Equations for Abelian Varieties

Abstract

In Chapter 4 we proved some criteria for a line bundle L on an abelian variety X to be very ample. The corresponding embedding ϕ _L :X → ℙ _N gives X the structure of a closed subvariety of ℙ _N . As such, X is the set of zeros of a homogeneous ideal I of polynomials in N +1 variables. Since the embedding ϕ _L is defined by means of a basis of theta functions of H ⁰(L), the polynomials of I may be considered as relations among these theta functions. According to classical terminology they are called theta relations. The subject of this chapter is to find a set of theta relations which generates the ideal I, and thus describes the subvariety X of ℙ _N completely in terms of equations.