Abelian varieties analogs of two results about algebraic curves

We characterize decomposable principally polarized abelian varieties of the form $E\times B$, with $E$ an elliptic curve, in two different ways, which are, surprisingly, completely analogous to classical results of curve theory concerning hyperelliptic curves. The first one is by the failure of a normal generation property, namely the generation in degree zero of a certain graded module over the symmetric algebra over $H^0(2\Theta)$. This appears to be the first result of this type in the realm of p.p.a.v.'s. The second characterization is by the failure of surjectivity of second order gaussian maps associated to line bundles corresponding to $6\Theta$, or, equivalently, by the fact that at some point, the line bundle corresponding to $3\Theta$ fails to separate $2$-jets. We also show that this last result is equivalent to an effective version of a theorem of Nakamaye characterizing the above decomposable abelian varieties as those computing the minimal Seshadri constant. Finally we propose some conjectural generalizations relating $p$-jets separation thresholds, higher gaussian maps sujectivity thresholds, and Seshadri constants.