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.