We extend the results of R.C. Gunning's paper "Some curves in abelian varieties", Inv. Math. 66 (1982), 377-389, including also degenerate cases of the original hypotheses. Gunning's characterization of Jacobi varieties in terms of trisecants of the Kummer variety leads to similar characterizations in terms of flexes of the Kummer variety.
In his paper [3], R.C. Gunning has given a new characterization of Jacobi varieties among all principally polarized abelian varieties, by using trisecants of the associated Kummer variety. The present paper is motivated by the link between Gunning's results and the -as yet unanswered -question about the Novikov Hypothesis. Our main statement is Theorem (3.1), which is just a more general version of the key result of [3], allowing also limit cases of the original assumptions. Section 3 is devoted to the proof of this statement. In particular, one obtains similar characterizations of jacobians by means of flexes instead of trisecants (cf Section 1).
After putting Novikov's Hypothesis in geometrical terms (cf (2.18)), its relationship with this version of Gunning's result becomes more apparent. The comparison suggests some intermediate questions which might be useful. We discuss this more closely in Section 2.
In Sections 1 and 2 we assume the groundfield k to be the field C of complex numbers; in the rest of the paper k is an algebraically closed field of arbitrary characteristic different from 2.