Abstract
Much has been wi-ritten on the possibility of extending the classical theory of linear equivalence on an algebraic curve to more general equivalence relations in which the group of all principle divisors is replaced by certain proper subgroups. Noether's theory of non-adjoint curves (see the end of this paper for references) is essentially a theory of this sort. Similarly the problem considered by a number of analysts of generalizing the classical Jacobi inversion theorem to differentials with singularities is in the same general direction. Severi's recent Funzioni Quasi Abeliane (which contains a large bibliography) includes a systematic account of what, in our language, is the equivalence relation on a curve with ordinary double points and cusps. Our present work deals with the algebraic aspects of the general problem; transcendental matters and generalized jacobian varieties will be treated later. For the general concepts involved, we refer to Zariski 1 and Chevalley.
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