Let \(L\) be an ample line bundle of polarization type \((1, d_ 2, \dots, d_ n)\) on the abelian \(n\)-fold \(X\). It is known that if \(n = 2\) then \(L\) is very ample iff \(d = d_ 2 \geq 5\), and \(L\) is globally generated (free) iff \(d \geq 3\). If \(n \geq 3\) not much is known.
In this paper the authors treat the case \(n = 3\). The general Ein-Lazarsfeld criterion for freeness of an ample line bundle \(L\) on a 3-fold \(X\) assumes the non-existence of some special curves on \(X\). The question is to replace this criterion by an effective one in the case of abelian 3-folds. Example 4.2 shows that no numerical criterion guarantees the freeness of \(L\): see proposition 4.3. However, the pair \((X,L)\) defines at least one principally polarized abelian 3-fold \(Y\) isogenous to \(X\). In this paper, the authors find such sufficient conditions for freeness of \(L\) only in terms of the fixed isogeny \(X \to Y\), and depending of the decomposition type of \(Y\) (being a product of 1, 2, or 3 irreducible abelian varieties): see theorem 1. – Next, the authors study sufficient conditions ensuring very ampleness of \(L\), by using the method of Comessatti for Jacobians with two principal polarizations, and Sakai’s version of Reider’s theorem yielding very ampleness for a large class of line bundles \(L\): see theorem 2. In particular, if the polarization is of type \((1,1,d)\), \(d \geq 13\), \(d \neq14\), then \(L\) is very ample.