High Quality Content by WIKIPEDIA articles! In mathematics, the theta-divisor ? is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of A of dimension dim A ? 1. Classical results of Bernhard Riemann describe ? in another way, in the case that A is the Jacobian variety J of an algebraic curve (compact Riemann surface) C. There is, for a choice of base point P on C, a standard mapping of C to J, by means of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q ? P. Then since J is an algebraic group, C may be added to itself k times on J, giving rise to subvarieties Wk. If g is the genus of C, Riemann proved that ? is a translate on J of Wg ? 1. He also described which points on Wg ? 1 are non-singular: they correspond to the effective divisors D of degree g ? 1 with no associated meromorphic functions other than constants.

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