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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?

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Though I don't have enough details to really write it out here, you'll be wanting to look at Mumford's "On the Equations Defining Abelian Varieties I, II, III", which are in volume I of his selected papers (as well as his website at the links provided), and also the wikipedia page here.

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