ag.algebraic geometry - equation for abelian varieties with a given polarization

Let $A$ be an abelian variety of dimension g and a polarization $L$ of type $(d_1,.....,d_g)$ (let alone the case $d_i=d_j,$ $forall i, j$). What is the degree of the generators of the homogeneous ideal of A projectively embedded via the sections of L?

I know that Gross and Popescu gave results for surfaces with L of type $(1,d)$ - for instance if $d>10$ the ideal is generated by quadrics - but what for other polarizations and most of all for higher dimensions? Is this known?