The investigation of the relationships between compact Riemann
surfaces (al gebraic curves) and their associated complex tori
(Jacobi varieties) has long been basic to the study both of Riemann
surfaces and of complex tori. A Riemann surface is naturally
imbedded as an analytic submanifold in its associated torus; and
various spaces of linear equivalence elasses of divisors on the
surface (or equivalently spaces of analytic equivalence elasses of
complex line bundies over the surface), elassified according to the
dimensions of the associated linear series (or the dimensions of
the spaces of analytic cross-sections), are naturally realized as
analytic subvarieties of the associated torus. One of the most
fruitful of the elassical approaches to this investigation has been
by way of theta functions. The space of linear equivalence elasses
of positive divisors of order g -1 on a compact connected Riemann
surface M of genus g is realized by an irreducible (g
-1)-dimensional analytic subvariety, an irreducible hypersurface,
of the associated g-dimensional complex torus J(M); this hyper 1
surface W- r;;;, J(M) is the image of the natural mapping Mg-
-+J(M), and is g 1 1 birationally equivalent to the (g -1)-fold
symmetric product Mg- jSg-l of the Riemann surface M."
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