solutions of this problem. The first solution requires us to know the image of some points of C in the Jacobian. This approach uses a method due to R. C. Gunning. The second solution determines the pull-back of the Picard bundle by a multiplication in / in terms of a module over the graded ring of theta sections. Here one uses a form of a theorem of D. Mumford on the equations defining abelian varieties projectively. is zero if / > 0 and Wn = nx.{n\3' ®¿?) is a locally free sheaf of rank ng . Let C ^ J be a universal abelian integral. Let Qn = ^*2^Cn ®^°\jxç ■ Then Qn is a family of invertible sheaves on C of degree n ■ g as [C: d] = g. If